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The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We indicate a strategy in order to construct bilinear multiplication algorithms of type Chudnovsky in large extensions of any finite field. In particular, using the symmetric version of the generalization of Randriambololona specialized on the elliptic curves, we show that it is possible to construct such algorithms with low bilinear complexity. More precisely, if we only consider the Chudnovsky-type algorithms of type symmetric elliptic, we show that the symmetric bilinear complexity of these algorithms is in \(O(n(2q)^{\log_q^\ast(n)})\) where \(n\) corresponds to the extension degree, and \(\log_q^\ast(n)\) is the iterated logarithm. Moreover, we show that the construction of such algorithms can be done in time polynomial in \(n\). Finally, applying this method we present the effective construction, step by step, of such an algorithm of multiplication in the finite field \(\mathbb{F}_{3^{57}}\). multiplication algorithm; bilinear complexity; elliptic function field; interpolation on algebraic curve; finite field Ballet, Stéphane; Bonnecaze, Alexis; Tukumuli, Mila, On the construction of elliptic Chudnovsky-type algorithms for multiplication in large extensions of finite fields, J. Algebra Appl., 0219-4988, 15, 1, 1650005, 26 pp., (2016) Number-theoretic algorithms; complexity, Structure theory for finite fields and commutative rings (number-theoretic aspects), Arithmetic theory of algebraic function fields, Elliptic curves, Cryptography On the construction of elliptic Chudnovsky-type algorithms for multiplication in large extensions of finite fields | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper, we prove that the constant terms of powers of a Laurent polynomial satisfy certain congruences modulo prime powers. As a corollary, the generating series of these numbers considered as a function of a \(p\)-adic variable admits an analytic continuation in a non-trivial way, as it was shown previously by Dwork for a class of hypergeometric series. congruences; finite fields; zeta-function Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.), Varieties over finite and local fields, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Dwork's congruences for the constant terms of powers of a Laurent polynomial | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The Hilbert scheme \(\mathrm{Hilb}^{p(t)} (\mathbb P^n)\) parametrizing closed subschemes of \(\mathbb P^n\) with Hilbert polynomial \(p(t)\) has been of great interest every since Grothendieck constructed it in the early 1960s. Early results include the connectedness theorem of \textit{R. Hartshorne} [Publ. Math., Inst. Hautes Étud. Sci. 29, 5--48 (1966; Zbl 0171.41502)] and smoothness of \(\mathrm{Hilb}^{p(t)} (\mathbb P^2)\) due to \textit{J. Fogarty} [Am. J. Math. 90, 511--521 (1968; Zbl 0176.18401)]. \textit{A. Reeves} and \textit{M. Stillman} showed that every non-empty Hilbert scheme contains a smooth Borel-fixed point [J. Algebr. Geom. 6, No. 2, 235--246 (1997; Zbl 0924.14004)] and \textit{A. P. Staal} classified those with exactly one such fixed point, which are necessarily smooth and irreducible [Math. Z. 296, No. 3--4, 1593--1611 (2020; Zbl 1451.14010)].
The main result classifies Hilbert schemes with two Borel-fixed points over a field \(k\) of characteristic zero. To describe the result, express the Hilbert polynomial \(p(t)\) in the form used by \textit{Gotzmann}, namely
\[
p(t) = \sum_{i=1}^m \binom{t+\lambda_i-i}{\lambda_i-1}
\]
where \(\lambda_1 \geq \lambda_2 \geq \dots \geq \lambda_m \geq 1\) [\textit{G. Gotzmann}, Math. Z. 158, 61--70 (1978; Zbl 0352.13009)]. Writing \(\mathbf{\lambda} = (\lambda_1,\dots,\lambda_m)\), the theorem lists for exactly which \(\mathbf{\lambda}\) the Hilbert scheme \(\mathrm{Hilb}^{p(t)} (\mathbb P^n)\) has two Borel-fixed points and further determines when it is (a) smooth, (b) irreducible and singular or (c) a union of two components. In each case the irreducible components are normal and Cohen-Macaulay and the singularities of the Hilbert scheme appear as cones over certain Segre embeddings of \(\mathbb P^a \times \mathbb P^b\). Since the writing of his paper, (a) \textit{A. P. Staal} [``Hilbert schemes with two Borel-fixed points in arbitrary characteristic'', Preprint, \url{arXiv:2107.02204}] has shown that the theorem is valid in all characteristics with a small modification when char \(k=2\) and (b) \textit{R. Skjelnes} and \textit{G. G. Smith} [J. Reine Angew. Math. 794, 281--305 (2023; Zbl 07640144)] have classified the smooth Hilbert schemes are described their geometry.
Despite the difficulty of the content, the paper is readably written. Section 1 gives preliminaries on Borel-fixed (strongly stable) ideals and the resolution of \textit{S. Eliahou} and \textit{M. Kervaire} [J. Algebra 129, No. 1, 1--25 (1990; Zbl 0701.13006)], while Section 2 identifies the tuples \(\mathbf{\lambda} = (\lambda_1,\dots,\lambda_m)\) corresponding to Hilbert schemes with two components. Section 3 uses the comparison theorem of \textit{R. Piene} and \textit{M. Schlessinger} [Am. J. Math. 107, 761--774 (1985; Zbl 0589.14009)] to compute the tangent space of the non-lexicographic Borel-fixed ideal \(I(\mathbf{\lambda})\) and give a partial basis for the second cohomology group of \(k[x_0,\dots,x_n]/I(\mathbf{\lambda})\). These are used in Section 4 where the main theorem is proved to describe the universal deformation space of \(I(\mathbf{\lambda})\) and hence the nature of singularities of the Hilbert schemes. Finally in Section 5 the author gives examples of Hilbert schemes with three Borel-fixed points. The last three examples relate to Hilbert schemes studied in the literature [\textit{S. Katz}, in: Zero-dimensional schemes. Proceedings of the international conference held in Ravello, Italy, June 8-13, 1992. Berlin: de Gruyter. 231--242 (1994; Zbl 0839.14001); \textit{D. Chen} and \textit{S. Nollet}, Algebra Number Theory 6, No. 4, 731--756 (2012; Zbl 1250.14004); \textit{D. Chen} et al., Commun. Algebra 39, No. 8, 3021--3043 (2011; Zbl 1238.14012)]. Hilbert scheme; singularities; Borel-fixed points; deformations of ideals Syzygies, resolutions, complexes and commutative rings, Parametrization (Chow and Hilbert schemes), Fine and coarse moduli spaces, Singularities of surfaces or higher-dimensional varieties Hilbert schemes with two Borel-fixed points | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper gives a full proof account of the state of the art of the theory of polylogarithms from a geometric point of view as advocated by the author. The case of the dilogarithm being known since some time by work of Gabrielov et al., Bloch, Wigner, Zagier,\( \dots\), the underlying paper focusses on the trilogarithm.
Classically the \(p\)-th polylogarithm \(\text{Li}_p (z)\) is defined as the analytic continuation of the expression \(\text{Li}_p(z)=\sum_{n=1}^\infty {z^n \over n^p}\), \(|z|\leq 1\). With \(\text{Li}_1(z)= - \log(1-z)\), one has the inductive formula \(\text{Li}_p(z)=\int^z_0 \text{Li}_{p-1} (z) {dt \over t}\) with its (multivalued) continuation to \(\mathbb{P}^1_\mathbb{C} \backslash \{0,1, \infty\}\). It turns out to be advantageous to consider modified functions \({\mathcal L}_p (z)={\mathcal R}_p \left( \sum^p_{j=0} {2^j B_j \over j!} (\log |z |)^j \cdot \text{Li}_{p-j} (z) \right)\), where the \(B_j\) are the Bernoulli numbers, and \({\mathcal R}_m\) denotes the real part for odd \(m\) and the imaginary part for even \(m\), \(\text{Li}_0 (z):= -{1 \over 2}\). Thus one has
\[
{\mathcal L}_2 (z)={\mathfrak I} \bigl( \text{Li}_2 (z) \bigr) + \arg (1-z) \cdot \log |z |,
\]
the Bloch-Wigner function. For the present paper the most interesting function becomes
\[
{\mathcal L}_3 (z)={\mathfrak R} \Bigl( \text{Li}_3 (z)-\log |z |\cdot \text{Li}_2 (z)+ \textstyle {{1\over 3}} \log^2 |z|\cdot \text{Li}_1(z) \Bigr).
\]
The functions \({\mathcal L}_p(z)\) are single-valued, real analytic on \(\mathbb{P}^1_\mathbb{C} \backslash \{0,1, \infty\}\) and continuous at \(0,1,\infty\). In particular, \({\mathcal L}_3 (0)= {\mathcal L}_3 (\infty) =0\) and \({\mathcal L}_3 (1)= \zeta_\mathbb{Q} (3)\), the Riemann zeta-function at \(s=3\). The functions \({\mathcal L}_p (z)\) admit a Hodge theoretic interpretation.
As a first goal one tries to find the generic functional equation for the (modified) trilogarithm. Motivated by results on the Bloch-Wigner function where the functional equations of the dilogarithm are related to the cross-ratio of four points in \(\mathbb{P}^1\), in the case of the trilogarithm one considers configurations of six (or seven) points in \(\mathbb{P}^2\). By an explicit geometric reasoning the main result is obtained: The generic functional equation for the trilogarithm \({\mathcal M}_3\) (a specific alternating sum of \({\mathcal L}_3\)'s) is a seven term identity for \({\mathcal M}_3\) on a configuration \((l_0, \dots, l_6)\) of seven points in \(\mathbb{P}^2\) that can be given explicitly in terms of the coordinates of the points. In particular, the Spence-Kummer relation may be derived. Furthermore, it is shown that the trilogarithm is determined by its functional equation.
Polylogarithms show up in other contexts, e.g. in connection with algebraic \(K\)-theory, motivic cohomology, characteristic classes, continuous cohomology, the Dedekind zeta function of an arbitrary number field, \(\dots\), etc. In some cases one can prove interesting results, e.g. for a number field \(F\) one can express (up to a non-zero rational factor) the value of \(\zeta_F (2)\) in terms of the dilogarithm \({\mathcal L}_2\) at specific values of its argument depending on the (complex) embeddings of \(F\). A similar result holds for \(\zeta_F (3) \) in terms of \({\mathcal L}_3\). For general \(\zeta_F (n)\), \(n=4,\dots\), Zagier stated the conjecture that they can be expressed, analogously to \(\zeta_F (2)\) and \(\zeta_F (3)\), in terms of \({\mathcal L}_n\). This fits very well in Beilinson's world where values of \(L\)-functions at special values of their arguments are given (up to non-zero rational factors) by the volume of the regulator map which is itself a map from algebraic \(K\)-groups to Deligne-Beilinson cohomology. As a matter of fact, the theory of polylogarithms is closely related to \(K\)-theory. One of its main building blocks is a certain complex \(\Gamma_F (n)\) (the existence of which was originally conjectured by Beilinson and Lichtenbaum) of the form:
\[
\Gamma_F (n): {\mathcal B}_n(F) @> \delta>> {\mathcal B}_{n-1} (F) \otimes F^\times @>\delta>> \cdots @>\delta>> {\mathcal B}_2 (F) \otimes \wedge^{n-2} F^\times @>\delta>> \wedge^n F^\times,
\]
where \({\mathcal B}_m (F) = \mathbb{Z} [\mathbb{P}^1 _F]/{\mathcal R}_m (F)\), with \({\mathcal R}_m (F) \subset \mathbb{Z} [\mathbb{P}^1_F]\) reflecting the functional equations of the classical \(m\)-polylogarithm. Here \({\mathcal B}_n (F)\) is placed in degree one, and the \(\delta\)'s are explicitly defined. For the \(K\)-groups of \(F\) one has \(K_n (F )_\mathbb{Q} = \text{Prim} H_n (GL_n(F), \mathbb{Q})\) and by the canonical filtration on \(H_n (GL_n (F), \mathbb{Q})\) implied by \(\text{Im} (H_n (GL_{n-1} (F), \mathbb{Q}) \to H_n (GL_n(F), \mathbb{Q}))\), one obtains a filtration \(K_n(F)_\mathbb{Q} \supset K_n^{(1)} (F)_\mathbb{Q} \supset K_n^{(2)} (F)_\mathbb{Q} \supset \cdots \). Let \(K_n^{[i]} (F)_\mathbb{Q}: = K_n^{(i)} (F)_\mathbb{Q}/K_n^{(i+1)} (F)_\mathbb{Q}\). Then one has Conjecture A: \(K_{2n-i}^{[n-i]} (F)_\mathbb{Q} = H^i (\Gamma_F(n) \otimes \mathbb{Q})\). On the other hand, Beilinson conjectured the existence of a mixed Tate category \({\mathcal M}_T(F)\) which should be Tannakian. Thus the formalism of Tannakian categories implies that \({\mathcal M}_T (F)\) is equivalent to the category of finite-dimensional representations of some graded pro-Lie algebra \(L(F)_\bullet = \oplus^{-\infty}_{i= -1} L(F)_i\). One may state Conjecture B: (i) \(L(F)_{\leq-2}\) is a free graded pro-Lie algebra such that the dual of the space of its degree \(-n\) generators is isomorphic to \({\mathcal B}_n (F)_\mathbb{Q}\); (ii) The dual map to the action of the quotient \(L(F)_\bullet /L(F)_{\leq -2}\) on the space of degree \(-(n-1)\) generators of \(L(F)_{\leq -2}\) is just the differential \(\delta: {\mathcal B}_n (F)_\mathbb{Q} \to({\mathcal B}_{n-1} (F)\otimes F^\times)_\mathbb{Q}\). It is shown that in Beilinson's world Conjecture A is equivalent to Conjecture B. Conjecture B has some deep consequences, e.g. its truth implies the truth of a conjecture of Bogomolov, and also of a conjecture due to Shafarevich which says that the commutant of \(\text{Gal} (\overline \mathbb{Q}/ \mathbb{Q})\) is a free profinite group.
Let \(F\) be an arbitrary field and define \(B_p(F): = \mathbb{Z} [\mathbb{P}^1_F \backslash \{0,1, \infty\}]/R_p(F)\), \(p\leq 3\), where the \(R_p(F)\) again reflect the functional equations of the classical \(p\)-th polylogarithm. One defines the complex \(B_3(F) \otimes \mathbb{Q}\) as follows: \(B_3(F)_\mathbb{Q} @>\delta>> (B_2(F) \otimes F^\times)_\mathbb{Q} @>\delta>> (\wedge^3 F^\times)_\mathbb{Q}\), with \(B_3 (F)_\mathbb{Q}\) placed in degree 1, and \(\delta \{x\} = [x] \otimes x\) and \(\delta ([x] \otimes y) = (1-x) \wedge x \wedge y\) for a generator \(\{x\}\) of \(B_3(F)\) and a generator \([x]\) of \(B_2(F)\). Then there are canonical maps \(c_1: K_5^{[2]} (F)_\mathbb{Q} \to H^1 (B_3(F) \otimes \mathbb{Q})\) and \(c_2: K_4^{[1]} (F)_\mathbb{Q} \to H^2 (B_3(F) \otimes \mathbb{Q})\). It is conjectured that \(c_1\) and \(c_2\) are isomorphisms. This should be related to results of Suslin on Milnor \(K\)-groups.
Other subjects discussed are duality of configurations, projective duality, and an explicit formula for the Grassmannian trilogarithm.
Many unsolved questions and deep conjectures remain. algebraic \(K\)-theory; values of \(L\)-functions; Beilinson conjecture; finite dimensional representations of graded pro-Lie algebra; polylogarithms; Bernoulli numbers; Riemann zeta-function; generic functional equation; trilogarithm; Bloch-Wigner function; Spence-Kummer relation; motivic cohomology; characteristic classes; Dedekind zeta function; number field; dilogarithm; regulator map; Deligne-Beilinson cohomology; mixed Tate category; Tannakian categories; duality of configurations; Grassmannian trilogarithm M. Prausa, \textit{epsilon: A tool to find a canonical basis of master integrals}, arXiv:1701.00725 [INSPIRE]. Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects), Applications of methods of algebraic \(K\)-theory in algebraic geometry, Étale and other Grothendieck topologies and (co)homologies, Zeta functions and \(L\)-functions of number fields, \(K\)-theory of global fields, Other functions defined by series and integrals Geometry of configurations, polylogarithms, and motivic cohomology | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(R\) be a commutative ring, \(G\) a finite abelian group. The group \(\mathrm{Gal}(R,G)\) of isomorphism classes of Galois extensions of \(R\) with group \(G\), as developed by Auslander, Goldman, Chase, Harrison and Rosenberg in the 1960s, has important connections with algebraic number theory: if \(R\) is the ring of integers of a number field \(K\), then by class field theory \(\mathrm{Gal}(R,G)\) is closely related to the class group of \(K\); more recently, work of Kersten, Michalichek and the author has found important and subtle connections with Leopoldt's and Vandiver's conjectures [c.f. \textit{I. Kersten} and \textit{J. Michalichek} [J. Number Theory 32, 371--386 (1989; Zbl 0709.11058)].
The main purpose of these notes is to describe \(\mathrm{Gal}(R,G)\) in a noncohomological way. Since \(\mathrm{Gal}(R,G)\) respects products in the second variable, the problem immediately reduces to \(G\) cyclic of prime power order \(p^n\), which we assume henceforth.
Chapter 0 gives a useful description of the basic theory and a description of \(\mathrm{Gal}(R,G)\) and the subgroup \(\mathrm{NB}(R,G)\) of Galois extensions with normal basis, when \(R\) is connected and contains \(p^{-1}\) and a primitive \(p^n\)-th root of unity \(\zeta\). This generalized Kummer theory goes back to the reviewer [Ill. J. Math. 15, 273--280 (1971; Zbl 0211.37102)] and \textit{A. Z. Borevich} [J. Sov. Math. 11, 514--534 (1979); transl. from Zap. Nauchn. Semin. LOMI Steklova 57, 8--30 (1976; Zbl 0379.13003)].
Chapter I then describes \(\mathrm{Gal}(R,G)\) when \(R\) contains \(p^{-1}\) but not \(\zeta\), by letting \(S=R[\zeta]\) and ``descending'' the Kummer theory over \(S\) to \(R\). This chapter is based on the author's paper [Trans. Am. Math. Soc. 326, 307--343 (1991; Zbl 0743.11060)], and has useful applications to number theory, as is shown in chapter IV.
Chapter II describes an alternate approach to obtaining \(\mathrm{Gal}(R,G)\) from \(\mathrm{Gal}(S,G)\) using corestriction, which leads to a proof that the map from \(\mathrm{NB}(R,G)\) to \(\mathrm{NB}(R/N,G)\) is surjective if \(N\subseteq\mathrm{Rad}(R)\).
In Chapter III the author specializes to number fields. He obtains the order of \(\mathrm{NB}(R,G)\) (as always, \(G\) is cyclic of order \(p^n)\) if \(p\) is odd and either \(R\) is a finite extension of \(\mathbb Q_p\) or \(R={\mathfrak O}_K(p^{-1})\), where \(K\) is a finite extension of \(\mathbb Q\) with ring of integers \({\mathfrak O}_K\). In the latter case, \(\mathrm{NB}(R,G)=O(1)p^{n(s+1)}\) where \(s\) is the number of pairs of nonreal complex embeddings of \(K\).
Chapter IV then presents the author's description of \(\mathrm{Gal}(R,\mathbb Z_ p)\) when \(R={\mathfrak O}_K(p^{-1})\), and its connection with Leopoldt's conjecture in cyclotomic fields. The results are adapted from the author's paper cited above, except for some additional information relating the theory with extensions obtained by adjoining torsion points of abelian varieties of CM type.
Chapter V studies \(\mathbb Z^p\)-extensions of the function field \(K\) of a variety defined over a number field. Among the results obtained is that all such extensions arise from \(\mathbb Z_p\)-extensions of the largest number field contained in \(K\). The results in this chapter are previously unpublished. The author views the approach in this chapter as an alternative approach to the geometric class field theory of \textit{N. Katz} and \textit{S. Lang} [Enseign. Math., II. Sér. 27, 285--314 (1981; Zbl 0495.14011)].
The final chapter gives an exposition of the author's paper in [Manuscr. Math. 64, 261--290 (1989; Zbl 0705.13004)] which reformulates and generalizes \textit{H. Hasse}'s description [J. Reine Angew. Math. 176, 174--183 (1936; Zbl 0016.05204)] using the ``Artin-Hasse exponential'', of those \(b\) in \(K^*/(K^*)^q\), \(q=p^n\), \(K\) a local field containing \(\mathbb Q_p[\zeta]\), so that the Kummer extension \(K[\root q \of {b}]\) is unramified over \(K\): this is equivalent to describe \(\mathrm{Gal}(R,G)\) where \(R\) is the valuation ring of \(K\). The chapter concludes with the (previously unpublished) construction of a generic \(G\)-Galois extension.
The author's work on \(\mathrm{Gal}(R,G)\) which is presented in this monograph is fundamental to the subject. These notes will become a standard reference in the area. extensions of function field; generic Galois extension; Kummer theory; Leopoldt's conjecture; cyclotomic fields; geometric class field theory C. Greither, Cyclic Galois extensions of commutative rings. Lecture Notes in Mathematics, vol. 1534. Springer, Berlin-Heidelberg-New York, 1992. Zbl0788.13003 MR1222646 Galois theory and commutative ring extensions, Research exposition (monographs, survey articles) pertaining to commutative algebra, Research exposition (monographs, survey articles) pertaining to number theory, Extension theory of commutative rings, Cyclotomic extensions, Integral representations related to algebraic numbers; Galois module structure of rings of integers, Ramification and extension theory, Ramification problems in algebraic geometry, Coverings in algebraic geometry Cyclic Galois extensions of commutative rings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For abelian varieties \(A\) and \(B\) defined over a number field \(K\), the authors consider the group \(\text{Ext}^1_K(A, B)\) of the isomorphism classes of the extensions of \(A\) by \(B\) in the category of the commutative group \(K\)-schemes. They prove that \(\text{Ext}^1_K(A, B)\) is a finite group, crediting this result to J. S. Milne and S. Ramachandran, and investigate the relation between such Ext-groups and modular parametrisations of elliptic curves defined over \(\mathbb{Q}\). If \(A\) and \(B\) are two elliptic curves over \(\mathbb{Q}\) associated, respectively, to the normalised new forms \(f(q)= \sum^\infty_{n=1} a_nq^n\) and \(g(q)= \sum^\infty_{n=1} b_n q^n\), then the exponent of the group \(\text{Ext}^1_{\mathbb{Q}}(A, B)\) is proved to divide the difference \(a_n- b_n\), for those odd \(n\) coprime to the product of the conductors of \(A\) and \(B\). For an elliptic curve \(E\) defined over \(\mathbb{Q}\), the authors reformulate the well-known result of \textit{F. Diamond}, \textit{M. Flach} and \textit{L. Guo} [Math. Res. Lett. 8, No. 4, 437--442 (2001; Zbl 1022.11023)], concerning the Bloch-Kato conjecture on the special value at \(s= 2\) of the symmetric square L-function \(L(\text{Sym}^2A, s)\), in terms of the groups \(\text{Ext}^1_{\mathbb{Q}}(E,E)\) and \(\text{Ext}^1_{\overline{\mathbb{Q}}}(E,E)\). abelian varieties; elliptic curves; extension groups; modular forms; congruence moduli; Bloch-Kato conjecture; L-function Elliptic curves over global fields, Congruences for modular and \(p\)-adic modular forms, Arithmetic ground fields for abelian varieties, Abelian varieties of dimension \(> 1\), Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols, Ext and Tor, generalizations, Künneth formula (category-theoretic aspects) Extensions of abelian varieties defined over a number field | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Es sei F ein algebraischer Funktionenkörper vom Transzendenzgrad t über K, \(R\subset K\) noetherscher Ring mit Quotientenkörper K, \(\Omega^ 1_{F/R}\) der Differentialmodul. [\textit{E. Kähler} [''Geometria aritmetica'', Ann. Mat. pura Appl., IV. Ser. 45 (1958; Zbl 0142.181); vgl. auch \textit{R. Berndt}, Math. Ann. 212, 249-270 (1975; Zbl 0291.12106)]. Die Differentialintegrität \(D(F| R)\subset \Omega^ 1_{F/R}\) und die Differentialintegrität relativ zum Differentendivisor \(D(F| {\mathfrak d}/R)\). Das Ziel dieser Arbeit ist die Berechnung von \(D(F| {\mathfrak d}/{\mathbb{Z}})\) im Falle eines Funktionenkörpers F einer elliptischen Kurve E über \({\mathbb{Q}}:\) Es ist \(D(F| {\mathfrak d}/{\mathbb{Z}})\) der von \(\omega =dx/f_ y\) erzeugte \({\mathbb{Z}}\)-Modul; hier ist f eine Minimalgleichung für E [vgl. \textit{A. Néron}, Publ. Math., Inst. Haut. Étud. Sci. 21 (1964; Zbl 0132.414)]. Der Beweis benutzt die von Néron (loc. cit.) angegebenen Gleichungen. Für die zu den Kongruenzuntergruppen \(\Gamma_ 0(N)\) von SL(2,\({\mathbb{Z}})\) gehörigen elliptischen Modulkurven erhält man hieraus: Die q-Entwicklung von \(\omega\) hat ganzzahlige Koeffizienten. modular curve; integral differentials; elliptic function field R. Radtke-Harder, Arithmetisch ganze Differentiale eines elliptischen Funktionenkörpers. Thesis. Hamburg, 1982. Analytic theory of abelian varieties; abelian integrals and differentials, Abstract differential equations, Morphisms of commutative rings, Elliptic curves Arithmetisch ganze Differentiale eines elliptischen Funktionenkörpers | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0695.00011.]
Given any s points \(P_ 1,...,P_ s\) in \({\mathbb{P}}^ 2\) and s positive integers \(m_ 1,...,m_ s\), let \(\Sigma_ n\) be the linear system of plane curves of degree \(n\) through \(P_ i\) with multiplicity at least \(m_ i (1\leq i\leq s)\). We give numerical bounds for the regularity of \(\Sigma_ n\) in the following cases: (a) the points \(P_ i\) are nonsingular points of an integral curve of degree \(d;\) (b) the \(P_ i\) are in general position; (c) the \(P_ i\) are in uniform position; (d) the \(P_ i\) are generic points of \({\mathbb{P}}^ 2\). We also study the sharpness of such bounds. regularity of linear systems of plane curves; fat points Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Divisors, linear systems, invertible sheaves, Families, moduli of curves (algebraic) Linear systems of plane curves through fixed ``fat'' points of \({\mathbb{P}}^ 2\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We study binomial \(D\)-modules, which generalize \(A\)-hypergeometric systems. We determine explicitly their singular loci and provide three characterizations of their holonomicity. The first of these is an equivalence of holonomicity and \(L\)-holonomicity for these systems. The second refines the first by giving more detailed information about the \(L\)-characteristic variety of a non-holonomic binomial \(D\)-module. The final characterization states that a binomial \(D\)-module is holonomic if and only if its corresponding singular locus is proper. binomial \(D\)-modules; \(A\)-hypergeometric systems Berkesch, Christine; Matusevich, Laura Felicia; Walther, Uli, Singularities and holonomicity of binomial \(D\)-modules, J. Algebra, 439, 360-372, (2015) Sheaves of differential operators and their modules, \(D\)-modules, Singularities in algebraic geometry, Other hypergeometric functions and integrals in several variables, Toric varieties, Newton polyhedra, Okounkov bodies Singularities and holonomicity of binomial \(D\)-modules | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(k\) be a number field, and \(X\) be a smooth projective \(k\)-variety. In this paper, the author considers a slightly modified set of adelic points
\[
X(\mathbb{A}_k)_\bullet : = \prod_{v \nmid \infty} X(k_v) \times \prod_{v \mid \infty} \pi_0\big( X(k_v) \big)
\]
where the factors at infinite places \(v\) in the usual set of adelic points are reduced to the set of connected components of \(X(k_v)\), and approaches the problem of determining the existence of a rational point on \(X\) using descent methods within \(X(\mathbb{A}_k)_\bullet\) via \textit{torsors under finite étale group schemes,} which can be viewed as generalizations of the \(n\)-Selmer group of an abelian variety. Before we discuss the main results, I would like to point out that the paper contains a nice overview of the descent via torsors under finite étale group schemes, which is an introduction suitable for those who are not familiar with this extent of a generalization of descent methods. One of the main results of this paper in the context of the Brauer-Manin obstruction is an improvement on \textit{S. Siksek} [``The Brauer-Manin obstruction for curves having split Jacobians'', J. Théor. Nombres Bordx. 16, No. 3, 773--777 (2004; Zbl 1076.14033)], which is a positive answer toward the author's conjecture that the Brauer-Manin obstruction is the only obstruction against rational points on curves -- Skorobogatov first formulated it as a question. Using the descent via torsors under all finite abelian étale group schemes, the author establishes that \textit{for all curves} \(C\) the subset \(\mathcal C(\mathbb{A}_k)_\bullet^{\text{f-ab}}\) consisting of points in \(\mathcal C(\mathbb{A}_k)_\bullet\) which survive the descent is equal to the Brauer set \(\mathcal C(\mathbb{A}_k)_\bullet^{\mathrm{Br}}\), and proves the conjecture in the descent context, provided that there is a nonconstant \(k\)-morphism from \(C\) into an abelian variety \(A\) such that the divisible part of the Shafarevich-Tate group \(\text{Ш}(k,A)\) of \(A\) is trivial and \(A(k)\) is finite. The author believes that this condition is satisfied for all curves of genus \(\geq 2\). Let us review the descent via torsors under finite group schemes. An \(X\)-torsor under a finite étale group scheme \(G\) is a smooth projective variety \(Y\) with the following commutative diagram such that \(Y\times G\) is identified with the fiber product \(Y\times_X Y\):
\[
\begin{tikzcd} Y\times G \rar["\mu"]\dar["\mathrm{pr}_1" '] & Y\dar["\pi"]\\ Y \rar["\pi" ']] & X \end{tikzcd}
\]
where \(\mu\) is a right action of \(G\) on \(Y\), \(\mathrm{pr}_1\) is the projection, and \(\pi\) is a finite étale morphism. Note that to a point \(P\) in \(X(k)\), we can associate an element in the Galois cohomology set \(\mathrm{H}^1(k,G)\) as the group scheme \(G\) acts on the fiber \(\pi\mathrm{Inv}(P)\). In other words, the \(X\)-torsor induces a map \(\phi_Y : X(k) \to \mathrm{H}^1(k,G)\), and hence, the commutative diagram:
\[
\begin{tikzcd} X(k) \dar\rar["\phi_Y"] & \mathrm{H}^1(k,G) \dar["\mathrm{res}"]\\ X(\mathbb{A}_k)_\bullet \rar["\delta" '] & \prod_v \mathrm{H}^1(k_v,G) \end{tikzcd}
\]
whose setting is quite analogous to that of the \(n\)-Selmer group of an abelian variety. The set \({\mathcal C}ov(X)\) is defined to be the subset consisting of elements \(Q\) in \(X(\mathbb{A}_k)_\bullet\) such that \(\delta(Q) \in \mathrm{Img(res)}\) for all \(X\)-torsors under all finite étale group schemes, and the subsets \(X(\mathbb{A}_k)_\bullet^{\text{f-sol}}\) and \(X(\mathbb{A}_k)_\bullet^{\text{f-ab}}\) are similarly defined with the solvable/abelian group schemes. Denoting by \(\overline{X(k)}\) the topological closure of \(X(k)\) in \(X(\mathbb{A}_k)_\bullet\), we have
\[
X(k) \subset \overline{X(k)} \subset X(\mathbb{A}_k)_\bullet^{\text{f-cov}} \subset X(\mathbb{A}_k)_\bullet^{\text{f-sol}} \subset X(\mathbb{A}_k)_\bullet^{\text{f-ab}}\subset X(\mathbb{A}_k)_\bullet.
\]
A significant part of this paper is for the development of a theory toward the occasions of equalities between these sets and toward the relationships with various Brauer sets, and as mentioned earlier, for the case of all curves, it is particularly fruitful.
For \(X\) being an abelian variety \(A\), the author takes a more natural generalization of \(n\)-descent; namely,
\[
0 \to \widehat{A(k)} \to \mathrm{Sel} \to T\, \text{ Ш}(k,A) \to 0
\]
where \(\widehat{A(k)} = A(k) \otimes \hat{\mathbb Z}\), \(\mathrm{Sel} = \lim\mathrm{Sel}^{(n)}\), and \(T\,\text{Ш}(k,A)\) is the Tate module of \(\text{Ш}(k,A)\). In this paper two main results for abelian varieties in this paper are introduced:
(1) For a set of finite places of \(k\) of good reduction for \(A\) and of density \(1\), we have canonical injective homomorphisms
\[
\widehat{A(k)} \to \widehat{\mathrm{Sel}(k,A)}\to \prod_{v \in S} A(\mathbb F_v)
\]
where the last map factors through \(\prod_{v \in S} A(k_v)\), and \(\widehat{A(k)} = \overline{A(k)}\) as images in \(\prod_{v \in S} A(k_v)\).
This is an improvement on \textit{J.-P. Serre} [Theorem 3, Sur les groupes de congruence des variétés abéliennes. II, Izv. Akad. Nauk SSSR, Ser. Mat. 35, 731--737 (1971; Zbl 0222.14025)];
(2) If \(Z \subset A\) is a finite subscheme of \(A\) over \(k\), then for a set \(S\) of places of \(k\) of density \(1\) the intersection \(Z(\mathbb{A}_k)_\bullet \cap \widehat{\mathrm{Sel}(k,A)}\) in \(\prod_{v \in S} A(k_v)/A(k_v)^0\) is the image of \(Z(k)\).
The second result means that for a finite subscheme \(Z\) of \(A\), the intersection \(Z(\mathbb{A}_k)_\bullet\cap\widehat{\mathrm{Sel}(k,A)}\) is the only obstruction against rational points on \(Z\).
The author also formulates an ``Adelic Mordell-Lang Conjecture'', and explains its implication on some subvarieties of \(A\):
\textit{Adelic Mordell-Lang Conjecture}: Let \(X \subset A\) be a subvariety not containing a translate of a nontrivial subgroup of \(A\). Then, there is a finite subscheme \(Z \subset X\) such that \(X(\mathbb{A}_k)_\bullet \cap \widehat{\mathrm{Sel}(k,A)} \subset Z(\mathrm{adel})_\bullet\).
This conjecture together with the second result for \(A\) implies that \(X(k)=Z(k)\), and the chain of adelic subsets shown above collapses to \(X(k)=X(\mathbb{A}_k)_\bullet^{\text{f-ab}}\). The author also remarks that the above conjecture is true when \(k\) is a global function field, \(A\) is ordinary, and \(X\) is not defined over \(k^p\) where \(p\) is the characteristic of \(k\). rational points; descent obstruction; covering; twist; torsor under finite group scheme; Brauer-Manin obstruction M. Stoll, ''Finite descent obstructions and rational points on curves,'' Algebra Number Theory, vol. 1, iss. 4, pp. 349-391, 2007. Curves of arbitrary genus or genus \(\ne 1\) over global fields, Rational points, Abelian varieties of dimension \(> 1\), Coverings of curves, fundamental group Finite descent obstructions and rational points on curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this article, we study the asymptotic cardinality of the set of algebraic points of fixed degree and bounded height of a surface defined over a number field, when the bound on the height tends to infinity. In particular, we show that this can be connected to the Batyrev-Manin-Peyre conjecture, i.e. the case of rational points, on some punctual Hilbert scheme. Our study shows that these associated Hilbert schemes provide, under certain conditions, new counterexamples to the Batyrev-Manin-Peyre conjecture. However, in the cases of \(\mathbb{P}^1 \times \mathbb{P}^1\) and \(\mathbb{P}^2\) detailed in this article, the associated Hilbert schemes satisfy a slightly weaker version of the Batyrev-Manin-Peyre conjecture. number theory; algebraic point; Hilbert scheme; heights Rational points, Varieties over global fields, Parametrization (Chow and Hilbert schemes), Heights Algebraic points of bounded height on a surface | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper a new class of planar curves, which properly contains a class of algebraic curves, is introduced. They are called P-curves, and are related with polynomial systems of ordinary differential equations. The main result is an extension to P-curves of the classical Bezout theorem. planar dynamical systems; P-curves; polynomial systems of ordinary differential equations; Bezout theorem Khovanskii, A.: Cycles of dynamical systems in the plane and rolle's theorem. Siberian math. J. 25, 502-506 (1984) Singularities of curves, local rings, Cycles and subschemes, Dynamical systems and ergodic theory Cycles of dynamical systems on the plane and Rolle's theorem | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors give a deterministic polynomial time algorithm for computing the zeta function of an arbitrary variety of fixed dimension over a finite field of small characteristic. An important property of the algorithm is the polynomial dependency of its complexity on the degree of the finite field. As the characteristic of the finite base field contributes polynomially to the running time, in practice the algorithm is expected to be limited to finite fields of small characteristic. An essential ingredient of the algorithm is given by Dwork's trace formula which relates an explicit Frobenius action on a certain \(p\)-adic cohomology space to the number of torus valued rational points of a given hypersurface over a finite field. The main result of the article reads as follows.
Theorem: There exist an explicit deterministic algorithm and an explicit polynomial \(Q(X)\) with the following property. Let \(\mathbb{F}\) be a finite field and assume that polynomials \(f_1, \ldots, f_r \in \mathbb{F}[X_1,\ldots,X_n]\) with total degrees \(d_1, \ldots, d_r\) respectively are given. The local zeta function of the affine scheme defined by the system of equations \(\{ f_i=0 \}_{i=1,\ldots,r}\) can be computed in a number of bit operations which is bounded by \(Q \big( 2^r p^n d^{n^2} a^n \big)\), where \(p\) denotes the characteristic of the finite field \(\mathbb{F}\), \(a=\log_p(\# \mathbb{F})\) and \(d=\sum_{i=1}^r d_i\).
The article is self-contained so that the results are accessible for non-experts. A concise introduction to the theory of additive character sums over finite fields, the analytic representation of characters and Dwork's cohomology theory is given. The theoretical part of the article culminates in a reproof of Dwork's trace formula. The algorithmic aspects of Dwork's theory are dealt with by giving precise estimates for the valuations of the power series coefficients that implicitly occur in the trace formula and by calculating in detail the complexity of the resulting algorithm. According to the authors the paper is written `primarily for theoretical computational interest'. No practical results are given.
Next we give the idea of the algorithm of Lauder and Wan. We use the notation of the above theorem. Applying the Weil conjectures, the computation of the zeta function comes down to the determination of the number of rational points over finitely many extension fields of \(\mathbb{F}\). In the following we denote the unique degree \(k\) extension of \(\mathbb{F}\) by \(\mathbb{F}_k\). We set \(q= \# \mathbb{F}\) and \(a=\log_p(q)\). Now suppose that we want to compute the number of \(\mathbb{F}_k\)-rational points of the scheme given by the system of equations \(\{ f_i=0 \}\). By an inclusion-exclusion argument involving the polynomials \(\prod_{i \in S} f_i\) where \(S \subseteq \{ 1, \ldots, r \}\) without loss of generality one can assume that the scheme, the zeta function of which is to be computed, is given by a hypersurface with defining polynomial \(f \in \mathbb{F}[X_1, \ldots, X_n]\). Let \(N_k^*\) denote the number of solutions of the equation \(f=0\) in the set \((\mathbb{F}_k^*)^n\). Using the fact that a torus decomposition of the set of rational points exists, by induction one is reduced to calculating the numbers \(N_k^*\). Using the classical theory of additive character sums it is straight forward to prove the formula \(\sum_{(x_0,\ldots,x_n) \in (\mathbb{F}_k^*)^{n+1}} \Psi_k \big( x_0f(x_1, \ldots,x_n) \big)=q^k N_k^*- ( q^k -1)^n\), where \(\{ \psi_k \}\) is a family of additive characters \(\Psi_k:\mathbb{F}_k \rightarrow \Omega^*\), which satisfy the compatibility \(\Psi_k=\Psi_1 \circ \mathrm{Tr}_{\mathbb{F}_k/\mathbb{F}}\), and \(\Omega\) is a universal \(p\)-adic domain. We note that in order to make algorithmic use of the latter formula a computational model for a suitable subring \(R\) of \(\Omega\) has to be chosen. The characters \(\Psi_k\) can be made explicit as follows. One defines a formal power series by setting \(\theta(z)=\mathrm{exp}(\pi z-\pi z^p)\) where \(\pi\) is chosen such that \(\pi^{p-1}=-p\). It can be shown that this power series converges on the closed unit disk. The functions \(\Psi_k=\Phi_k \circ \omega\), where \(\omega\) is the Teichmüller map and \(\Phi_k(z)=\prod_{i=0}^{ak-1} \theta(z^{p^i})\) form a compatible system of additive characters in the above sense.
In order to do computations it is convenient to express the character sum \(\sum \Psi_k \big( x_0f(x_1, \ldots,x_n) \big)\) in terms of the trace of a certain operator on an infinite dimensional linear subspace of the formal \(p\)-adic power series ring \(R[[X_0,\ldots,X_n]]\). For technical reasons an additional indeterminate \(X_0\) is needed. For \(\mu=(\mu_0, \ldots, \mu_n) \in \mathbb{Z}_{\geq 0}^{n+1}\) we set \(X^{\mu}=X_0^{\mu_0} \ldots X_n^{\mu_n}\). One considers the space of formal power series \(L_{\Delta}\) which consists of the infinite sums in the monomials \(X^{\mu}\) where \(\mu\) ranges over the cone \(\Delta\) generated by the exponents of non-zero terms of the polynomial \(X_0 f(X_1, \ldots, X_n)\). One writes \(X_0f=\sum_{\mu} a_{\mu} X^{\mu}\), where the sum ranges over the support of \(X_0f\), and defines a formal power series by setting \(F^{(a)}(X)=\prod_{\mu} \prod_{s=0}^{a-1} \theta \big( \omega(a_{\mu}^{p^s}) X^{p^s\mu} \big)\). The power series \(F^{(a)}\) is an element of the ring \(L_{\Delta}\). The latter ring is stable under the map \(\alpha_a=\Gamma_p^a \circ F^{(a)}\) where the latter is the composition of the multiplication by the formal power series \(F^{(a)}\) followed by Dwork's left inverse of the `Frobenius' map. The operator \(\alpha_a\) is nuclear. Thus, the trace \(\mathrm{Tr}(M_a)\) of a matrix \(M_a\), which represents the operator \(\alpha_a\) with respect to the basis given by the monomials \(X^{\mu}\), is well-defined. A trace formula can be deduced from the above character sum formula by an explicit power series computation. The latter trace formula, Dwork's fundamental theorem, is given by \((q^k-1)^{n+1} \mathrm{Tr}(M_a^k) = q^k N_k^*- ( q^k -1)^n\). Lauder and Wan suggest in their paper to compute the matrix \(M_a\) up to a certain precision using explicit power series expansions, which allows them to determine an exact value for \(N_k^*\) using Dwork's formula. In a non-trivial calculation they show that there exists a meaningful modular reduction of the trace formula. As a consequence, they are able to give an explicit algorithm for the computation of the number of rational points on a hypersurface over a finite field. point counting; zeta function; varieties over finite fields Lauder A., ''Counting Points on Varieties over Finite Fields of Small Characteristic.'' (2001) Number-theoretic algorithms; complexity, Varieties over finite and local fields, Computational aspects of higher-dimensional varieties, Finite ground fields in algebraic geometry, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Applications to coding theory and cryptography of arithmetic geometry, Cryptography Counting points on varieties over finite fields of small characteristic | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this note the author gives an explicit formula for the \(b\)-function of certain prehomogeneous vector spaces associated with flag manifolds of the general linear group. These prehomogeneous vector spaces are related to Eisenstein series of the general linear group and the studied \(b\)-functions describe the possible poles of those series. flag manifold; general linear group; prehomogeneous vector space; b-function Sato, F.: B-functions of prehomogeneous vector spaces attached to flag manifolds of the general linear group. Comment. math. Univ. st. Pauli 48, 129-136 (1999) Prehomogeneous vector spaces, Forms of degree higher than two, Analytic theory (Epstein zeta functions; relations with automorphic forms and functions), Grassmannians, Schubert varieties, flag manifolds, Linear algebraic groups over the reals, the complexes, the quaternions \(b\)-functions of prehomogeneous vector spaces attached to flag manifolds of the general linear group | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The idea of an order function originated with the second author [\textit{R. Pellikaan}, J. Stat. Plann. Inference 94, No.~2, 287-301 (2001; Zbl 0981.94055)] as a way to consider one-point Goppa codes using ring theory, without using algebraic geometry. An order function generalizes the order of pole of a rational function along a prime divisor. The concepts of order function and order domain were then generalized by \textit{R. Matsumoto} and \textit{S. Miura} [``On construction and generalization of algebraic geometry codes'', in: Proc. Algebr. Geom., Number Theory, Coding Theory and Cryptography, Univ. Tokyo 2000, 3-15 (2000)], and by \textit{M. E. O'Sullivan} [Finite Fields Appl. 7, No. 2, 293-317 (2001; Zbl 1027.94032)]. Here, the authors further generalize these concepts, with an order structure consisting of an algebra \(R\) (order domain) over a field and a map (order function), satisfying certain properties, from \(R\) to a well-ordered set, which then gets a semigroup structure from the order function. If this semigroup is finitely generated, then the authors use Hilbert functions to show that the rank of this semigroup is equal to the dimension of the ring \(R\). Following \textit{M. E. O'Sullivan} [loc. cit.], the authors extend the theory of Gröbner bases to order domains. They also study the behavior of order domains under formation of factor rings, extension of scalars, and tensor product. order domain; order function; order structure; Gröbner basis; one-point Goppa codes Geil O., Pellikaan R.: On the structure of order domains. Finite Fields Appl. \textbf{8}, 369-396 (2002). Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Geometric methods (including applications of algebraic geometry) applied to coding theory, General structure theory for semigroups, Valuations and their generalizations for commutative rings, Applications to coding theory and cryptography of arithmetic geometry On the structure of order domains | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We construct an analog of the classical theta function on an abelian variety for the closed 4-dimensional symplectic manifolds that are \(T^2\)-bundles over \(T^2\) with the zero Euler class. We use our theta functions for a canonical symplectic embedding of these manifolds into complex projective spaces (an analog of the Lefschetz theorem). theta function; symplectic embedding D. V. Egorov, Theta functions on fiber bundles of two-dimensional tori with zero Euler class, Sibirsk. Mat. Zh. 50 (2009), no. 4, 818 -- 830 (Russian, with Russian summary); English transl., Sib. Math. J. 50 (2009), no. 4, 647 -- 657. Theta functions and abelian varieties, Compact complex surfaces Theta functions on \(T^2\)-bundles over \(T^2\) with the zero Euler class | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this interesting paper the author deals with the ABC-conjecture and the asymptotic Fermat conjecture over global fields. He also states some conjectures about elliptic curves and corresponding Galois representations which are related to these two conjectures. Furthermore, he proves some of them in the case of function fields. Fermat equations; modular curves; ABC-conjecture; asymptotic Fermat conjecture; elliptic curves; Galois representations; function fields G. Frey, ''On ternary equations of Fermat type and relations with elliptic curves,'' in Modular Forms and Fermat's Last Theorem, New York: Springer-Verlag, 1997, pp. 527-548. Elliptic curves over global fields, Higher degree equations; Fermat's equation, Elliptic curves On ternary equations of Fermat type and relations with elliptic curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper we consider the explicit solutions of some classical finite-dimensional integrable systems linearized on unramified coverings of generalized Jacobian varieties and apply such solutions to integrate ellipsoidal billiards.
Section 1 shows how a generalized Abel-Jacobi mapping arises as a reduction of some classical integrable systems (the Euler top in the space and asymptotic geodesic motion on an ellipsoid) to quadratures.
In Sections 2 and 3 we recall the procedure of inversion of classical and generalized Abel-Jacobi mappings in a self-contained form and give theta-functional expressions for the so-called \textit{Wurzelfunktionen} (root functions) defined on coverings of abelian varieties\dots{} In Section 4, we use the Wurzelfunktionen to obtain solutions for asymptotic geodesic motion on an ellipsoid and the problem of the Euler top motion in space in terms of generalized theta-functions. Then we compare the latter solutions with those found by Jacobi.
In Section 5, following the same approach, we easily derive solutions for ellipsoidal billiards and their generalizations naturally arising as limit cases of geodesic motion on a quadric of higher dimension. In addition, for the first time we give solution to a billiard on an ellipsoidal layer\dots{}
Finally, Section 6 considers the extreme case of degeneration of Jacobians, when the corresponding theta-functions reduce to tau-functions\dots{} In addition, we describe a remarkable relation between the change of phases of the billiard solution under impacts and the change of geometric phases of solitons of the KdV equation corresponding to collision of solitons. integrable systems; Jacobian varieties; integration of ellipsoidal billiards; Abel-Jacobi mapping; Euler top; asymptotic geodesic motion; ellipsoidal layer; theta-functions; KdV equation; solitons [InlineMediaObject not available: see fulltext.]\textbf{65}(2), 133-194 (2010) (Russian). English translation: Dragović, V., Radnović, M.: Integrable billiards and quadrics. \textit{Russ. Math. Surv}., \textbf{65}(2), 319-379 (2010) Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), , Abelian varieties and schemes, Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions Classical integrable systems and billiards related to generalized Jacobians | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems \textit{G. Ellingsrud} and \textit{Ch. Peskine} [Invent. Math. 95, No, 1, 1-11 (1989; Zbl 0676.14009)] proved a theorem which states that there are only a finite number of components in the Hilbert scheme parameterizing smooth surfaces in \(\mathbb{P}^4\) not of general type. On the other hand, \textit{R. Braun} and \textit{G. Fløystad} [Compos. Math. 93, No. 2, 211-229 (1994; Zbl 0823.14021)] proved that a smooth surface in \(\mathbb{P}^4\) not of general type has degree \(d\leq 105\).
In this note, the author proves that there is only a finite number of components in the Hilbert scheme of surfaces in \(\mathbb{P}^r\) parameterizing integral surfaces of degree \(d\). Then he deduces that a smooth surface in \(\mathbb{P}^4\) not of general type has degree \(d\leq 105\). Thus he obtains a simpler and quicker proof of the above theorem. The proof in this note essentially relies on Castelnuovo's theory and on a lower bound for the genus of the generic hyperplane section of a smooth surface in \(\mathbb{P}^4\). Enriques' classification; Hilbert scheme; Castelnuovo's theory; surfaces in \(\mathbb{P}^4\) Gennaro, V, A note on smooth surfaces in \({\mathbb{P}}^4\), Geometriae Dedicata, 71, 91-96, (1998) Parametrization (Chow and Hilbert schemes), Surfaces of general type, Surfaces and higher-dimensional varieties, Projective techniques in algebraic geometry, Low codimension problems in algebraic geometry A note on smooth surfaces in \(\mathbb{P}^4\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a general smooth projective algebraic curve of genus \(g\geq2\) over \(\mathbb C\); more precisely, suppose that \(X\) is Petri, that is the multiplication map \(H^0(L)\otimes H^0(L^*\otimes K)\to H^0(K)\) is injective for every line bundle \(L\) on \(X\). Denote by \(G(\alpha;n,d,k)\) the moduli space of \(\alpha\)-stable coherent systems on \(X\) of type \((n,d,k)\); here \(\alpha\) is a positive real number and a coherent system of type \((n,d,k)\) is a pair \((E,V)\), where \(E\) is a vector bundle on \(X\) of rank \(n\) and degree \(d\) and \(V\) is a linear subspace of \(H^0(E)\) of dimension \(k\). These moduli spaces have been described in some detail for \(k\leq n\) (for the most complete results, see [Int. J. Math. 18, No. 4, 411--453 (2007; Zbl 1117.14034]), but much less is known for \(k>n\).
In this paper, the author obtains some results for \(k>n\). Define \(\beta:=g-(n+1)(n-d+g)\); this is both the dimension of \(G_d^n=G(1,d,n+1)\) (the classical space of linear systems of degree \(d\) and (projective) dimension \(n\)) and the ``expected dimension'' of \(G(\alpha;n,d,n+1)\). Theorem 1 asserts (1) if \(\beta<0\), then \(G(\alpha;n,d,k)=\emptyset\), and, if \(0\leq\beta<g\) or \(\beta=g\) and \(n\) does not divide \(g\), then (2)/(3) \(G(\alpha;n,d,k)\) is independent of \(\alpha>0\) (in the language of [Int. J. Math. 14, No. 7, 683--733 (2003; Zbl 1057.14041)], there are no ``flips''), (4) \((E,V)\in G(\alpha;n,d,k)\) if and only if \((E,V)\) is generically generated and \(H^0{}(E{'}^{*}{})=0\), where \(E'\) is the (sheaf-theoretic) image of the evaluation map \(V\otimes{\mathcal O}\to E\) and (5) if \((E,V)\in G(\alpha;n,d,k)\), then \(E\) is a stable bundle. Note that, in this theorem, \(\beta\) is not the``expected dimension'' of \(G(\alpha;n,d,k)\) except when \(k=n+1\). Theorem 2 concerns the case \(k=n+1\); its main assertions are that, if \(\beta\leq g\), then (1) \(G(\alpha;n,d,n+1)\neq\emptyset\) if and only if \(\beta\geq0\), (2) \(G(\alpha;n,d,n+1)\) is independent of \(\alpha>0\) and (3) if \(\beta>0\), then \(G(\alpha;n,d,n+1)\) is smooth and irreducible of dimension \(\beta\) and the generic element is generated. Moreover, if \(\beta=0\), then \(G(\alpha;n,d,n+1)\) is isomorphic to \(G_d^n\) and is therefore a finite set with \(g!\prod_{i=0}^n{i!\over(g-d+n+i)!}\) points. The main assertions of Theorem 3 are that, if \(g\geq n^2-1\) and \(\beta\geq0\), then (1) \(G(\alpha; n,d,n+1)\neq\emptyset\) for all \(\alpha>0\) and (2) there exist coherent systems \((E,V)\) with \((E,V)\) \(\alpha\)-stable for all \(\alpha>0\) and \(E\) stable, and the set of all such \((E,V)\) forms a smooth, irreducible variety of dimension \(\beta\) (the assumption \(\beta\leq g\) is no longer needed). An application is given to the stability of the pullback of the tangent bundle under a morphism from \(X\) to a projective space.
Section 5 concerns the dual span construction introduced by \textit{D. Butler} in an unpublished paper [\url{arXiv:alg-geom/9705009}] (see also [Int. J. Math. 14, op.cit., section 5.4]). In particular, Theorem 6 asserts that, if \(d<g+n_1\leq g+n_2\) and \(\alpha>0\), then \(G(\alpha;n_1,d,n_1+n_2)\neq\emptyset\) if and only if \(G(\alpha;n_2,d,n_1+n_2)\neq\emptyset\), while Theorem 7 relates the smoothness of these spaces at corresponding points. Finally, section 6 includes further results for the case \(n=2\) and in particular for \(n=g=2\).
The results concerning the case \(k=n+1\) have been substantially extended by \textit{U. N. Bhosle}, the author and the reviewer [Manuscr. Math. 126, No. 4, 409--441 (2008; Zbl 1160.14021)]. Algebraic curve; coherent systems; moduli space; stability; Brill-Noether loci Brambila-Paz, L., Non-emptiness of moduli space of coherent systems, Int. J. Math., 19, 7, 777-799, (2008) Vector bundles on curves and their moduli Non-emptiness of moduli spaces of coherent systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The aim of this paper is to study, by using the technique of \(s\)- differents developed by the author in his forthcoming paper ``On the differential class of an algebraic function field'' (to appear in Rev. Roum. Math. Pures Appl.), the link between the genus and Riemann-Roch theorem for algebraic function fields of one variable with arbitrary constant field as considered by F. K. Schmidt in 1936 via the notion of different divisor, and the same notions investigated by A. Weil in 1938 with the use of differentials. field extension; genus; Riemann-Roch theorem; algebraic function fields; different divisor; differentials Differential algebra, Field extensions, Algebraic functions and function fields in algebraic geometry On the notion of a differential in the theory of algebraic functions with arbitrary constant field | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A tau function of the 2D Toda hierarchy can be obtained from a generating function of the two-partition cubic Hodge integrals. The associated Lax operators turn out to satisfy an algebraic relation. This algebraic relation can be used to identify a reduced system of the 2D Toda hierarchy that emerges when the parameter \(\tau\) of the cubic Hodge integrals takes a special value. Integrable hierarchies of the Volterra type are shown to be such reduced systems. They can be derived for positive rational values of \(\tau\). In particular, the discrete series \(\tau=1,2,\dots\) correspond to the Volterra lattice and its hungry generalizations. This provides a new explanation to the integrable structures of the cubic Hodge integrals observed by Dubrovin et al. in the perspectives of tau-symmetric integrable Hamiltonian PDEs. tau function; Toda hierarchy; Lax operators; cubic Hodge integrals Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry, Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds Cubic Hodge integrals and integrable hierarchies of Volterra type | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper proves an effective version of a theorem of \textit{J.-H. Evertse} and \textit{R. Ferretti} [in: Schlickewei, Hans Peter et al., Diophantine approximation. Festschrift for Wolfgang Schmidt. Vienna, Austria, 2003. Wien: Springer, Dev. Math. 16, 175--198 (2008; Zbl 1153.11032)], in the function field case of characteristic zero.
The main theorem of the paper is as follows. Let \(K\) be the function field of a nonsingular variety \(V\) over an algebraically closed field of characteristic zero. Let \(X\) be a smooth projective subvariety in \(\mathbb P^N\), and let \(n=\dim X\). For \(i=1,\dots,q\) let \(Q_i\) be a homogeneous polynomial of degree \(d_i\) in \(K[X_0,\dots,X_N]\). Assume that the supports of the divisors \((Q_i)\) do not contain \(X\), and that the intersection of any \(n+1\) of them with \(X\) is empty. Let \(S\) be a finite set of prime divisors on \(V\), and let \(\varepsilon>0\). Then there is an effectively computable, proper algebraic subset \(W_\varepsilon\) of \(X\) and effectively computable constants \(C_\varepsilon\) and \(C'_\varepsilon\), such that the inequality
\[
\sum_{i=1}^q \sum_{\mathfrak p\in S} \frac{\lambda_{\mathfrak p,Q_i}(\mathbf x)}{d_i} \leq (n+1+\varepsilon) h(\mathbf x) + C_\varepsilon'
\]
holds for all \(\mathbf x\in X(K)\) such that \(\mathbf x\notin W_\varepsilon\) and \(h(\mathbf x)>C_\varepsilon\). Here \(\lambda_{\mathfrak p,Q_i}\) is a Weil function for the divisor \((Q_i)\) on \(X\) at the place \(\mathfrak p\).
The theorem is proved using the methods of Evertse and Ferretti (Chow forms, Chow weights, and Hilbert weights), together with an effective version of Schmidt's Subspace Theorem due to the second author [Math. Z. 246, No. 4, 811--844 (2004; Zbl 1051.11041)].
The paper also introduces a simplification of the proof, using an elementary lemma (Lemma 14) in place of an arithmetic geometry lemma due to S. Zhang. Schmidt's Subspace theorem; function field; diophantine approximation; Chow form; Hilbert weight; degree of contact M. Ru and J. T.-Y. Wang, An effective Schmidt's subspace theorem for projective varieties over function fields, Int. Math. Res. Not. IMRN 3 (2012), 651--684. Schmidt Subspace Theorem and applications, Arithmetic varieties and schemes; Arakelov theory; heights, Value distribution theory in higher dimensions An effective Schmidt's subspace theorem for projective varieties over function fields | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper introduces schemes called ``correspondence scrolls'' and gives a general study of them. For a closed subscheme \(Z\) in \(\Pi _{i=1}^n {\mathbb A}^{a_i+1}\), defined by a multigraded ideal \(I\subset A:={\Bbbk}[x_{i,j} : 1\le i \le n, 0 \le j \le a_i]\), and for \(\mathbf{b}=(b_1, \ldots , b_n) \in \mathbb{N}_+^n\), consider the homomorphism \({\Bbbk}[z_{i,\alpha }] \rightarrow A/I\) which sends a variable \(z_{i, \alpha }\) to the monomial \(x_i^\alpha\), of degree \(b_i\). Here \(x_i^ {\alpha }\) denotes \(x_{i,0}^{\alpha _0} \cdots x_{i, a_i}^{\alpha _{a_i}}\). The kernel of the above map defines a closed projective subscheme \(C(Z, {\mathbf b}) \subset { \mathbb P}^N\) (\(N= \sum \binom{a_i+b_i}{a_i}-1)\), called \textit{correspondence scroll}. This definition includes classical correspondences as well as interesting non-classical ones: rational normal scrolls, double structures which are degenerate \(K3\) surfaces, degenerate Calabi-Yau threefolds, etc. Many invariants or properties of correspondence scrolls are studied: dimension, degree, nonsingularity, Cohen-Macaulay and Gorenstein property and others. The paper is very well written and invites to further research. rational normal scroll; Veronese embedding; join variety; multiprojective space; variety of complexes; variety of minimal degree; double structure; \(K3\) surface; Calabi-Yau scheme; Gorenstein ring; Gröbner basis \(n\)-folds (\(n>4\)), Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Linkage, complete intersections and determinantal ideals, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Rational and ruled surfaces, \(K3\) surfaces and Enriques surfaces, Calabi-Yau manifolds (algebro-geometric aspects), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Determinantal varieties, Rational and unirational varieties, Projective techniques in algebraic geometry Correspondence scrolls | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The higher Chow groups of a regular scheme were defined by Bloch, furnishing a geometric interpretation for the higher \(K\)-groups. The sheaves of higher \(K\)-groups recover the full ordinary Chow groups, including torsion: formula of Bloch-Quillen, spectral sequence of Gersten. In the paper these relationships are generalized to the higher Chow groups. Relative Chow groups, with respect to a closed subscheme, and relative higher Chow groups are defined, to which the relationships extend as well. For divisors, the relative Chow group is expressed in terms of Picard groups, using the generalized Bloch formula. algebraic \(K\)-theory; Bloch-Quillen formula; Gersten spectral sequence; sheaves of higher \(K\)-groups; torsion; higher Chow groups; relative Chow group; Picard groups S. E. Landsburg, ''Relative Chow groups,'' Illinois J. Math., vol. 35, iss. 4, pp. 618-641, 1991. \(K\)-theory of schemes, Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects), Applications of methods of algebraic \(K\)-theory in algebraic geometry, Algebraic cycles Relative Chow groups | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The main achievement of this paper is the construction of a new six-dimensional irreducible symplectic manifold \(\widetilde{\mathcal M}\). The author shows that this variety has \(b_ 2=8\). This implies that \(\widetilde{\mathcal M}\) cannot be deformed into one of the known irreducible symplectic varieties, not even up to birational equivalence.
According to the Bogomolov decomposition of compact Kähler manifolds with torsion first Chern class, there are three types of ``building blocks'': complex tori, Calabi-Yau varieties, and irreducible symplectic manifolds. Compared with the other two classes, examples of irreducible symplectic manifolds are quite scarce. Similar to previous constructions, the new example is obtained as an irreducible factor in the Bogomolov decomposition of a symplectic desingularisation of a moduli space of sheaves on an Abelian surface. According to results of Yoshioka, such a construction can be successful only if the moduli space contains points which correspond to strictly semi-stable sheaves.
The construction of \(\widetilde{\mathcal M}\) is the following. Let \(J\) be the Jacobian of a genus-two curve, \(\Theta\) a Theta divisor and \(\eta\) the orientation class of \(J\). Using methods from an earlier paper of the author [J. Reine Angew. Math. 512, 49--117 (1999; Zbl 0928.14029)], it can be shown that there exists a symplectic desingularisation \(\widetilde{\mathcal M}_ v \rightarrow {\mathcal M}_ v\) of the moduli space \({\mathcal M}_ v\) of \(\Theta\)-semi-stable torsion free sheaves on \(J\) with Mukai vector \(v=2-2\eta\). The variety \(\widetilde{\mathcal M} \subset \widetilde{\mathcal M}_ v\) is the fibre over \((0,\widehat{0})\) of a natural locally trivial fibration \(\widetilde{\mathcal M}_ v \rightarrow J\times \widehat{J}\). After verifying that \(\widetilde{\mathcal M}\) is symplectic and of dimension six, the author shows that \(\widetilde{\mathcal M}\) is simply connected and has \(b_ 2=8\). He defines a refinement of S-equivalence in order to obtain a moduli theoretic interpretation of a subset of \(\widetilde{\mathcal M}_ v\). This allows him to use the generalised Lefschetz hyperplane theorem to gain information about the topology of \(\widetilde{\mathcal M}\). Ricci flat Kähler metric; hyper-Kähler; Hilbert scheme; holomorphic symplectic form; moduli space; Mukai vector; Bogomolov decomposition. K.\ G. O'Grady, A new six-dimensional irreducible symplectic variety, J. Algebraic Geom. 12 (2003), no. 3, 435-505. Global theory of symplectic and contact manifolds, Kähler manifolds, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Compact complex \(n\)-folds, Global differential geometry of Hermitian and Kählerian manifolds, Hyper-Kähler and quaternionic Kähler geometry, ``special'' geometry, Jacobians, Prym varieties, Topological aspects of complex manifolds A new six-dimensional irreducible symplectic variety | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We describe the classical Hasse principle for the existence of nontrivial zeros for quadratic forms over number fields, namely, local zeros over all completions at places of the number field imply nontrivial zeros over the number field itself. We then go on to explain more general questions related to the Hasse principle for nontrivial zeros of quadratic forms over function fields, with reference to a set of discrete valuations of the field. This question has interesting consequences over function fields of \(p\)-adic curves. We also record some open questions related to the isotropy of quadratic forms over function fields of curves over number fields. Hasse principle; function fields of \(p\)-adic curves Parimala, R.: A Hasse principle for quadratic forms over function fields. Bull. amer. Math. soc. (N.S.) 51, No. 3, 447-461 (2014) Quadratic forms over general fields, Algebraic functions and function fields in algebraic geometry, Arithmetic theory of algebraic function fields A Hasse principle for quadratic forms over function fields | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let K be a local field of characteristic 0 (i.e. complete with respect to a discrete valuation) with perfect residue field k of characteristic \(p>0.\) Let \(\bar K\) be the algebraic closure of K and \(G=Gal(\bar K/K)\). Let \(K_ 0\) be the quotient field of the ring of Witt vectors W(k) and let \(e=[K:K_ 0]\) be the absolute ramification index of K. Denote by \(\sigma\) the Frobenius automorphism over k, W(k) and \(K_ 0.\)
Let MF be the category whose objects are finite dimensional k-modules M equipped with a filtration \(M=M^ 0\supset M^ 1\supset...\supset M^ p=0\) by k-submodules, \(0\leq i\leq p\), and \(\sigma\)-semilinear maps \(\phi^ i: M^ i\to \bar M\) such that \(\ker (\phi^ i)=M^{i+1}\) and \(\phi^ 0(M^ 0)+\phi^ 1(M^ 1)+...+\phi^{p-1}(M^{p-1})=M\) and whose morphisms are morphisms of filtered modules which commute with all \(\phi^ i\), \(0\leq i<p\). Denote by Rep(K) the category of \({\mathbb{F}}_ p[G]\)-modules of finite length over \({\mathbb{F}}_ p\) and let \(\phi\) Rep(K) be the category of pairs \((H^ 0,H)\), where \(H\in Rep(K)\) and \(H^ 0\) is a trivial G-submodule in H. The morphisms in \(\phi\)Rep(K) are morphisms of filtered G-modules \((H^ 0,H)\to (H^ 0_ 1,H_ 1)\) modulo the subgroups of the morphisms which factorize \((H^ 0,H)\to (0;H/H^ 0)\to (H^ 0_ 1,H^ 0_ 1)\to (H^ 0_ 1,H_ 1)\), where the first and the last are uniquely determined.
The author proves the existence of a faithfully exact functor \(\tilde U: MF\to \phi Rep(K)\) and partially characterizes its image. This functor is a slight modification of the functor \(U_ S\) of \textit{J. M. Fontaine} and \textit{G. Laffaille} [Ann. Sci. Ec. Norm. Super., IV. Sér. 15, 547-608 (1982; Zbl 0579.14037)].
Now let K be a field of algebraic numbers and X be a smooth, proper scheme over the ring of integers of K and \(h^{ij}\) the Hodge numbers of \(X\otimes {\mathbb{C}}\). Using the above results and the comparison between crystalline and étale cohomology the author (and independently J. M. Fontaine) succeeded in proving that, for \(K={\mathbb{Q}}\), \(h^{ij}=0\) for \(i+j\leq 3\), \(i\neq j\), and for \(K={\mathbb{Q}}(\sqrt{-3})\), \(h^{ij}=0\) for \(i+j\leq 2\), \(i\neq j\). triviality of Hodge numbers; crystalline cohomology; ring of Witt vectors; proper scheme over the ring of integers \(p\)-adic cohomology, crystalline cohomology, Schemes and morphisms, Galois cohomology, Witt vectors and related rings Modular crystalline representations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(k\) be an algebraically closed field, and let \(P_ 1,\ldots,P_ s\) be points in general position in \(\mathbb{P}^ 2(k)\) (no three on a line), and let \(m_ 1,\ldots,m_ s\) be positive integers. The paper concerns the Hilbert function of the coordinate ring of the points \(P_ i\) with multiplicities \(m_ i\), respectively. The first main result provides a characterization for the points to lie on an irreducible conic. The second one gives, in case \(m_ i=2\) for all \(i\), a relation between the index of regularity (the degree where the Hilbert function stabilizes to the Hilbert polynomial) and the number of points lying on a conic. points in general position; Hilbert function; multiplicities; index of regularity; Hilbert polynomial; number of points lying on a conic Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Projective techniques in algebraic geometry, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Enumerative problems (combinatorial problems) in algebraic geometry Some remarks on the Hilbert function of a zero-cycle in \(\mathbb{P}^ 2\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(d\) and \(g\) be integers, and consider the Hilbert scheme \(H(d,g)\) parametrizing smooth, irreducible projective space curves of degree \(d\) and genus \(g\). The author proves that if \(H(d,g)\) is nonempty, then it contains a generically smooth component of the ``expected'' dimension. Moreover, the cohomological properties of a general curve in this component is studied. Hilbert scheme of smooth connected space curves; degree; genus Kleppe, J.O.: On the existence of nice components in the Hilbert Scheme \(\text{H}(d,g)\) of Smooth Connected Space Curves. Boll. U.M.I (7) 8-B, 305-326 (1994) Families, moduli of curves (algebraic), Plane and space curves, Parametrization (Chow and Hilbert schemes), Vector bundles on curves and their moduli On the existence of nice components in the Hilbert scheme \(H(d,g)\) of smooth connected space curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For a morphism \(f:X\to Y\) with \(Y\) being nonsingular, the Ginzburg-Chern class of a constructible function \(\alpha\) on the source variety \(X\) is defined to be the Chern-Schwartz-MacPherson class of the constructible function \(\alpha\) followed by capping with the pull-back of the Segre class of the target variety \(Y\). We give some generalizations of the Ginzburg-Chern class even when the target variety \(Y\) is singular and discuss some properties of them. bivariant theory; Chern-Schwartz-MacPherson class; constructible function; Riemann-Roch formula Yokura, S.: Generalized Ginzburg--Chern classes. Séminaires et congrès 10, 429-442 (2005) Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Other homology theories in algebraic topology Generalized Ginzburg-Chern classes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This article explores vanishing theorems and Brauer-Hasse-Noether exact sequences for the cohomology of higher-dimensional fields. If we consider \(k\) a finite field, a \(p\)-adic field, or a number field. Let \(K\) be a finite extension of the Laurent series field in \(m\) variables \(k((x_1,\dots,x_m))\). When \(r\) is an integer and \(l\) is a prime number, the author considers the Galois module \(\mathbb Q_{l}/\mathbb Z_{l}(r)\) over \(K\), and proves several vanishing theorems for its cohomology. In the particular case in which \(K\) is a finite extension of the Laurent series field in two variables \(k((x_1,x_2))\), the author also proves exact sequences that play the role of the Brauer-Hasse-Noether exact sequence for the field \(K\) and that involve some of the cohomology groups of \(\mathbb Q_{l}/\mathbb Z_{l}(r)\) which do not vanish. Galois cohomology; Bloch-Kato conjecture; Laurent series fields in two or more variables; function fields in two or more variables; singularities; finite base fields; \(p\)-adic base fields; global base fields; Hasse principle; Brauer group; Brauer-Hasse-Noether exact sequence Galois cohomology, Varieties over finite and local fields, Varieties over global fields, Galois cohomology, Galois cohomology, Finite ground fields in algebraic geometry, Local ground fields in algebraic geometry, Global ground fields in algebraic geometry, Arithmetic ground fields for surfaces or higher-dimensional varieties, Singularities of surfaces or higher-dimensional varieties Vanishing theorems and Brauer-Hasse-Noether exact sequences for the cohomology of higher-dimensional fields | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(V\) denote an open irreducible subset of the Hilbert-scheme of \(\mathbb P^r\) parametrizing irreducible and smooth (if \(r\geq 3)\) resp. nodal (if \(r=2)\) curves of genus \(g\) and \(\pi\colon V\to \mathcal M_ g\) the natural map into the moduli space of curves of genus \(g\). \(V\) is called to have the expected number of moduli if \(\dim \pi(V)=\min(3 g-3, 3g - 3+\rho(g,r,n))\), where \(\rho(g,r,n)\) is the Brill-Noether number. It is the aim of the paper to construct such families of curves. For plane curves the result is quite complete. It is shown that for all \(g\) and \(n\geq 5\) such that \(n-2\leq g\leq \binom{n-1}{2}\) there is an irreducible component of the family of plane irreducible nodal curves of degree \(n\) and genus \(g\), having the expected number of moduli. In the range \(n-2\leq g\leq 3n/2-3\) (resp. \(2n-4\leq g\leq \binom{n-1}{2}\) this result was known (resp. independently proved) by \textit{P. Griffiths} and \textit{J. Harris} [Duke Math. J. 47, 233--272 (1980; Zbl 0446.14011)] (resp. by Coppens).
For \(r\geq 3\) the result is: for all \(g\) and \(n\geq r+1\) such that \(n-r\leq g\leq(r(n-r)-1)/(r-1)\) (resp. \(n-3\leq g\leq 3n-18\) if \(r=3)\) there is an open set of an irreducible component of the Hilbert scheme of \(\mathbb P^r\) parametrizing smooth irreducible curves of degree \(n\) and genus \(g\), which has the expected number of moduli. An immediate consequence is a special case of a theorem of \textit{D. Eisenbud} and \textit{J. Harris} [Invent. Math. 74, 371--418 (1983; Zbl 0527.14022)] on the existence of very ample line bundles of a prescribed type.
The proofs of the theorems are given by induction, the induction step being roughly as follows: Start with a particular curve \(C\) in \(\mathbb P^n\) whose existence is known, construct a new curve by adding a particular rational curve \(\gamma\) and get a new curve \(C'\) by flat smoothing of \(C\cup \gamma\). Hilbert-scheme; moduli space of curves Sernesi E. '' On the existence of certain families of curve .'', Inv. Math. 75 ( 1984 ), 125 - 171 . MR 728137 | Zbl 0541.14024 Families, moduli of curves (algebraic) On the existence of certain families of curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In the paper we discuss 7-dimensional orbits in \(\mathbb{C}^4\) of two families of nilpotent 7-dimensional Lie algebras; this is motivated by the problem on describing holomorphically homogeneous real hypersurfaces. Similar to nilpotent 5-dimensional algebras of holomorphic vector fields in \(\mathbb{C}^3 \), the most part of algebras considered in the paper has no Levi non-degenerate orbits. In particular, we prove the absence of such orbits for a family of decomposable 7-dimensional nilpotent Lie algebra (31 algebra). At the same time, in the family of 12 non-decomposable 7-dimensional nilpotent Lie algebras, each containing at least three Abelian 4-dimensional ideals, four algebras has non-degenerate orbits. The hypersurfaces of two of these algebras are equivalent to quadrics, while non-spherical non-degenerate orbits of other two algebras are holomorphically non-equivalent generalization for the case of 4-dimensional complex space of a known Winkelmann surface in the space \(\mathbb{C}^3\). All orbits of the algebras in the second family admit tubular realizations. homogeneous manifold; holomorphic function; vector field; Lie algebra; abelian ideal Almost homogeneous manifolds and spaces, Holomorphic functions of several complex variables, Lie algebras of vector fields and related (super) algebras, Families, moduli of curves (algebraic), Ideals and multiplicative ideal theory in commutative rings On degeneracy of orbits of nilpotent Lie algebras | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0676.00006.]
Let F be an algebraically closed field of characteristic 0, G a reductive linear algebraic group defined over F, V a finite dimensional representation over F which has an element with trivial G-stabilizer, F[V] the symmetric algebra on V and F(V) its field of fractions. Let \(F(V)^ G\) be the field of G-invariant elements of F(V). According to Bogomolov's result \(F(V)^ G\) is up to stable isomorphism independent of the choice of V. This allows to talk about the invariant field of G without specifying V, denote it simply by F(G). In the paper under review some connections between invariant fields of reductive linear groups and division algebras are discussed.
The main result of the paper is the following Theorem 1. Suppose \(\phi\) : G\({}'\to G\) is a surjective homomorphism of linear connected reductive algebraic groups with central kernel cyclic of order n. Then \(F(G')\) is stably isomorphic to F(G)(A) for some central simple algebra \(A| F(G)\) of exponent n, where F(G)(A) is the function field of the Brauer- Severi variety defined by A. Furthermore some results are presented for some specific groups: \(SL_ n\), Spin groups of odd degree and \(SO_ n\). It should be noted the importance of the result about \(SL_ n\) and its images. Theorem 2. Let \(D=UD(F,n,2)\) be the generic division algebra over F of degree n in two variables, Z the center of D and A a division algebra in the class r[D]\(\in Br(Z)\). Set \(G_ r\) to be \(SL_ n(F)/C_ r\) where \(C_ r\) is the central cyclic subgroup of order r, let V be a \(G_ r\)-representation over F with an element with trivial stabilizer and \(F(G_ r)\) the invariant field of \(G_ r\) on F(V). Then \(F(G_ r)\) is stably isomorphic to Z(A), where Z(A) is the function field of the Brauer-Severy variety defined by A. finite dimensional representation; symmetric algebra; stable isomorphism; invariant fields; reductive linear groups; division algebras; function field; Brauer-Severi variety David J. Saltman, Invariant fields of linear groups and division algebras, Perspectives in ring theory (Antwerp, 1987) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 233, Kluwer Acad. Publ., Dordrecht, 1988, pp. 279 -- 297. Division rings and semisimple Artin rings, Other matrix groups over rings, Vector and tensor algebra, theory of invariants, Representation theory for linear algebraic groups, Brauer groups of schemes, Galois cohomology, Rational and unirational varieties Invariant fields of linear groups and division algebras | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A prehomogeneous vector space \((G,\rho,V)\) consists of a connected linear algebraic group \(G\) defined over a field \(K\) of characteristic 0, \(V\) a finite dimensional \(\overline K\)-vector space with \(K\)-structure \(V_ K\) and a \(K\)-rational representation of \(G\) on \(V\) such that there exists a proper algebraic subset \(S\) of \(V\) such that \(V_{\overline K}\backslash S_{\overline K}\) is a single \(\rho(G_{\overline K})\)-orbit. For every prehomogeneous vector space (p.v.) \((G,\rho,V)\) one defines its ``castling transform'' \((\tilde G,\tilde\rho,\tilde V)\). Sato and Kimura have shown that \((G,\rho,V)\) is a p.v. iff \((\tilde G,\tilde\rho,\tilde V)\) is a p.v. Similarly, given any mathematical object associated with a p.v. \((G,\rho,V)\) and having certain properties, one is interested in determining properties the corresponding object associated with the castling transform \((\tilde G,\tilde\rho,\tilde V)\) has. The paper under review considers zeta functions associated with \((G,\rho,V)\) and \((\tilde G,\tilde\rho,\tilde V)\) for a local field \(K\) of characteristic 0. It is shown (lemma 2.2) that these zeta-functions satisfy certain functional equations involving meromorphic coefficient functions \(\Gamma^ \mu_{i,i^*}\) and \(\tilde\Gamma^ \mu_{i,i^*}\). The main result of this paper (theorem 3) gives a complete and simplified proof of Shintani's formula relating the \(\Gamma^ \mu_{i,i^*}\) and \(\tilde\Gamma^ \mu_{i,i^*}\). --- As a corollary a relation between the corresponding \(b\)-functions is obtained. The paper contains a thorough discussion of the prerequisites entering Shintani's formula as well as a complete proof of this formula. gamma function; prehomogeneous vector space; castling transform; zeta functions; Shintani's formula; \(b\)-functions Sato, F.; Ochiai, H., Castling transforms of prehomogeneous vector spaces and functional equations, Comment. math. univ. st. Pauli, 40, 61-82, (1991) Homogeneous spaces and generalizations, Gamma, beta and polygamma functions, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Castling transforms of prehomogeneous vector spaces and functional equations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For a wildly ramified p-extension E/F of algebraic function fields of one variable in an algebraically closed field k of characteristic p with Galois group G, Nakajima obtained two exact sequences which determined implicitly the structure of the holomorphic semisimple differentials as k[G]-module. In this paper, in many cases, e.g., if there is a fully ramified prime, the structure is determined explicitly.
Analogous results are obtained for p-extensions of \({\mathbb{Z}}_ p\)-fields of CM-type. In the latter situation, if E/F is unramified, the structure of the minus part of the p-class group of E is determined as \({\mathbb{Z}}_ p[G]\)-module. algebraic function fields; holomorphic semisimple differentials; p- extensions of \({\mathbb{Z}}_ pfields\) of CM-type; p-class group G. Villa and M. Madan,Structure of semisimple differentials and p-class groups in \(\mathbb{Z}\) p -extensions. Manuscripta Mathematica57 (1987), 315--350. Cyclotomic extensions, Arithmetic theory of algebraic function fields, Iwasawa theory, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Algebraic functions and function fields in algebraic geometry Structure of semisimple differentials and p-class groups in \({\mathbb{Z}}_ p\)-extensions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(R\) be a complete discrete valuation ring with quotient field of characteristic zero and algebraically closed residue field \(k\) of characteristic \(p>0\). Let \(\phi: \mathbb{R} \to \mathbb{R}\) be the unique lifting of the Frobenius of \(k\), and define \(\delta: \mathbb{R} \to \mathbb{R}\) by \(\delta x =\frac{\phi (x) - x^p}{p}\). Let \(G/R\) be a smooth group scheme of finite type. The author defines a \(\delta\)-formal function of order \(\leq n\) on \(G(R)\) to be an \(R\) valued function which can locally (in the Zariski topology of \(G\)) be expressed as a polynomial in affine coordinate functions and (\(n\) or fewer) iterates of \(\delta\). These form a ring \(O^n(G)\). For each \(n\), these rings have comultiplications, coinverses, and counits coming from the group scheme structure of \(G\). A coherent family \(\mathcal J\) of ideals \(J_n \subset O^n(G)\) respecting these cooperations determines a subgroup of \(G(R): \mathcal J^{\text{sol}} = \{x \in G(R) \mid f(x)=0\) for all \(f \in J_n\), \(n \geq 0\}\).
The author calls such subgroups \(\delta\)-subgroups. The author calls a subgroup \(\Gamma \subset G(R)\) Zariski dense modulo \(p\) if the image of \(\Gamma\) in the closed fibre \(G_0(k)\) of \(G/R\) is Zariski dense. The main result of the paper is the following theorem:
Let \(G/R\) be such that \(G_0/k\) is a simple algebraic group such that the adjoint representation of \(G_0\) on \(\text{Lie}(G_0)\) is irreducible. Let \(\Gamma\) be a \(\delta\)-subgroup which is Zariski dense modulo \(p\). Then \(\Gamma = G(R)\).
The proof depends heavily on the author's previous work on \(p\) formal groups and arithmetic jet theory. complete discrete valuation ring; lifting of the Frobenius; characteristic \(p\); smooth group scheme; \(p\) formal groups; arithmetic jet theory Buium, A.: Differential subgroups of simple algebraic groups over p-adic fields. Amer. J. Math. 120, 1277-1287 (1998) Differential algebra, Linear algebraic groups over local fields and their integers, Derivations, actions of Lie algebras, Group schemes, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure Differential subgroups of simple algebraic groups over \(p\)-adic fields | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems See the joint review of Parts I and II [Nova Acta Soc. Sci. Upsal., IV. Ser. 14, No. 3, 40 S. (1947)] in Zbl 0029.24902. function fields; plane curves of genus one; exceptional points Nagell, T.: [3] ''Les points exceptionnels sur les cubiques planes du premier genre'', II, ibid. Nova Acta Reg. Soc. Sci. Upsaliensis, Ser. IV, 14, 1946, No. 3. Algebraic functions and function fields in algebraic geometry, Special algebraic curves and curves of low genus Les points exceptionnels sur les cubiques planes du premier genre | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(q = p^e\), where \(p > 0\) is the characteristic of the finite field \({\mathbb F}_q\). The author obtains equidistribution results on \(p^n\)-torsion of group schemes of abelian varieties of genus \(g\) over \({\mathbb F}_q\) in large \(q\)-approach.
In a previous paper [Contemp. Math. 463, 1--7 (2008; Zbl 1166.11018)], the author treated the case of hyperelliptic curves \(C_g\) of fixed genus \(g\) over a finite field, that is, finite subgroups \(\mathrm{Jac}(C_g)[N]\) of the Jacobeans of \(C_g\), and the results obtained there have been found to be deeply connected with results of Cohen-Lenstra type for quadratic function fields [\textit{H. Cohen} and \textit{H. W. Lenstra jun.}, Lect. Notes Math. 1068, 33--62 (1984; Zbl 0558.12002)].
In earlier article \textit{B. Cais} et al. [J. Inst. Math. Jussieu 12, No. 3, 651--676 (2013; Zbl 1284.14055)] fixed a base field \({\mathbb F}_q\) and describe a probability distribution on isomorphism classes of principally quasipolarized \(p\)-divisible groups over the field.
The author is primarily interested in characterizing the automorphism group schemes of principally quasipolarized Barsotti-Tate groups \(BT_n\) of level \(n\). Let \(N \geq 3\) be an integer relatively prime to \(p\), let \({\mathcal A}_g\) be the moduli space, over \({\mathbb F}_q\), of \(g\)-dimensional abelian varieties equipped with degree-1 polarization and symplectic level-\(N\) structure. Let \({\mathcal X} \to {\mathcal A}_g\) be the universal principally polarized abelian scheme. The flavor of the results is suggested by the following extraction from the authors main theorem.
Let \(\xi\) be a geometric isomorphism class of principally quasipolarized \(BT_n\); let \({\mathcal A}_{g,\xi}\) be the locus where \({\mathcal X}[p^n]\) is geometrically isomorphic to \(\xi\). We will show that there is a finite group \(A(\xi)\) whose conjugacy classes \(A(\xi)^{\natural}\) parametrize, for suitable fields \({\mathbb F}_q\), \({\mathbb F}_q\)-rational forms of \(\xi\). We essentially show (see Theorem 4.5 for details) that, for each \(\alpha \in A(\xi)^{\natural}\), the proportion of elements of \((X, \lambda) \in {\mathcal A}_{g,\xi}({\mathbb F}_q)\) for which \((X, \lambda)[p^n] \cong {\xi }^{\alpha}\) approaches \(1/{\natural}Z_{ A(\xi)}( \alpha)\) in a surprisingly uniform way. abelian variety; group scheme; finite field; arithmetic statistics Positive characteristic ground fields in algebraic geometry, Algebraic moduli of abelian varieties, classification, Formal groups, \(p\)-divisible groups, Group schemes The distribution of torsion subschemes of abelian varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We study several integrable Hamiltonian systems on the moduli spaces of meromorphic functions on Riemann surfaces (the Riemann sphere, a cylinder and a torus). The action-angle variables and the separated variables (in Sklyanin's sense) are related via a canonical transformation, the generating function of which is the Abelian type integral of the Seiberg-Witten differential over the spectral curve. Integrable systems; action-angle variables; separation of variables; moduli space Takasaki K., J. Geom. Phys. 47 pp 1-- (2003) Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Relationships between algebraic curves and integrable systems An integrable system on the moduli space of rational functions and its variants. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(R\) be a commutative ring with identity and \(M\) be a unitary \(R\)-module. In this paper, we obtain a scheme \((\mathcal{X}(M),\mathbb{O}_{\mathcal{X}(M)})\) over the primary-like spectrum \(\mathcal{X}(M)\) of \(M\) and prove that \((\mathcal{X}(M),\mathbb{O}_{\mathcal{X}(M)})\) is a Noetherian scheme when \(R\) is a Noetherian ring. Zariski topology; sheaf of rings; scheme Schemes and morphisms, Theory of modules and ideals in commutative rings Primary-like submodules and a scheme over the primary-like spectrum of modules | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper studies and proves uniqueness of the Deligne-Lusztig curves \(X\) and corresponding function fields constructed via the Ree groups over \(\mathbb{F}_ q\), where \(q\) is an odd power of 3.
The curves are particular in two senses. First of all, the number of \(\mathbb{F}_ q\)-rational points on \(X\), being \(1+q^ 3\), is the largest a curve of genus \(g=g(X)={3\over 2}q_ 0(q-1)(q+q_ 0+1)\), \((q_ 0=3^ s)\) can have. Secondly, they have very large groups of automorphisms compared to \(g(X)\), their sizes \(q^ 3(q-1)(q^ 3+1)\) exceed by far the Hurwitz upper bound \(84(g-1)\) valid in characteristic zero.
The curves and corresponding function fields are of interest in the theory of error correcting codes via the Goppa construction. It is possible to construct codes of length \(q^ 3\) over \(\mathbb{F}_ q\), such that dimension+minimum distance \(\geq 1+q^ 3-g\). The large groups of automorphisms equip the resulting geometric Goppa codes with correspondingly large symmetry. number of rational points; Deligne-Lusztig curves; function fields; large groups of automorphisms; Goppa codes HP Johan~P. Hansen and Jens~Peter Pedersen, \emph Automorphism groups of Ree type, Deligne-Lusztig curves and function fields, J. Reine Angew. Math. \textbf 440 (1993), 99--109. Algebraic functions and function fields in algebraic geometry, Geometric methods (including applications of algebraic geometry) applied to coding theory, Arithmetic ground fields for curves, Curves over finite and local fields, Finite ground fields in algebraic geometry, Arithmetic theory of algebraic function fields Automorphism groups of Ree type, Deligne-Lusztig curves and function fields | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(C\) be a space curve of degree \(d\), not lying on a surface of degree \(s-1\), and of maximal genus \(G(d,s)\). For \(d\geq(s-1)^ 2+1\), \(G(d,s)\) has been determined by \textit{L. Gruson} and \textit{C. Peskine} [in Algebraic Geometry, Proc., Tromsø Symp. 1977, Lect. Notes Math. 687, 31-59 (1978; Zbl 0412.14011)]. They also describe the curves \(C\).
In this paper, the author describes the curves \(C\) for \(d=(s-1)^ 2-r\), \(0\leq r\leq s-4\), and the family of these curves in the Hilbert scheme. The value of \(G(d,s)\) in this range has been computed by \textit{Ph. Ellia} [J. Reine Angew. Math. 413, 78-87 (1991; Zbl 0711.14015)]. linkage; deficiency module; space curve; maximal genus; Hilbert scheme Plane and space curves, Linkage, Parametrization (Chow and Hilbert schemes) On certain curves of maximal genus in \(\mathbb{P}{}^ 3\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0547.00033.]
The aim of this paper is the use of characteristic series and a Godbillon-Vey type algorithm to solve problems concerning analytic dynamical systems, related to limit cycles. characteristic series; Godbillon-Vey type algorithm; analytic dynamical systems; limit cycles Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems, Cycles and subschemes, Local analytic geometry Cycles limites, étude locale | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We consider the Schwarz maps with monodromy groups isomorphic to the triangle groups \((2,4,4)\) and \((2,3,6)\) and their inverses. We apply our formulas to studies of mean iterations. Schwarz map; theta function; mean iteration Classical hypergeometric functions, \({}_2F_1\), Theta functions and abelian varieties Schwarz maps associated with the triangle groups \((2,4,4)\) and \((2,3,6)\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems \textit{F. Severi} claimed in the 1920s that the Hilbert scheme \({\mathcal H}_{d,g,r}\) of smooth irreducible non-degenerate curves \(C \subset \mathbb P^r\) of degree \(d\) and genus \(g\) is irreducible for \(d \geq g+r\) [Vorlesungen über algebraische Geometrie. Übersetzung von E. Löffler. Leipzig u. Berlin: B. G. Teubner (1921; JFM 48.0687.01)]. \textit{L. Ein} proved Severi's claim for \(r=3\) and \(r=4\) [Ann. Sci. Éc. Norm. Supér. (4) 19, No. 4, 469--478 (1986; Zbl 0606.14003); Proc. Symp. Pure Math. 46, 83--87 (1987; Zbl 0647.14012)], but there are counterexamples of various authors for \(r \geq 6\).
It has been suggested by \textit{C. Ciliberto} and \textit{E. Sernesi} [in: Proceedings of the first college on Riemann surfaces held in Trieste, Italy, November 9-December 18, 1987. Teaneck, NJ: World Scientific Publishing Co. 428--499 (1989; Zbl 0800.14002)] that Severi intended irreducibiity of the Hilbert scheme \({\mathcal H}^{\mathcal L}_{d,g,r} \subset {\mathcal H}_{d,g,r}\) of curves whose general member is linearly normal: indeed, the counterexamples above arise from families whose general member is not linearly normal.
Here the authors prove irreducibility for \(g+r-2 \leq d \leq g+r\) (the Hilbert scheme is empty for \(d > g+r\) by Riemann-Roch) and for \(d=g+r-3\) under the additional assumption that \(g \geq 2r+3\). This extends work of \textit{C. Keem} and \textit{Y.-H. Kim} [Arch. Math. 113, No. 4, 373--384 (2019; Zbl 1423.14028)]. Hilbert scheme; linear series; linearly normal curves Parametrization (Chow and Hilbert schemes), Families, moduli of curves (algebraic) On the Hilbert scheme of linearly normal curves in \(\mathbb{P}^r\) of relatively high degree | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper considers the algebraic function \(\hat{F}(z):=(-1\pm\sqrt{1-z^2})/z^2\), and the exact set of periodic points of \(\hat{F}(z)\) in \(\bar{\mathbb{Q}}\). First set up some relevant notations. Let \(K={\mathbb{Q}}(\sqrt{-d}),\) where \(-d\equiv 1\pmod 8\) of odd conductor \(f\). Let \(d_K\) be the discriminant of \(K\) and set \(-d=d_Kf^2\). Let \(\Omega_f\) be the ring class field of \(K\).
Theorem 1. The exact set of periodic points of \(\hat{F}(z)\) in \(\bar{\mathbb{Q}}\) consists of the coordinates of certain solutions \((x,y)=(\pi,\xi)\) of the Fermat curve \(x^4+y^4=1\) in ring class fields \(\Omega_f\) of imaginary quadratic fields \(K\).
This result is obtained by showing that the \(2\)-adic function \(F(z)=(-1+\sqrt{1-z^4})/z^2\) is the lift of the Frobenius automorphism on the coordinate \(\pi\) for which \(|\pi|_2<1\), for any \(d\equiv 7\pmod 8\), considered as elements of the maximal unramified extension \(K_2\) of \({\mathbb{Q}}_2\). More precisely, the periodic points of \(F(z)\) in the disc \(|\pi|_2<1\) are \(z=0\) and roots \(\pi\) of the normal polynomial \(b_d(x)\) (the minimal polynomial of solutions \((\pi_d,\xi_d)\) of the Fermat equation in \(\Omega_f\)) of degree \(2h(-d),\) where \(h(-d)\) is the class number of the order \(R_{-d}\) of discrminant \(-d\) in \(K\).
As a corollary, this result gives an interpretation of Deuring's class number formulas in the case of \(p=2\).
Corollary. With \(K={\mathbb{Q}}(\sqrt{-d})\), \(-d=d_kf^2\equiv 1\pmod 8\) with \(f\) odd, and \(2=\mathfrak{p}_2\bar{\mathfrak{p}}_2\). Let \(\mathcal{D}_n\) be the set of negative discriminants \(-d\equiv 1\pmod 8\) for which the Frobenius automorphism \(\tau\) has order \(n\) in the Galois group \(\mathrm{Gal}(\Omega_f/K)\). Then for any \(n>1\), there is the class number formula \[\sum_{-d\in{\mathcal{D}}_n} h(-d)=\sum_{k|n} \mu(n/k)2^k.\] periodic points; algebraic function; class number formula; modular function; ring class fields Algebraic functions and function fields in algebraic geometry, Higher degree equations; Fermat's equation, Elliptic curves over local fields, Complex multiplication and moduli of abelian varieties Periodic points of algebraic functions and Deuring's class number formula | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(K_0(\mathcal{V}_k)\) be the Grothendieck group of \(k\)-varieties. The first and third author [``Devissage and localization for the Grothendieck spectrum of varieties'', Preprint, \url{arXiv:1811.08014}] have constructed a higher algebraic \(K\)-theory spectrum \(K(\mathcal{V}_k)\) such that \(\pi_0 K(\mathcal{V}_k) = K_0(\mathcal{V}_k)\). In this paper we construct non-trivial classes in the higher homotopy groups of \(K(\mathcal{V}_k)\) when \(k\) is finite or a subfield of C. To do this we give a recipe for lifting motivic measures \(K_0(\mathcal{V}_k) \rightarrow K_0(\mathcal{E})\) to maps of spectra \(K(\mathcal{V}_k) \rightarrow K(\mathcal{E})\). We consider two special cases: the classical local zeta function, thought of as a homomorphism \(K_0(\mathcal{V}_{\mathbb{F}_q}) \rightarrow K_0(\mathrm{End}(\mathbb{Q}_\ell))\), and the compactly-supported Euler characteristic, thought of as a homomorphism \(K_0(\mathcal{V}_{\mathbb{C}}) \rightarrow K_0(\mathbb{Q})\). We use lifts of these motivic measures to prove that the Grothendieck spectrum of varieties contains nontrivial geometric information in its higher homotopy groups by showing that the map \(\mathbb{S} \rightarrow K(\mathcal{V}_k)\) is nontrivial in higher dimensions when \(k\) is finite or a subfield of C, and, moreover, that when \(k\) is finite this map is not surjective on higher homotopy groups. \(K\)-theory; zeta function; \(K\)-theory of varieties; Grothendieck group of varieties; motivic measure Applications of methods of algebraic \(K\)-theory in algebraic geometry, Zeta functions and \(L\)-functions, Motivic cohomology; motivic homotopy theory, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects), Grothendieck groups (category-theoretic aspects), Stable homotopy theory, spectra Derived \(\ell\)-adic zeta functions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This article shows the geometry behind the well-known reduction formula by Barnes and Bailey [\textit{W. N. Bailey}, Q. J. Math., Oxf. II. Ser. 10, 236--240 (1959; Zbl 0087.28204)] that implies the factorization of Appell's generalized hypergeometric series into a product of two Gauss' hypergeometric functions.
The main results are formulated in the following theorem
Theorem. (a) Every Appell hypergeometric function with rational parameters satisfying the quadric property can be obtained as period integral of a suitable holomorphic two-form on the generalized Kummer variety of two superelliptic curves.
(b) The Multivariate Clausen Identity of Barnes and Baily follows entirely from geometry, that is, from the existence of an isotrivial fibration and a second non-isotrivial fibration on the generalized Kummer variety of two superelliptic curves.
The authors first construct a surface of general type as minimal nonsingular model of a product-quotient surface with only rational double points from a pair of superelliptic curves of genus \(2r-1\) with \(r\in{\mathbb{N}}\). Then they show that this generalized Kummer variety is equipped with two fibrations with general fiber of genus \(2r-1\). Next, the authors prove that the Multivariate Clausen Identity realizes the equality of periods of a holomorphic two-form evaluated over a suitable two-cycle using the structure of either of the two constructed fibrations. Kummer surfaces; product-quotient surfaces; special function identities; hypergeometric functions; fibrations; superelliptic curves Variation of Hodge structures (algebro-geometric aspects), Fibrations, degenerations in algebraic geometry, \(K3\) surfaces and Enriques surfaces, Surfaces of general type, Appell, Horn and Lauricella functions Special function identities from superelliptic Kummer varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0528.00004.]
This is mainly an expository article of previous results by Ph. Griffiths and the author. Poincaré had introduced the notion of normal function associated to a ''sufficiently general'' curve C of an algebraic surface. An important fact was that viceversa given a normal function the homology class of C can be reconstructed. Griffiths has generalized Poincaré methods to an algebraic manifold X of any dimension, getting for every k a group \(N^ k(X)\) of ''normal functions'' and an associated group of integral cohomology classes of degree 2k, clearly related to the group of algebraic cycles of X of codimension k. Precisely one of the main results of this theory [cf. \textit{P. A. Griffiths}, Am. J. Math. 101, 94-131 (1979; Zbl 0453.14001), and the author, Invent. Math. 33, 185-222 (1976; Zbl 0329.14008) and Ann. Math., II. Ser. 109, 415-476 (1979; Zbl 0446.14002)] is the following: consider the normal function associated to a smooth projective morphism \(f: X\to S\) where S is a non singular algebraic curve and the ''analogous'' notion for the completions: \(\bar f:\) \(\bar X\to \bar S\). There is a subgroup of \(N^ k(\bar X)\) (defined by a differential equation), the group of horizontal normal functions, whose cohomology classes are exactly those of type (k,k) in \(H^{2k}(\bar X,{\mathbb{Z}})\). The precise statement requires several definitions. Relations with the Hodge conjecture are obvious. group of algebraic cycles of codimension k; intermediate Jacobians; normal function; horizontal normal functions; Hodge conjecture S. Zucker, Intermediate Jacobians and normal functions , Topics in Transcendental Algebraic Geometry (Princeton, N.J., 1981/1982), Ann. of Math. Stud., vol. 106, Princeton University Press, Princeton, NJ, 1984, pp. 259-267. Transcendental methods, Hodge theory (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Transcendental methods of algebraic geometry (complex-analytic aspects), Picard schemes, higher Jacobians Intermediate Jacobians and normal functions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The following four concepts on a commutative ring with unity A are shown to be equivalent: (1) a flat surjection, i.e., a ring homomorphism \(A\to A/I\) where A/I is flat as an A-module, (2) a closed subset \(Y\subseteq X=Spec(A)\) such that the sheaf induced on Y from the structure sheaf \(\tilde A\) on X yields an affine scheme, (3) a torsion class of type I BSP [cf. the author and \textit{T. S. Shores}, Commun. Algebra 9, 1161-1214 (1981; Zbl 0574.13009)] and (4) an exact torsion functor on the category of A-modules. bounded splitting property; affine scheme; flat surjection; BSP; torsion functor Torsion theory for commutative rings, Local structure of morphisms in algebraic geometry: étale, flat, etc., Injective and flat modules and ideals in commutative rings Four algebraic spectral properties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This is a very useful survey of results from the authors book [\textit{A. Neeman}, ``Triangulated categories'', Am. Math. Stud. 148, Princeton University Press (2001; Zbl 0974.18008)]. It starts with a presentation of the main classical results on compactly generated triangulated categories, and two applications of the theory are given: to the unbounded derived categories of quasi-coherent sheaves on a quasi-projective noetherian scheme, and to the stable category of a group algebra. Then the author presents the motivation for the introduction of well generated triangulated categories, which are regarded as a large cardinal generalization of the compactly geenrated ones. At the end of the survey, the dual of Brown representability is discussed, and some recent progress on this open problem is presented. scheme; derived category; localization; \(K\)-theory Neeman, A.: A survey of well generated triangulated categories. In: Representations of Algebras and Related Topics. Fields Inst. Commun., vol. 45, pp. 307--329. Am. Math. Soc., Providence (2005). MR2146659 (2006d:18004) Derived categories, triangulated categories, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], \(K\)-theory of schemes A survey of well generated triangulated categories | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors study the elliptic-curve single-scalar multiplication over finite fields, i.e. given a finite field \(k\) (the ground field), an elliptic curve \(E\) (with small parameters), an integer \(n\) (the scalar) and a point \(P\in E(k)\), they identify today's fastest methods to compute the point \(nP\) on \(E\). Due to the well-known cryptographic applications, tables and figures are given for \(160\), \(256\) and \(512\)-bit scalars, expressing the necessary number of multiplications per bit as a function of the \textbf{I}/\textbf{M} ratio, i.e. the number of multiplications needed to provide an inversion in the ground field.
In order to do this, the authors first consider the problem of adding two points (or doubling a point). They look at twelve different coordinate systems: Projective, Jacobian (these two systems are also considered in the particular, faster, case \(a_4=-3\)), Doubling (\textit{resp.} Tripling)-oriented Doche/Icart/Kohel, Montgomery, Jacobi intersections, Jacobi quartics, Hessian, Edwards and inverted Edwards. For each system, the relation with the classical Weierstrass model and the corresponding (affine) coordinates is given. Note that certain systems do not provide a model for every elliptic curve.
Since the precomputation of some little multiples \(2P,3P,5P,7P,\cdots,mP\) is necessary, they look for the optimal odd \(m\) (always less than \(31\)), constructing for each \(m,n\) an ``addition-subtraction'' chain, that allows a fast computation of \(nP\). In order to do this, they combine ``windows'' techniques, and average over many random scalars of given size to get the best choice.
Finally, the authors consider four cases, allowing zero, one, two or three inversions in the ground field (typically, an inversion is needed when one wants to give the affine coordinates of a point). Then they compare their respective performances when the ratio \textbf{I}/\textbf{M} varies.
All the results are summarized in three tables, and the fastest ones in three figures, giving a clear account of what is known about this problem nowadays. Note also that all these results are updated at the address \url{http://hyperelliptic.org/EFD}. elliptic curves; fast addition; scalar multiplication; coordinate systems Daniel J. Bernstein and Tanja Lange, Analysis and optimization of elliptic-curve single-scalar multiplication, Finite fields and applications, Contemp. Math., vol. 461, Amer. Math. Soc., Providence, RI, 2008, pp. 1 -- 19. Cryptography, Finite ground fields in algebraic geometry, Computational aspects of algebraic curves, Elliptic curves Analysis and optimization of elliptic-curve single-scalar multiplication | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems \textit{C. Rosseau} [Lect. Notes Math. 753, 623-659 (1979; Zbl 0433.32003)] has shown that classical or standard function theory of n variables is no other than intuitionistic function theory of one variable over \(C^{n- 1}\). Similar works have been done by the author [Publ. Res. Inst. Math. Sci. 22, 801-811 (1986; Zbl 0614.03058)] in the realm of Sato hyperfunctions and by \textit{G. Takeuti} and \textit{S. Titani} [Ann. Pure Appl. Logic 31, 307-339 (1986; Zbl 0615.03048)] in the realm of complex manifolds. The main purpose of this paper is to pursue similar results in the arena of algebraic geometry. Since we would like to do so in an intuitionistically valid manner, we reconstruct some rudiments of algebraic geometry, using the complete Heyting algebra of radical ideals in place of the space of prime ideals with Zariski topology as the starting point of our scheme theory. Heyting valued set theory; algebraic geometry; complete Heyting algebra of radical ideals; scheme theory Nonclassical and second-order set theories, Intuitionistic mathematics, Foundations of algebraic geometry Some connections between Heyting valued set theory and algebraic geometry. Prolegomena to intuitionistic algebraic geometry | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(S\) be a connected Dedekind scheme, \(X\) a connected scheme and \(f:X \to S\) a faithfully flat morphism of finite type with a section. The author proves that \(X\) has a fundamental group scheme. This is defined to be the \(S\)-group scheme in the initial object for the category \(\mathcal{P}(X)\) whose objects are triples \((Y,G,y)\) where \(Y \to X\) is an fpqc-torsor over a finite and flat \(S\)-group scheme \(G\) and \(y\) is an \(S\)-valued point of \(Y\). He also proves that for an \textit{affine} scheme \(X\), the fundamental group of \(X_{\text{red}}\) is a closed subgroup of the fundamental group of \(X\). These results generalize former results of the author, and also of \textit{M. A. Garuti} [Proc. Am. Math. Soc. 137, No. 11, 3575--3583 (2009; Zbl 1181.14053)]. torsor; fundamental group scheme Antei M.: The fundamental group scheme of a non reduced scheme. Bull. Sci. Math. 135(5), 531--539 (2011) Group schemes, Positive characteristic ground fields in algebraic geometry The fundamental group scheme of a non-reduced scheme | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Given \(c \in \mathbb Q\) and a positive integer \(N\), let \(f_c\) be the endomorphism of the affine line \(\mathbb A^1_{\mathbb Q}\) defined by \(f_c (x) = x^2 +c\), and let \(f^N_c\) be the \(N\)-fold composition of \(f_c\). If \(f^{-N}_c\) denotes the \(N\)-fold preimage, the set of rational preimage of \(a \in \mathbb A^1 (\mathbb Q)\) is given by
\[
\bigcup_{N \geq 1} f^{-N}_c (a) (\mathbb Q) = \{ x_0 \in \mathbb A^1 (\mathbb Q) \mid f^N_c (x_0) = a \; \text{ for some } \; a \in \mathbb A^1 (\mathbb Q) \} .
\]
This paper is concerned with the problem of bounding the number of rational points that eventually landing at the origin after iteration. Subject to the validity of the Birch-Swinnerton-Dyer conjecture and some other related conjectures for the \(L\)-series of a special abelian variety and using a number of modern tools for locating rational points on higher genus curves, the authors prove that the maximum number of rational iterated preimages is six. They also provide further insight into the geometry of the preimage curves. quadratic dynamical systems; arithmetic geometry; rational points Faber, Xander; Hutz, Benjamin: On the number of rational iterated pre-images of the origin under quadratic dynamical systems, (2008) Rational points, Arithmetic aspects of modular and Shimura varieties, Computer solution of Diophantine equations, Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets On the number of rational iterated preimages of the origin under quadratic dynamical systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In the paper under review, the authors give several optimal inequalities of Szpiro's type for curves of genus \(g\geq 2\) defined over a function field.
Precisely, let \(K\) be the function field of a non-singular complete curve \(C\) of genus \(b\) defined over an algebraically closed field. For a smooth algebraic curve \(Y\) of genus \(g\geq 2\) defined over \(K\), the authors considered the associated relatively minimal fibration \(f: X\to C\) of genus \(g\) whose generic fiber is \(Y/K\), where \(X\) is a smooth surface and any fiber contains no \((-1)\)-curves. When \(f\) is semistable and non-trivial, Szpiro's inequalities bound the number \(N\) of the singular points of the fibers and the degree of the discriminant \(\mathcal{D}_{Y/K}\).
Denote by \(F_1,\dots,F_s\) the singular fibers of \(f\), and let \(s_1\) be the smallest number of \(F_i\) whose Jacobian \(J(F_i)\) is non-compact. The authors present three improvements of Szpiro's inequalities. The first states that if \(\text{char} k=0\), then
\[
N<(4g+2)(1-\frac{g_f}{g})(2b-2+s_1)
\]
where \(q_f=h^{0,1}(X)-b\) is the relative irregularity of \(f\). If \(q_f\geq1\), then
\[
N\leq 4(g-q_f)(2b-2+s_1).
\]
The second states that if \(\text{char} k=p>0\), then
\[
N<2g(4g+2)(2b-2+s)
\]
which is independent of \(p\). The third one states that if \(f: X\to C\) is a non-trivial hyperelliptic fibration over an algebraically closed field \(k\) of characteristic \(0\), let \(K=k(C)\) and let \(Y=X\times_C\mathrm{Spec}(K)\) be the generic fiber of \(f\). Then
\[
\deg\mathcal{D}_{Y/K}\leq\frac{4g+2}{g}((g-q_f)(2b-2)+\deg\mathcal{N}_{Y/K}),
\]
where \(\mathcal{N}_{Y/K}\) is the conductor divisor of \(Y/K\). Szpiro inequality; curve over function field; slope inequality; Arakelov inequality; discriminant Tan, S-L; Xu, W-Y, On Szpiro inequality for semistable families of curves, J. Number Theory, 151, 36-45, (2015) Families, moduli of curves (algebraic), Arithmetic varieties and schemes; Arakelov theory; heights On Szpiro inequality for semistable families of curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems By a scheme of degree m is meant a scheme of relative position of ovals in the projective plane of a real curve of degree m. A scheme of degree m is of type I if any curve of degree m with the given scheme divides its complexification and is of type II if all curves of degree m with the given scheme do not divide their complexifications. A scheme is called maximal if it is not part of a larger scheme of the same degree. \textit{V. A. Rokhlin} [Usp. Mat. Nauk 33, No.5(203), 77-89 (1978; Zbl 0437.14013)] made the conjecture that a maximal scheme is of type I. This assertion is valid for schemes of degree \(\leq 7\). In the paper under review the author proves that there exists a maximal scheme of degree 8, which is of type II. So the Rokhlin conjecture is disproved. projective real curve; scheme of degree m; position of ovals; complexification Projective techniques in algebraic geometry, Special algebraic curves and curves of low genus, Real algebraic and real-analytic geometry Counterexamples to a conjecture of Rokhlin | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0722.00006.]
The Hartshorne-Rao module of a curve \(C\subset\mathbb{P}^ 3_ k\) is defined as \(M_ C=\sum_{n\in\mathbb{Z}}H^ 1({\mathcal I}_ C(n)))\), where \({\mathcal I}_ C\) denotes the ideal sheaf of \(C\). Let \(C_ i\), \(i=1,2\), be two curves. Is there a construction of a curve \(C\) (in terms of \(C_ i)\) such that \(M_ C\) is isomorphic to some shift of \(M_{C_ 1}\oplus M_{C_ 2}\)? This is solved by \textit{P. Schwartau} in his unpublished thesis ``Liaison addition and monomial ideals'' [Ph. D. thesis, Brandeis University 1982), see also \textit{J. Stückrad} and \textit{W. Vogel} [``Buchsbaum rings and applications. An interaction between algebra, geometry, and topology'', VEB Deutscher Verlag der Wissenschaft (Berlin 1986; Zbl 0606.13017); published simultaneous by Springer Verlag)] for a reproduction of his arguments.
In this paper the authors generalize Schwartau's result in several senses: (1) There is a liaison addition in any codimension in \(\mathbb{P}^ n_ k\), \(n\geq 3\), of schemes of mixed codimensions. --- (2) There is an addition of any number of schemes. --- (3) There is no need that the schemes are attached via a complete intersection (as in Schwartau's result).
The basic idea is the construction of a scheme \(Z\) consisting set- theoretically of the union of certain given schemes such that the cohomology of \(Z\) is related to those of the given schemes in terms of a long exact sequence. This far reaching generalization yields, in the case the underlying scheme is arithmetically Cohen-Macaulay, that the cohomology of \(Z\) is the shift of the direct sum of the involved schemes. There are a number of applications of this clever technique to the construction of arithmetically Buchsbaum schemes, the double linkage, Hilbert functions, etc. decomposition of Hartshorne-Rao module; arithmetical Cohen-Macaulay scheme; liaison addition; arithmetically Buchsbaum schemes; double linkage; Hilbert functions Linkage, Linkage, complete intersections and determinantal ideals A generalized liaison addition | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems On pages 338 and 339 in his first notebook, Ramanujan defined remarkable product of theta-functions \(a_{m,n}\) and also recorded eighteen explicit values depending on two parameters \(m\) and \(n\). All these values have been established by \textit{B. C. Berndt} et al. [Proc. Edinb. Math. Soc., II. Ser. 40, No. 3, 583--612 (1997; Zbl 0901.33007)]. In this paper, we establish a new general formulae for the explicit evaluations of \(a_{3m,3}\) and \(a_{m,9}\) by using \(P-Q\) mixed modular equation and values for certain class invariant of Ramanujan. Using these formulae, we calculate some new explicit values of \(a_{3m,3}\) for \(m = 2,7,13,17,25,37\) and \(a_{m,9}\) for \(m = 17,37\). modular equation; class invariants; remarkable product of theta-function Theta series; Weil representation; theta correspondences, Basic hypergeometric functions in one variable, \({}_r\phi_s\), Dedekind eta function, Dedekind sums, Theta functions and abelian varieties The explicit formulae and evaluations of Ramanujan's remarkable product of theta-functions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a smooth projective variety over an algebraic closed field of characteristic zero and \(D\) be an effective divisor on \(X\). There are various obstruction theories attached to \((X,D)\): the topological obstruction to the deformation of the fundamental class of \(D\) as a Hodge class with values in \(H^2(\mathcal{O}_X)\) (see e.g. [\textit{K. Kodaira}, Complex manifolds and deformation of complex structures. Transl. from the Japanese by Kazuo Akao. Berlin: Springer (2005; Zbl 1058.32007)]), the geometric obstruction to the deformation of \(D\) as an effective Cartier divisor of a first order infinitesimal deformation of \(X\) with values in \(H^1(\mathcal{N}_{D/X})\) ([\textit{A. Grothendieck} ``FGA'', Secretariat Math. Paris (1962; Zbl 0239.14002)]).
In this article, the authors give a ``local'' topological obstruction theory with values in the local cohomology group \(H^2_D(\mathcal{O}_X)\) and compare it with the obstruction theories mentioned above. As an interesting application, they give examples of first order deformations \(X_t\) of \(X\) where the class \([D]\) deforms as a Hodge class but \(D\) does not lift as an effective Cartier divisor of \(X_t\). obstruction theories; Hodge locus; semi-regularity map; deformation of linear systems; Noether-Lefschetz locus Infinitesimal methods in algebraic geometry, Local cohomology and algebraic geometry, Transcendental methods, Hodge theory (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Algebraic cycles, Variation of Hodge structures (algebro-geometric aspects) Local topological obstruction for divisors | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Relations beween the occurring singularity structure and the decomposition of the local zeta function in families of Calabi-Yau \(n\)-folds containing singular fibres are studied. Here the local zeta function for a smooth projective variety \(X\) over \(\mathbb{F}_p\) is defined s follows:
\[
\varsigma(X|\mathbb{F}_p, t):=\exp(\sum\limits_{r\in \mathbb{N}}\# X(\mathbb{F}_{p^r})\frac{t^r}{r}).
\]
Properties about the local zeta functions at good primes are listed. The singularities occurring in some \(1\)--dimensional and \(2\)--dimensional families (families of Fermat-type Calab-Yau \(n\)-folds) are analysed in detail. Examples of \(2\)--parameter families are especially studied. Here the singularity analysis provides correct predictions for the changes of degree in the decomposition of the zeta--function when passing to singular fibres. zeta function; Calabi-Yau \(n\)-fold; singularity Calabi-Yau manifolds (algebro-geometric aspects), Varieties over finite and local fields, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Computational aspects of higher-dimensional varieties, Fibrations, degenerations in algebraic geometry, Singularities of surfaces or higher-dimensional varieties Zeta functions for families of Calabi-Yau \(n\)-folds with singularities | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The Morel-Voevodsky \(\mathbb{P}^1\)-stable \(\mathbb{A}^1\)-homotopy category \(\mathcal{SH}(k)\) of a field \(k\) offers a way to construct interesting (co)homology theories of algebraic varieties. Two theories introduced before its invention, algebraic \(K\)-theory and higher Chow groups (a.k.a. motivic cohomology) of smooth \(k\)-varieties, are representable in \(\mathcal{SH}(k)\), whereas algebraic cobordism is the prime example, due to Voevodsky, of a (co)homology theory created through its representing \(\mathbb{P}^1\)-spectrum \(\mathsf{MGL}\). It is built out of Thom spaces \(\mathrm{Th}(\gamma_k)\) of tautological bundles \(\gamma_k \to \mathrm{colim}_{n\to \infty} \mathrm{Gr}(k,n)\) on infinite Grassmann varieties and constitutes a close relative of the topological complex cobordism spectrum \(\mathrm{MU}\). Motivic stable (co)homotopy represented by the motivic sphere spectrum \(\mathbf{1}\in \mathcal{SH}(k)\) may be regarded as the ``initial'' example in the sense that all \(\mathbb{P}^1\)-spectra in \(\mathcal{SH}(k)\) are canonically modules over \(\mathbf{1}\).
One major technique used in the very difficult (at least in these four examples) task of computing (co)homology groups for a given theory is to build a filtration on its representing \(\mathbb{P}^1\)-spectrum. If its associated graded is more accessible, the filtration leads to a spectral sequence whose initial term may be computed using algebraic means, at least in a certain range. Often the way of building a filtration applies to every \(\mathbb{P}^1\)-spectrum naturally, thus leading to a filtration on the whole homotopy category \(\mathcal{SH}(k)\). Voevodsky's slice filtration (which leads to ``the'' motivic spectral sequence for algebraic \(K\)-theory on smooth \(k\)-varieties) and Morel's homotopy \(t\)-structure are two prominent examples.
This article introduces another filtration on \(\mathcal{SH}(k)\), the \textit{Chow} \(t\)-structure. Its nonnegative part is the subcategory \(\mathcal{SH}(k)_{c\geq 0}\) generated under colimits and extensions by Thom spectra \(\mathrm{Th}(\xi)\) of virtual vector bundles \(\xi\in K(X)\) over smooth proper \(k\)-varieties. Theorem 3.14, the main result of this article, exhibits a close relation between the Chow \(t\)-structure and the algebraic cobordism spectrum \(\mathsf{MGL}\): If \(E\in \mathcal{SH}(k)_{c\geq 0}\), then the homotopy groups of its nonpositive truncation \(\tau_{c\leq 0}E = E_{c=0}\) can be canonically identified with Ext-groups of the \(\mathsf{MGL}\)-homology of \(E\) in the category of \(\mathrm{MU}_\ast\mathrm{MU}\)-comodules:
\[
\pi_{2w-s,w}\tau_{c\leq 0}E \cong \mathrm{Ext}_{\mathrm{MU}_\ast\mathrm{MU}}^{s,2w}(\mathrm{MU}_\ast,\mathsf{MGL}_{2\ast,\ast}E)
\]
In this and the following results, the characteristic of \(k\) is implicitly inverted if it is positive. Taking \(E=\mathbf{1}\) the motivic sphere spectrum reproduces the \(E^2\)-term of the topological Adams-Novikov spectral sequence. Hence the Chow \(t\)-structure provides a vast generalization (to arbitrary base fields and coefficients) of work by Gheorghe, Wang, and Xu on the \(2\)-completed motivic sphere spectrum over the complex numbers [\textit{B. Gheorghe}, Doc. Math. 23, 1077--1127 (2018; Zbl 1407.55007)], which in turn allowed amazing progress in classical homotopy theory: Isaksen, Wang, and Xu used it to extend the range in which the infamous stable homotopy groups of spheres are known by a factor of about \(\tfrac{3}{2}\). [\textit{D. C. Isaksen} et al., Proc. Natl. Acad. Sci. USA 117, No. 40, 24757--24763 (2020; Zbl 1485.55017)].
One consequence of Theorem 3.14 reconstructs the motivic Adams spectral sequence (based on motivic cohomology with \(\mathbb{Z}/p\)-coefficients) for \(\mathbb{P}^1\)-spectra \(E=E_{c=0}\) in the Chow heart in an essentially algebraic fashion. The authors provide computational evidence for the power of this approach by addressing the resulting spectral sequence for the motivic sphere spectrum \(\mathbf{1}\) over \(\mathbb{C}\), \(\mathbb{R}\), and finite fields. motivic stable homotopy theory; stable homotopy groups of spheres; Adams-Novikov spectral sequence Motivic cohomology; motivic homotopy theory, Derived categories, triangulated categories, Stable homotopy of spheres The Chow \(t\)-structure on the \(\infty\)-category of motivic spectra | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The residual scheme \({\text{res}}_Y X\) of the scheme \(X\) with respect to \(Y\) in the projective space is studied. Properties and characterizations of its hypersurface section are obtained. scheme; graded rings and modules Schemes and morphisms, Graded rings and modules (associative rings and algebras) Residual schemes and general hypersurface sections | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The goal of the article is to explore the structure of singularities that occur in generic fibres in positive characteristic. The author determines which rational double points do and which do not occur on generic fibres.
Let \(k\) be an algebraically closed ground field of characteristic \(p>0\), and suppose \(f:S \rightarrow B\) is a morphism between smooth integral schemes. Then the generic fiber \(S_\eta\) is a regular scheme of finite type over the function field \(E=\kappa(\eta).\) In characteristic \(0,\) this implies that \(S_\eta\) is smooth over \(E\). The absolute Galois group \(G=\operatorname{Gal}(\bar E/E)\) acts on the geometric generic fiber \(S_{\bar\eta}\) with quotient isomorphic to \(S_\eta\), so to understand the generic fiber it suffices to understand the geometric generic fiber which is again smooth over an algebraically closed field, together with its Galois action.
The situation is more complicated in characteristic \(p>0\). The reason is that over nonperfect fields the notion of regularity is weaker than the notion of geometric regularity, which coincides with formal smoothness. Here it easily happens that the geometric generic fiber \(S_{\bar\eta}\) acquires singularities. As is proved by Bombieri and Mumford, there are quasi-elliptic fibrations of \(p=2\) and \(p=3\), which are analogous to elliptic fibrations but have a cusp on the geometric generic fiber. A proper morphism \(f:S\rightarrow B\) of smooth algebraic schemes is called a \textit{quasifibration} if \(\mathcal O_B=f_\ast(\mathcal O_S)\) and if the generic fiber \(S_\eta\) is not smooth. Quasi-fibrations involve some fascinating geometry and offer new freedom to achieve geometrical constructions that are impossible in characteristic \(0\).
Nonsmoothness of the generic fiber \(S_\eta\) leads to unusual complications. However, singularities appearing on the geometric generic fiber \(S_{\bar\eta}\) are not arbitrary. First, they are locally of complete intersection; hence many powerful methods from commutative algebra apply. They also satisfy far more restrictive conditions, and the goal of the paper is to analyze these. Hirokado started an analysis, characterizing those rational double points in odd characteristic that appear on geometric fibres. His approach was to study the closed fibres \(S_b\) \((b\in B)\) and their deformation theory. In this article, the author looks at the generic fibre \(S_\eta\) and work over the function field \(\kappa(\eta).\) The author works in the following abstract setting: Given a field \(F\) in characteristic \(p>0\) and a subfield \(E\) such that the field extension \(E\subset F\) is purely inseparable. Then the author considers \(F\)-schemes \(X\) of finite type that descend to regular \(E\)-schemes \(Y\), that is, \(X\backsimeq Y\otimes_E F.\) The first results of such schemes are: In codimension 2, the local fundamental groups are trivial and the torsion of the local class groups are \(p\)-groups. Moreover, the Tjurina numbers are divisible by \(p\), the stalks of the Jacobian ideal have finite projective dimension, and the tangent sheaf \(\Theta_X\) is locally free in codimension \(2\). These conditions give strong conditions on the singularities. The first main result of the article is a surprising restriction on the cotangent sheaf: If an \(F\)-scheme \(X\) descends to a regular scheme then, for each point \(x\in X\) of codimension \(2\), the stalk \(\Omega^1_{X/F,x},\) contains an invertible direct summand.
As an application of these results the author determines which rational double points appear on surfaces descending to regular schemes and which do not. It turns out that the situation is most challenging in characteristic 2: Besides the \(A_n\)-singularities, which behave as in characteristic \(0\), there are the following isomorphism classes: \(D_n^r\) with \(0\leq r\leq \lfloor n/2\rfloor-1\) and \(E_6^0, E_6^1,E_7^0,\dots,E_7^3,E_8^0,\dots,E_8^4.\) Notice that all the members of the list have a tangent sheaf that are locally free, although there are other rational double points with locally free tangent sheaf.
The author sets up notation and gives some elementary examples and results. He analyzes \(F\)-schemes that \(X\) descend to regular \(E\)-schemes \(Y\), and treats the local fundamental groups. He proves that integer-valued invariants are multiples of \(p\), and he treats the finite projective dimension of sheaves obtained from the cotangent sheaf \(\Omega^1_{X/F}.\)
This article is written in an easy to understand language, it is more or less self contained with good references when needed. Finally, it treats a computable field of singularity theory which is of importance to the study in positive characteristic. singularities on generic fibers; absolute Galois group; positive characteristic; descent; descends to regular scheme; quasi-fibration Stefan Schröer, Singularities appearing on generic fibers of morphisms between smooth schemes, Michigan Math. J. 56 (2008), no. 1, 55 -- 76. Arithmetic ground fields (finite, local, global) and families or fibrations, Fibrations, degenerations in algebraic geometry Singularities appearing on generic fibers of morphisms between smooth schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \({\mathfrak O}=W(k)\) be the ring of Witt vectors over a perfect field k of characteristic \(p>0.\) Let \({\mathcal G}_{{\mathfrak O}}\) and \({\mathcal H}_ k\) the subcategories of the categories of finite commutative group schemes over \({\mathfrak O}\) and finite Honda systems, respectively, consisting of all objects which are annihilated by multiplication by p. Fontaine proved that these categories are antiequivalent when \(p\neq 2\); when \(p=2\), the subcategories of unipotent objects in \({\mathcal G}_{{\mathfrak O}}\) and \({\mathcal H}_ k\) are antiequivalent. His arguments were based on his classification of formal and p-divisible groups over \({\mathfrak O}\). The author defines a subcategory \({\mathcal G}^*_{{\mathfrak O}}\) of \({\mathcal G}_{{\mathfrak O}}\). Assuming in the case \(p=2\) that k is algebraically closed, he constructs an antiequivalence \({\mathcal G}^*_{{\mathfrak O}}\to {\mathcal H}_ k\). Since \({\mathcal G}^*_{{\mathfrak O}}={\mathcal G}_{{\mathfrak O}}\) when \(p\neq 2\), this gives a new construction for the Fontaine antiequivalence with \(p\neq 2\). When \(p=2\), the restriction of the antiequivalence \({\mathcal G}^*_{{\mathfrak O}}\to {\mathcal H}_ k\) on the unipotent objects gives the Fontaine antiequivalence. Arguments are based on the Dieudonné theory for finite commutative group schemes over k. finite Honda systems; finite commutative group schemes --------, Honda systems of group schemes of period \(p\) , Math. USSR-Izv. 30 (1988), 419-453. Formal groups, \(p\)-divisible groups, Group schemes Honda systems of group schemes of the period p | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper deals with an intimate connection between Picard's theorem and elliptic finite-gap solutions of completely integrable systems. The authors show that elliptic finite-gap potentials of Hill's equation are precisely those for which all solutions for all spectral parameters are meromorphic functions in the independent variables complementing a classical theorem of Picard. Moreover, the authors construct the hyperelliptic Riemann surface associated with a finite-gap potential (not necessarily elliptic) which is elegant and new. Picard's theorem; elliptic finite-gap solutions; completely integrable systems; Hill's equation; hyperelliptic Riemann surface F.GESZTESYand R.WEIKARD,\textit{Picard potentials and Hill's equation on a torus}, Acta Math. 176 (1996), no. 1, 73--107.http://dx.doi.org/10.1007/BF02547336.MR1395670 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems, Relationships between algebraic curves and integrable systems Picard potential and Hill's equation on a torus | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Fix a pair \(\mathcal{D}\subset \mathcal{C}\) of posets, considered as categories. Define the total cofiber \(\Gamma(X)\) of a diagram \(X:\mathcal{C}\to\mathcal{SS}ets_\ast\) of pointed simplicial sets as the strict cofiber of the map \(\text{hocolim}_{\mathcal D}(X)\to \text{hocolim}_{\mathcal C}(X)\). Analogously, the total cofiber of a diagram of pointed topological spaces is defined. By applying the definition levelwise, we obtain a total cofiber functor for diagrams of spectra in the sense of \textit{A. K. Bousfield} and \textit{E. M. Friedlander} [Geom. Appl. Homotopy Theory, II, Proc. Conf., Evanston 1977, Lect. Notes Math. 658, 80-130 (1978; Zbl 0405.55021)]. The total cofiber functor has been used by the author in his studies of quasi-coherent sheaves on projective toric varieties as a replacement of the global section functor and its derived functors. If \(\mathcal{C}\) is the poset of non-empty faces of a polytope and \(\mathcal{D}\) the subposet of proper faces and \(X\) is a \(\mathcal{C}\)-diagram of pointed spaces the author constructed a spectral sequence
\[
E^2_{p,q}= \lim{}^{n-p}\widetilde{H}_q(X;\mathbb{Z})\to \widetilde{H}_{p+q}(\Gamma(X);\mathbb{Z})
\]
where \(n\) is the dimension of the polytope [\textit{T. Hüttemann}, K-Theory 31, 101--123 (2004; Zbl 1068.55015)]. This spectral sequence can be interpreted as a device for comparing the homotopy limit of \(X\) with the total cofiber. The present paper investigates this relationship more closely. For a diagram \(X\) of pointed spectra the author constructs a map
\[
\widetilde{\Gamma}_X:\text{holim}_{\mathcal C}X\to \Hom_{\mathcal{SS}ets_\ast} (\Gamma(N(\mathcal{C}\downarrow -)_+), \widetilde{\Gamma}(X))
\]
from the homotopy limit into a mapping space and characterizes all pairs of posets \(\mathcal{D}\subset\mathcal{C}\) for which \(\widetilde{\Gamma}_Y\) is a stable weak equivalence of spectra. Here \(N\) is the nerve functor and \(\tilde{\Gamma}(Y)\) is the levelwise fibrant replacement of \(\Gamma(X)\). As an application he extends the spectral sequence above to more general diagrams and derives homotopy spectral sequences for the total cofiber. homotopy limit; homotopy colimit; closed model category; toric varieties; spectral sequences T Hüttemann, Total cofibres of diagrams of spectra, New York J. Math. 11 (2005) 333 Stable homotopy theory, spectra, Spectral sequences in algebraic topology, Toric varieties, Newton polyhedra, Okounkov bodies, Homotopy extension properties, cofibrations in algebraic topology, Cohomotopy groups, General topology of complexes Total cofibres of diagrams of spectra | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Topological structure of translation-invariant noncommutative Yang-Mills theories are studied by means of a cohomology theory, the so-called \(\star\)-cohomology, which plays an intermediate role between de Rham and cyclic (co)homology theory for noncommutative algebras and gives rise to a cohomological formulation comparable to Seiberg-Witten map. translation-invariant star product; noncommutative Yang-Mills; spectral triple; Chern character; Connes-Chern character; family index theory; topological anomaly; BRST Yang-Mills and other gauge theories in quantum field theory, Anomalies in quantum field theory, Riemann-Roch theorems, Chern characters, Finite-dimensional groups and algebras motivated by physics and their representations, Motivic cohomology; motivic homotopy theory \(\star\)-cohomology, Connes-Chern characters, and anomalies in general translation-invariant noncommutative Yang-Mills | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this article we study various forms of \(\ell\)-independence (including the case \(\ell =p\)) for the cohomology and fundamental groups of varieties over finite fields and equicharacteristic local fields. Our first result is a strong form of \(\ell\)-independence for the unipotent fundamental group of smooth and projective varieties over finite fields. By then proving a certain `spreading out' result we are able to deduce a much weaker form of \(\ell\)-independence for unipotent fundamental groups over equicharacteristic local fields, at least in the semistable case. In a similar vein, we can also use this to deduce \(\ell\)-independence results for the cohomology of smooth and proper varieties over equicharacteristic local fields from the well-known results on \(\ell\)-independence for smooth and proper varieties over finite fields. As another consequence of this `spreading out' result we are able to deduce the existence of a Clemens-Schmid exact sequence for formal semistable families. Finally, by deforming to characteristic \(p\), we show a similar weak version of \(\ell\)-independence for the unipotent fundamental group of a semistable curve in mixed characteristic. local function fields; cohomology; \(L\)-functions; motives; unipotent fundamental groups Curves over finite and local fields, Étale and other Grothendieck topologies and (co)homologies, Motivic cohomology; motivic homotopy theory Around \(\ell\)-independence | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Development of signature schemes with short signatures is a quite relevant non-trivial challenge. The most perspective existing schemes are multivariate schemes and schemes based on Weil pairing. But the cryptographic tools used in these schemes are still not supported by most cryptographic software, this complicates their use in practice. We propose three methods of shortening standard ElGamal-type signatures and analyze how these methods affect the security. Applying all three methods to the GOST signature scheme with elliptic curve subgroup of order \(q\in(2^{255},2^{256})\) may reduce the signature size from 512 to 320 bits providing sufficient security and acceptable (for non-interactive protocols) signing and verifying time. short signature scheme; ElGamal-type signature scheme; GOST; provable security Authentication, digital signatures and secret sharing, Cryptography, Applications to coding theory and cryptography of arithmetic geometry On methods of shortening ElGamal-type signatures | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper gives a purely algebraic proof that the Fourier transform preserves the rigidity index of irreducible regular holonomic \(\mathcal{D}_{\mathbb{P}^1}[*\{\infty\}]\)-modules. \(\mathcal{D}\)-modules; Fourier transform; rigid local systems Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Forms and linear algebraic groups Rigidity index preservation of regular holonomic \(\mathcal{D}\)-modules under Fourier transform | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In the paper under review the representation of a group of automorphisms into the \(\ell\)-adic cohomology of Dwork hypersurfaces is studied. This yields a decomposition of the \(\ell\)-adic cohomology. This decomposition yields a factorization of the zeta function of Dwork hypersurfaces. This factorization is finer than the factorization obtained by the reviewer [Algebra Number Theory 1, No. 4, 421--450 (2007; Zbl 1166.14016)]. zeta function factorization; Dwork hypersurfaces; isotypic decomposition; arithmetic mirror symmetry DOI: 10.1016/j.ffa.2010.10.003 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Varieties over finite and local fields, Finite ground fields in algebraic geometry Isotypic decomposition of the cohomology and factorization of the zeta functions of Dwork hypersurfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be an integral scheme over a connected Dedekind scheme \(S\) with \(f: X \to S\) a faithfully flat morphism of finite type. Let \(x \in X(S)\). \textit{C. Gasbarri} defined in his paper [Duke Math. J. 117, No. 2, 287--311 (2003; Zbl 1026.11057)] the abelianalized fundamental scheme \(\pi_1(X,x)\). This is a generalization of Nori's definition of the fundamental group scheme of a reduced connected and proper scheme \(X\) over a prefect field \(k\), which is a generalization of the classical étale fundamental group.
Assume that the existence of the Picard scheme \(\mathrm{\mathbf Pic}_{X/S}\), \(\mathrm{\mathbf Pic}^0_{X/S}\) as an open \(S\)-group subscheme of \(\mathrm{\mathbf Pic}_{X/S}\) is defined such that for any point \(s \in S\), \(\mathrm{\mathbf Pic}^0_{X/S} \times_S k(s) \simeq \mathrm{\mathbf Pic}^0_{X_s/k(s)}\). It exists with assumptions that \(\mathrm{\mathbf Pic}_{X/S}\) is separated over \(S\) and \(\mathrm{\mathbf Pic}^0_{X_s/k(s)}\) are smooth and have the same dimensions for all \(s \in S\). When \(\mathrm{\mathbf Pic}^0_{X/S}\) is a projective abelian scheme, set \(\mathrm{\mathbf Alb}_{X/S} := (\mathrm{\mathbf Pic}^0_{X/S} )^*\) and call it the Albanese scheme of \(X \to S\). Let \(\mathrm{\mathbf Pic}^{\tau}_{X/S}\) be an open scheme of \(\mathrm{\mathbf Pic}_{X/S}\) defined by \(\cup_n n^{-1} (\mathrm{Pic}^0_{X/S})\) where \(n:\mathrm{\mathbf Pic}_{X/S} \to \mathrm{\mathbf Pic}_{X/S}\) is the multiplication by \(n\) homomorphism. Let \(\mathrm{NS}^{\tau}_{X/S}\) be the \(S\)-scheme representing the quotient sheaf associated with respect to the fpqc topology to the functor \(T \mapsto \mathrm{\mathbf Pic}_{X/S}^{\tau}(T)/\mathrm{\mathbf Pic}^0_{X/S}(T)\).
The main result of this paper is that there is an exact sequence of commutative group schemes:
\[
0 \to (\mathrm{\mathbf NS}^{\tau}_{X/S})^{\wedge} \to \pi_1(X,x)^{\mathrm{ab}} \to \pi_1(\mathrm{\mathbf Alb}_{X/S}, 0_{\mathrm{\mathbf Alb}_{X/S}}) \to 0.
\]
As a corollary, when \(f: C \to S\) is a smooth and projective curve with integral geometric fibres, the natural morphism \(\pi_1(C, x)^{\mathrm{ab}} \to \pi_1(J,0_J)\) is an isomorphism where \(J\) is the Jacobian of \(C\). fundamental group scheme; Albanese scheme Antei, M., \textit{on the abelian fundamental group scheme of a family of varieties}, Israel J. Math., 186, 427-446, (2011) Group schemes, Picard groups, Coverings in algebraic geometry On the abelian fundamental group scheme of a family of varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(A\) be an \(n\times n\) complex matrix and \(A = \mathfrak{R}(A) + \mathfrak{I}(A)\) be its Cartesian decomposition, where \(\mathfrak{R}(A) = \frac{A + A^{*}}{2}\) and \(\mathfrak{I}(A)= \frac{A - A^{*}}{2i}\). The numerical range of \(A\) is defined as
\[
W(A) = \{ \xi^{*}A\xi : \xi \in \mathbb{C}^{n} \text{ and } \xi^{*} \xi = 1\}.
\]
The Toeplitz-Hausdorff theorem asserts that the numerical range \(W(A)\) is a convex set in the Gaussian plane \(\mathbb{C}\). Given a point \(z \in W(A)\), the inverse numerical range problem is to find a unit vector \(\xi\) so that \(z = \xi^{*}A \xi\). This problem was solved in the case of a generic \(2 \times 2\) matrix. The main result of this paper answers this question for the boundary points and points on the boundary generating curve of \(W(A)\) when the matrix \(A\) is a \(3 \times 3\) canonical symmetric matrix for which \(\mathfrak{I}(A)\) is a diagonal matrix with distinct eigenvalues. inverse numerical range; determinantal representation; elliptic curve; kernel vector function Norms of matrices, numerical range, applications of functional analysis to matrix theory, General ternary and quaternary quadratic forms; forms of more than two variables, Inverse problems in linear algebra, Elliptic curves Inverse numerical range and determinantal representation | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Moser averaging techniques are used to obtain classical and quantum Birkhoff canonical forms for the 2D perturbed harmonic oscillator. Emphasis is laid on the perturbed semiclassical quantum Hamiltonians \(H=H_{0}+h^{2} H_{2}+h^{3}H_{3}+ \dots\), where \(H_{0} = -(h^{2}/2) (\partial^{2} / \partial x_{1}^{2} + \partial^{2} / \partial x_{2}^{2}) + (1/2)(x_{1}^{2}+x_{2}^{2})\) and \(H_{i}, i\geq2\) are semiclassical pseudodifferential operators of order zero. One also obtains the asymptotic expansion of the spectral measure of \(H\) in powers of \(h\) with terms expressible in terms of the quantum Birkhoff canonical form. In the final section, other settings to which the developed techniques can be applied are shortly described. harmonic oscillator; toric system; Birkhoff canonical form; Morse functions; integrable systems Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol'd diffusion, Asymptotic distributions of eigenvalues in context of PDEs, Toric varieties, Newton polyhedra, Okounkov bodies Canonical forms for perturbations of the harmonic oscillator | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(C\) be a curve (i.e. a locally C. M. subscheme equidimensional of dimension 1) in \(\mathbb{P}^n_k\) over an algebraically closed field \(k\) of characteristic 0, of degree \(d\) and let \(\Gamma=C\cap H\) be the generic hyperplane section. Let \(I\) (resp. \(J)\) be the homogeneous ideal of \(C\) (resp. \(\Gamma)\) in the polynomial ring \(S=k[x_1, \dots, x_{n+1}]\) (resp. \(R =k [x_1,\dots, x_n])\). It is known that, if \(C\) is reduced and irreducible, then \(\Gamma\) consists of points in uniform position. Assume instead that \(\Gamma\) is not in uniform position; then \(\Gamma\) contains a subscheme \(\Gamma'\) of degree \(d'\) such that \(h_{\Gamma'}(s) <\min \{d',h_\Gamma(s)\}\), for some \(s\), where \(h\) is the Hilbert function. One can ask if, in this situation, \(C\) contains a curve \(C'\) whose generic hyperplane section is \(\Gamma'\). In this paper we give a positive answer to this question under some additional hypotheses on \(\Gamma\):
Theorem. Assume that, for some \(s>0\), both \(J_s\) and \(J_{s+1}\) have a GCD \(F\) of positive degree and let \(\Gamma'= V(F) \cap \Gamma\). Then:
(a) there exists a curve \(C'\cap C\) whose generic hyperplane section is \(\Gamma'\);
(b) there exists a hypersurface \(T\subset \mathbb{P}_k^n\) whose generic hyperplane section is \(V(F)\);
(c) \(C'\) is equal to \(C\cap T\) up to zero-dimensional components. Hilbert function; generic hyperplane section Strano, R.: Curves and their hyperplane sections. J. Pure Appl. Algebra \textbf{152}(1-3), 337-341 (2000). Commutative algebra, homological algebra and representation theory (Catania/Genoa/Rome, 1998) Plane and space curves, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Curves and their hyperplane sections | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems \textit{R. Zomorrodian} [Trans. Am. Math. Soc. 288, 241--255 (1985; Zbl 0565.20029)] proved that a nilpotent subgroup \(G\) of the automorphism group of an algebraic curve defined over the complex numbers has size at most \(16(g-1)\), where \(g\) is the genus of the curve. He also proved that if the equality holds, then \(g-1\) is a power of \(2\). Conversely, if \(g-1\) is a power of \(2\), then there exists an algebraic curve of genus \(g\) that admits an automorphism group of order \(16(g -1)\) which is a nilpotent 2-group.
In this paper, the authors partially extend these results to the positive characteristic case (Theorem 1.1). They also provide examples of infinite families of algebraic curves defined over a field of positive characteristic with a nilpotent group of automorphisms attaining the bound \(16(g-1)\) (Section 4).
Their approach uses function field theory, and it is completely different from the original approach of Zomorrodian, based on the method of Fuchsian groups. function field; Hurwitz genus formula; nilpotent group; positive characteristic Algebraic functions and function fields in algebraic geometry, Automorphisms of curves On nilpotent automorphism groups of function fields | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors compute the supports of the perverse cohomology sheaves of the Hitchin fibration for \(\operatorname{GL}_n\) over the locus of reduced spectral curves. In contrast to the case of meromorphic Higgs fields they find additional supports at the loci of reducible spectral curves. Their contribution to the global cohomology is governed by a finite twist of Hitchin fibrations for Levi subgroups. The corresponding summands give non-trivial contributions to the cohomology of the moduli spaces for every \(n\geq2\). A key ingredient is a restriction result for intersection cohomology sheaves that allows them to compare the fibration to the one defined over versal deformations of spectral curves. This paper is organized as follows: Section 1 is an introduction to the subject and summarizes the main results. In Section 2 the authors set up notation and conventions. In Section 3 they recall the main result from [\textit{L. Migliorini} and \textit{V. Shende}, Algebr. Geom. 5, No. 1, 114--130 (2018; Zbl 1406.14005)], that constrains the potential supports of their perverse cohomology sheaves in terms of higher discriminants. The symplectic structure of Hitchin fibrations allows them to describe these in terms of the action of Jacobians of spectral curves. Section 4 combines the Ngô support theorem and results on compactified Jacobians and concerns the fact that the supports have to be partition strata. Section 5 is devoted to comparison with versal families. Here the authors prove the restriction result for IC-sheaves and show that it applies to the Hitchin fibration by computing the Kodaira-Spencer map for the universal family of spectral curves. Section 6 concerns the fact that the partition strata are supports. Here the authors use the Cattani-Kaplan-Schmid complex for the versal family to translate the problem of determining the generic fibers into a combinatorial problem that they can then solve. They describe the monodromy of these summands, in order to prove that the new summands contribute to the global cohomology of the Hitchin fibration for any \(n\geq2\). cohomology sheaves; Hitchin fibration; spectral curves; supports; Higgs fields Vector bundles on curves and their moduli, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Sheaves in algebraic geometry A support theorem for the Hitchin fibration: the case of \(\operatorname{GL}_n\) and \(K_C\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The special linear group \(\text{SL}_{n+1}(K)\) is the simply connected group and the projective linear group \(\text{PGL}_{n+1}(K)\) the adjoint group of Lie type \(A_n\). They are distinguished sections of the (reductive) general linear group \(\text{GL}_{n+1}(K)\) (admitting the same root system). We shall introduce corresponding connected algebraic groups resp. finite groups for each Lie type (to indecomposable root systems). These will be called universal (or general) groups of the given type. The universal groups involve the simply connected and adjoint groups and, in certain respects, appear to be better behaved (automorphisms, Schur multipliers, character tables).
For example, the general linear group \(\text{GL}_{n+1}(q)\), the unitary group \(\text{U}_{n+1}(q)\), the conformal symplectic group \(\text{CSp}_{2n}(q)\), the special Clifford group \(\text{G}^+_{2n+1}(q)\) to the nonsingular quadratic form of dimension \(2n+1\) and Witt index \(n\) (over the finite field \(K=\mathbb{F}_q\) with \(q\) elements) are the universal groups of type \(A_n\), \({^2A_n}\), \(C_n\), \(B_n\), respectively. For other types one obtains new ``classical groups''.
When \(K\) is algebraically closed, the universal groups are characterized in terms of the related simply connected and adjoint groups, as reductive groups with connected centres. In the finite case the groups are constructed via certain canonical Frobenius morphisms. Here, a corresponding group-theoretic description is more intricate (even for the general linear group). The underlying building presents a geometric link between related groups, the group of automorphisms generated by the root groups being understood as the nucleus of the family of groups of Lie type. special linear groups; projective linear groups; general linear groups; connected algebraic groups; root systems; universal groups; adjoint groups; unitary groups; conformal symplectic groups; special Clifford groups; reductive groups; Frobenius morphisms; root groups; groups of Lie type DOI: 10.1112/S0024610799008066 Linear algebraic groups over finite fields, Classical groups (algebro-geometric aspects) Simply connected, adjoint and universal groups of Lie type | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Here we apply the so-called Horace method for zero-dimensional schemes to error-correcting codes on complete intersections. In particular, we obtain sharper estimates on the minimum distance. Zero-dimensional scheme; Reduced complete intersection; Error-correcting code Ballico E., Fontanari C.: The Horace method for error-correcting codes. Appl. Algebra Eng. Commun. Comput. 17(2), 135--139 (2006) Geometric methods (including applications of algebraic geometry) applied to coding theory, Projective techniques in algebraic geometry, Complete intersections The Horace method for error-correcting codes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(Q\to Y\) be a family of 2-dimensional quadrics over a 3-dimensional base with \(Q\) and \(Y\) smooth and consider its relative Fano scheme of lines \(\rho: M\to Y\). In this paper, a criterion for the smoothness of \(M\) is given and the bounded derived category \(\mathcal D^b(M)\) of coherent sheaves on \(M\) is studied.
Let \(D_r\subset Y\) be the locus of quadrics of corank at least \(r\). Under the assumptions that the generic fibre of \(Q\to Y\) is smooth, \(D_3=\emptyset\), and \(D_2\) consists of finitely many points \(y_1,\dots,y_N\) all of which are ordinary double points of \(D_1\), the author proves that \(M\) is smooth and there is a semiorthogonal decomposition
\[
\mathcal D^b(M)=\bigl\langle \mathcal D^b(X^+),\mathcal D^b(Y,\mathcal B_0),\{\mathcal O_{\Sigma_i^+}\}_{i=1}^N\bigr\rangle
\]
obtained as follows. For \(i=1,\dots,N\), the fibre \(\rho^{-1}(y_i)\) is the union of two planes and \(\mathbb P^2\cong \Sigma_i^+\subset M\) is chosen as one of them. Then it is shown that the structure sheaves \(\mathcal O_{\Sigma_i^+}\) form a completely orthogonal exceptional collection. The Stein factorisation \(M\to X\to Y\) of \(\rho\) consists of a generically conic bundle \(M\to X\) and the double cover \(X\to Y\) ramified over \(D_1\). Let \(M^+\) be the flip of \(M\) in the planes \(\Sigma_i^+\). The author proves that the induced map \(M^+\to X\) factors as \(M^+\to X^+\to X\) where \(M^+\to X^+\) is a \(\mathbb P^1\)-fibration and \(X^+\to X\) is a small resolution of singularities. It follows that \(\mathcal D^b(X^+)\) can be embedded into \(\mathcal D^b(M)\) via the pull-back along \(M^+\to X^+\) followed by the canonical embedding \(\mathcal D^b(M^+)\subset \mathcal D^b(M)\). Finally, \(\mathcal B_0\) is the sheaf of the even part of the Clifford algebra associated to the quadric fibration \(Q\to Y\) and it is shown that \(\mathcal D^b(Y,\mathcal B_0)\) is equivalent to the left orthogonal complement of \(\mathcal D^b(X^+)\) in \(\mathcal D^b(M^+)\). The results are used by \textit{C. Ingalls} and the author in [``On nodal Enriques surfaces and quartic double solids'', \url{arXiv:1012.3530}] to provide a description of the derived category of a nodal Enriques surface. quadric fibration; Fano scheme of lines; derived category; semiorthogonal decomposition; Clifford algebra A. Kuznetsov, Scheme of lines on a family of 2-dimensional quadrics: geometry and derived category, math. AG/arXiv:1011. 4146. Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Brauer groups of schemes, Quadratic spaces; Clifford algebras, Fibrations, degenerations in algebraic geometry, Families, moduli, classification: algebraic theory, Grassmannians, Schubert varieties, flag manifolds Scheme of lines on a family of 2-dimensional quadrics: geometry and derived category | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper proves equidistribution of periodic points under regular automorphisms on \(\mathbb A^n\). This generalizes earlier results for Hénon maps on \(\mathbb A^n\), as well as providing a non-archimedean counterpart to [\textit{T.-C. Dinh} and \textit{N. Sibony}, ``Density of Positive Currents and Dynamics of Hénon type automorphism of \(\mathbb C^k\)'', Preprint, \url{arXiv:1203.5810}]. More precisely, let \(K\) be a number field, \(v\) a place, \(f:\mathbb A^n \to \mathbb A^n\) an affine regular automorphism, and \(x_m \in \mathbb A^n(\overline K)\) a sequence of \(f\)-periodic points that are generic (i.e. any infinite subsequence is Zariski-dense). Letting \(\mu_m\) be the discrete probability measure supported on the Galois orbit of \(x_m\), Theorem A shows that \(\mu_m\) converges weakly to an \(f\)-invariant probability measure on the \(v\)-adic Berkovich projective space.
The author actually introduces a generalization of affine regular automorphisms: a commuting pair \(S = \{f_1,f_2\}\) of polynomial self-maps on \(\mathbb A^n\) is \textit{strongly regular} if (1) \(f_i\)'s do not have a common indeterminacy point in \(\mathbb P^n\), (2) iterates \(f_i^m\) and \(f_1\circ f_2\) satisfy various degree conditions, and (3) \(f_i\)'s satisfy the height growth condition \(h(f_i(P))\gg h(P)\). The technical heart of the paper is Theorem 6.5: if \(S = \{f_1,f_2\}\) is strongly regular and in addition \(\deg f_1 = \deg f_2\), then a sequence of adelic metrics converges uniformly, say to \(\|\cdot \|_S\). To construct this metric, the author shows that \(\{f_1^m, f_2^m\}\) is also strongly regular (Corollary 6.4), so \((f_1^m, f_2^m)\) defines a morphism \(\phi_m: \mathbb P^n \rightarrow \mathbb P^{2n}\). Then the pullback via \(\phi_m\) of the \(v\)-adic sup norm on the degree-one line bundle is adelic for each \(m\). By extending the theory of Green functions and good reduction developed for affine regular automorphisms [\textit{S. Kawaguchi}, Algebra Number Theory 7, No. 5, 1225--1252 (2013; Zbl 1302.37067)] to the case of strongly regular pairs, the uniform convergence is proved.
As a result of Theorem 6.5, \textit{X. Yuan}'s theory [Invent. Math. 173, No. 3, 603--649 (2008; Zbl 1146.14016)] immediately implies that given a strongly regular pair \(S = \{f_1,f_2\}\) such that \(\deg f_1 = \deg f_2\) and a generic sequence \(x_m\) which is small with respect to \(\|\cdot \|_S\), the Galois orbits of \(x_m\)'s are equidistributed (Theorem B). As for an affine regular automorphism \(f\), Proposition 7.1 shows that there exist \(l_1\) and \(l_2\) such that \(\deg f^{l_1} = \deg f^{-l_2}\). The author also proves that the \(f\)-periodic points over \(\overline K\) are Zariski-dense (Theorem C), by adapting the proof for polarizable morphisms [\textit{N. Fakhruddin}, J. Ramanujan Math. Soc. 18, No. 2, 109--122 (2003; Zbl 1053.14025)]. From these results, Theorem A follows from Theorem B. equidistribution; affine regular automorphism; small point; periodic point; canonical height; Green function; adelic metric Fornæss, J.-E.: Dynamics in several complex variables. CBMS Regional Conference Series in Mathematics, vol. 87. American Mathematical Society, Providence, RI (1996) Heights, Applications to coding theory and cryptography of arithmetic geometry, Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables, Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps, Height functions; Green functions; invariant measures in arithmetic and non-Archimedean dynamical systems The equidistribution of small points for strongly regular pairs of polynomial maps | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Der Verf. untersucht die Eigenschaften der transzendenten Funktionen, welche auf der zerschnittenen \textit{Riemann}schen Fläche eindeutig und bis auf etwaige Pole stetig sind und überdies zu beiden Seiten der Querschnitte ein durch die folgenden beiden Gleichungen bestimmtes Verhalten zeigen:
\[
\left.\begin{aligned} \log \varphi (S_i z) & = \log \varphi (z) + \sum_j \alpha_{ij} u_j+ \beta_i, \\ \log \varphi (T_i z) & = \log \varphi (z) + \sum_j \gamma_{ij} u_j + \delta_i \end{aligned}\right\} \quad (i= 1,2, \dots, p).
\]
Hierin bedeutet \(S_i z, T_i z\) den Übergang von der einen Seite der \(2p\) Querschnitte auf die andere und \(u_1, u_2, \dots, u_p\) die \textit{Riemann}schen Normalintegrale erster Gattung. Die \(2p^2 + 2p\) Konstanten \(\alpha_{ij}, \beta_{ij}, \gamma_{ij}, \delta_i\) unterliegen \(p + 1\) linearen Gleichungen, sind aber sonst willkürlich. Solche Funktionen werden allgemein als Faktorfunktionen, wenn sie nur eine einfache Nullstelle und keine Unendlichkeitsstelle haben, als Primfunktionen, wenn sie gar keine Null- und Unendlichkeitsstelle haben, als akzidentelle Faktorfunktionen bezeichnet.
Es wird zunächst gezeigt, daß\ die angegebenen Funktionen stets aus Exponentialgrößen der Form
\[
e^{\int VdU}
\]
zusammengesetzt werden können, worin \(V\) ein Elementarintegral zweiter Gattung, \(U\) ein Integral erster Gattung bedeutet; tritt zu \(V\) ein Integral erster Gattung hinzu, so erhält die Funktion einen akzidentellen Faktor. Es werden sodann die wichtigsten Eigenschaften der Primfunktionen entwickelt und ihre Beziehungen zur \textit{Riemann}schen Theta-Funktion, zur \textit{Klein}schen Primform und zu der Transzendente \(T_{\xi \eta} (x_1, x_2, \dots, x_p)\) von \textit{Clebsch} und \textit{Gordan} dargelegt. Theta functions; Abelian integrals; prime function Theta functions and curves; Schottky problem Prime functions on a \textit{Riemann} surface. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors exhibit a relationship between twisted elliptic general for \(K3\) surfaces and the sporadic Mathieu group \(M_{24}\). elliptic genus; superconformal algebra; moonshine; Mathieu group; Jacobi form; mock theta function T. Eguchi and K. Hikami, \textit{Note on twisted elliptic genus of K}3 \textit{surface}, \textit{Phys. Lett.}\textbf{B 694} (2011) 446 [arXiv:1008.4924] [INSPIRE]. \(K3\) surfaces and Enriques surfaces, Elliptic genera, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Representations of sporadic groups Twisted elliptic genus for \(K3\) and Borcherds product | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For a smooth projective variety \(X\) over \(\mathbb C\), let \(H^*(X)\) and \(\operatorname{CH}^*(X)\) be the cohomology ring and the Chow ring with coefficients in \(\mathbb Q\), respectively. The grading operator \(h \colon H^*(X) \rightarrow H^*(X)\) is given by multiplication with \(k-n\) in degree \(k\). For an element \(a \in H^2(X)\) such that \(e_a^k \colon H^{n-k}(X) \rightarrow H^{n+k}(X)\) is an isomorphim for all \(1 \leq k \leq n\), where \(e_a\) denotes the cup product with \(a\), there exists a unique operator \(f_a\) such that \([e_a,f_a] = h\). The Néron-Severi Lie algebra of \(X\) is the Lie algebra \(\mathfrak g_{\mathrm{NS}}(X)\) generated by the so-called Lefschetz triples \((e_a,f_a,h)\) with \(a \in H^{1,1}(X,\mathbb Q)\).
In the present paper, the author conjectures (Conjecture 1.2) that if \(X\) is a hyperkähler variety of \(K3^{[2]}\)-type, then there exists a map \(\varphi\colon \mathfrak g_{\mathrm{NS}}(X) \rightarrow \operatorname{CH}^*(X \times X)\) lifting the natural inclusion \(\mathfrak g_{\mathrm{NS}}(X) \hookrightarrow \operatorname{End}_\mathbb Q H^*(X)\) through the cycle class map \(\operatorname{CH}^*(X \times X) \rightarrow \operatorname{End}_\mathbb Q H^*(X)\). The map \(\varphi\) is given explicitly by \(\varphi(e_a) = \Delta_*(a)\), \(\varphi(f_a) = F_a\) and \(\varphi(h) = H\), where \(\Delta \colon X \rightarrow X \times X\) is the diagonal embedding and \(F_a \in \operatorname{CH}^3(X \times X)\), \(H \in \operatorname{CH}^4(X \times X)\) are defined using Markman's canonical lift \(L \in \operatorname{CH}^2(X \times X)\) of the cohomology class \(\mathfrak B\) associated to the Beauville-Bogomolov bilinear form on \(H^2(X)\) [\textit{E. Markman}, J. Algebr. Geom. 29, No. 2, 199--245 (2020; Zbl 1439.14123)].
The conjecture was proven by [\textit{G. Oberdieck}, Comment. Math. Helv. 96, No. 1, 65--77 (2021; Zbl 1462.14040)] in the case that \(X = S^{[2]}\), where \(S\) is a projective K3 surface. The conjecture is re-proven in the present paper by showing that the given lifts \(F_a, H\) agree with the lifts provided by Oberdieck in terms of Nakajima operators (Proposition 1.1, Proposition 3.21). A weaker form of the conjecture is proven in the case that \(X\) is the Fano variety of lines of a smooth cubic fourfold.
The author then moves on to prove that two decompositions of \(\operatorname{CH}^*(X)\) agree in the case that \(X\) is a hyperkähler variety of \(K3^{[2]}\)-type with a lift \(L\) of \(\mathfrak B\) satisfying certain relations (Theorem 1.7). The first decomposition is the eigenspace decomposition with respect to the operator \(H_* \in \operatorname{End}_\mathbb Q(\operatorname{CH}^*(X)\), while the second decomposition is the Fourier decomposition given by the correspondence \(e^L \in \operatorname{CH}^*(X \times X)\) studied by [\textit{M. Shen} and \textit{C. Vial}, The Fourier transform for certain hyperkähler fourfolds. Providence, RI: American Mathematical Society (AMS) (2016; Zbl 1386.14025)]. Chow ring; hyper-Kähler manifold; Hilbert scheme of points; Fano variety of lines Holomorphic symplectic varieties, hyper-Kähler varieties, (Equivariant) Chow groups and rings; motives, Algebraic cycles, \(4\)-folds The Chow ring of hyperkähler varieties of \(K3^{[2]}\)-type via Lefschetz actions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We present a new effective Nullstellensatz with bounds for the degrees which depend not only on the number of variables and on the degrees of the input polynomials but also on an additional parameter called the geometric degree of the system of equations. The obtained bound is polynomial in these parameters. It is essentially optimal in the general case, and it substantially improves the existent bounds in some special cases.
The proof of this result is combinatorial, and relies on global estimates for the Hilbert function of homogeneous polynomial ideals. In this direction, we obtain a lower bound for the Hilbert function of an arbitrary homogeneous polynomial ideal, and an upper bound for the Hilbert function of a generic hypersurface section of an unmixed radical polynomial ideal. effective Nullstellensatz; geometric degree of the system of equations; Hilbert function M. Sombra, ''Bounds for the Hilbert function of polynomial ideals and for the degrees in the Nullstellensatz,'' \textit{J. Pure Appl. Algebra}, \textbf{117/118}, 565-599 (1996). Relevant commutative algebra, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Symbolic computation and algebraic computation Bounds for the Hilbert function of polynomial ideals and for the degrees in the Nullstellensatz | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Generalized algebraic-geometry codes are linear codes which generalize the well-known geometric Goppa codes, since their construction makes use of places which are not necessarily of degree one. In this paper authors determine the \(n\)-automorphism group of generalized algebraic-geometry codes associated with rational, elliptic and hyperelliptic function fields. Such group is, up to isomorphism, a subgroup of the automorphism group of the underlying function field. Further they calculate some examples of \(q\)-ary linear codes for \(q=5\), 7 in the hyperelliptic case. These examples show that it is possible to construct generalized algebraic-geometry codes with a nontrivial \(n\)-automorphism group. geometric Goppa codes; Finite fields; algebraic function fields; generalized algebraic-geometry codes Picone, A.; Spera, A. G.: Automorphisms of hyperelliptic GAG-codes. Discrete math. 309, No. 2, 328-340 (2009) Geometric methods (including applications of algebraic geometry) applied to coding theory, Algebraic coding theory; cryptography (number-theoretic aspects), Applications to coding theory and cryptography of arithmetic geometry Automorphisms of hyperelliptic GAG-codes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors carry out calculations of the local Donaldson-Thomas theory of curves, which refers to all Donaldson-Thomas invariants on the total space of a rank two bundle over a curve relative to surfaces determined by the fibers over fixed points in the curve. By comparing these results to those obtained by \textit{J. Bryan} and \textit{R. Pandharipande} [J. Am. Math. Soc. 21, No. 1, 101--136 (2008; Zbl 1126.14062)] on the local Gromov-Witten theory of curves, they establish the Gromov-Witten/Donaldson-Thomas correspondence for the local theory of curves. This provides a rich class of new examples where the GW/DT correspondence is verified, and these techniques are likely to play a basic role in the general proof of the GW/DT correspondence for 3-folds. The calculations are achieved by localization and degeneration methods, and the results relate both the Gromov-Witten and Donaldson-Thomas invariants to the quantum cohomology of the Hilbert scheme of points in the plane. Gromov-Witten; Donaldson-Thomas; Hilbert scheme A. Okounkov and R. Pandharipande, \textit{The local Donaldson-Thomas theory of curves}, math.AG/0512573. Enumerative problems (combinatorial problems) in algebraic geometry, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) The local Donaldson-Thomas theory of curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a smooth complex projective variety of dimension \(n \geq 2\) and an ample base point free linear system \(|V|\) on \(X\). A \textit{bad zero-scheme} for the pair \((X,V)\) is defined by the condition that the sublinear system of elements of \(|V|\) by it contains only reducible or non--reduced elements. In particular one can fix the length of the zero-scheme, say \(t\), and focus on elements of the Hilbert scheme of zero-schemes of length \(t\) of a particular form, for example reduced. This leads the authors to the definition of the \textit{(reduced) \(t\)-bad locus} of the pair \((X,V)\). A count on dimensions on the incidence correspondence correspondence defining these loci provides a bound on the dimension of the reduced \(b_0\)-bad locus, being \(b_0\) the minimal length for which it is non-empty. The bound is reached only when \(n=2\) (see Thm. 3.4) and a description of these surfaces is provided (in particular a classification if \(|V|\) is very ample). Part of these results are extended in Section 4 to the non-reduced case while in Section 5 a new invariant \(s\) is introduced and studied. A library of examples is provided in Section 6. linear systems; reducible or non-reduced divisors; bad loci Divisors, linear systems, invertible sheaves, Special surfaces The variety of bad zero-schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems After introducing Tropical Algebraic Varieties, we give a polyhedral description of tropical hypersurfaces. Using TOPCOM and GAP, we show that there exist 59 types of two dimensional tropical quadric surfaces. We also show a criterion for a quadric hypersurface to be non-degenerate in terms of a tropical rank. tropical algebraic variety; tropical rank; regular subdivision Real algebraic and real-analytic geometry, Polytopes and polyhedra On tropical quadric surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(K/k\) be a field extension of transcendence degree 1 such that \(k\) is algebraically closed in \(k\). A prime divisor \(P\) is a discrete valuation ring of rank 1 on \(K\) denoted as a function \(v_ P\). The collection of all such \(P\) is the Riemannian surface of \(K/k\), \(R(K/k)\) or \(R(K)\). A function \(x \in K\) has \(P\) as a pole of order \(v_ P(x)\) if \(v_ P(x) < 0\). In this paper the author examines the question: ``For a finite sequence of negative natural numbers \(n_ 1,n_ 2, \dots, n_ t\) and a sequence of prime divisors \(P_ 1, \dots, P_ t\) of \(K\) with \(v_{P_ i} (x_ i) = n_ i\), \(x_ i \in K\), is it possible to find a function \(x \in K\) such that \(v_{P_ i} (x-x_ i) \geq 0\), for \(i = 1,2, \dots, t\), and \(v_ P(x) \geq 0\) for all other \(P \in R(K)\)?'' The answer is positive for a rational function field, but not true in general.
The author shows that the deviation of an algebraic function field of one variable from a rational function field, relative to the above question, is exactly the invariant of \(K/k\) which one usually calls the ``genus'' of \(K/k\). The author also introduces two other invariants of \(K\) and discusses some classical results from the theory of algebraic function fields. genus; prime divisor; discrete valuation ring of rank 1; algebraic function field of one variable; invariants Arithmetic theory of algebraic function fields, Algebraic functions and function fields in algebraic geometry On the genus of an algebraic function field | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Here we describe the recent development of study of degenerate families of Riemann surfaces. This field is located at the boundary area where topology, algebraic geometry, complex analysis and other fields overlap.
In \S1, we sketch the brief history from the birth of Kodaira's theory to the present studies, where our two main frameworks are monodromy and Morsification. The topics in \S\S 2 and 3 are concerned with the localization of the signature. In particular, mainly describe in \S2 the topological approaches which are related to the Meyer function, and describe in \S3 the algebro-geometric approaches which are related to algebraic surfaces of general type. In \S4, we sketch the recent topics of Lefschetz fibrations from the topological viewpoint. Kodaira's theory; Meyer function; algebraic surfaces of general type; Lefschetz fibrations Ashikaga, T.; Endo, H.: Various aspects of degenerate families of Riemann surfaces, Sūgaku expo. 19, No. 2, 171-196 (2006) Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Families, moduli of curves (analytic) Various aspects of degenerate families of Riemann surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We classify coherent modules on \(k [x, y]\) of length at most 4 and supported at the origin. We compare our calculation with the motivic class of the moduli stack parametrizing such modules, extracted from the Feit-Fine formula. We observe that the natural torus action on this stack has finitely many fixed points, corresponding to connected skew Ferrers diagrams. coherent sheaves; finite length modules; Grothendieck ring of varieties; Hilbert scheme of points; torus actions Stacks and moduli problems, Parametrization (Chow and Hilbert schemes) On coherent sheaves of small length on the affine plane | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A new trend in algebraic independence theory, initiated by an idea of A. Durand in 1977 -- see [\textit{M. Langevin} (ed.) and \textit{M. Waldschmidt} (ed.), Cinquante ans de polynômes. Fifty years of polynomials. Lecture Notes in Mathematics, 1415. Berlin etc.: Springer-Verlag (1990; Zbl 0683.00011)], relates the transcendence degree of a field generated by complex numbers with simultaneous approximation measures for the generators. See for instance the author's papers [J. Number Theory 64, No. 2, 291--338 (1997; Zbl 0901.11026)] and chap.~4,6,8 of [\textit{Yu. V. Nesterenko} (ed.) and \textit{P. Philippon} (ed.), Introduction to algebraic independence theory. Lecture Notes in Mathematics. 1752. Berlin: Springer (2001; Zbl 0966.11032)]. To a large extent, this connection is still a conjectural one, apart from the case of small dimension or small codimension where the author proved the expected statements [\textit{P. Philippon}, J. Number Theory 81, No. 2, 234--253 (2000; Zbl 1096.11504)]. Here, he settles in any dimension or codimension the analogous problem in the case where the field of complex algebraic numbers is replaced by the algebraic closure of a field \(k(z)\) of rational fractions over an algebraically closed field \(k\). From his main result he derives transfer lemmas which have applications in the study of multiplicity estimates. approximation; function field; transfer lemma; algebraic independence; Hilbert function DOI: 10.1142/S1793042111004502 Approximation in non-Archimedean valuations, Algebraic functions and function fields in algebraic geometry Functional approximations of curves in projective space | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The Galois cohomology group \(H^ 3(K, {\mathbb{Q}}/{\mathbb{Z}}(2))\) is studied for an algebraic function field K in one variable over an algebraic number field F. A description of \(H^ 3\)(K, \({\mathbb{Q}}/{\mathbb{Z}}(2))\) analogous to the classical description of Br(F) is given. Hasse principles for \(H^ 3(K, {\mathbb{Q}}/{\mathbb{Z}}(2))\) are proved; for example, it is shown that the map \(H^ 3(K, {\mathbb{Q}}/{\mathbb{Z}}(2))\to \prod_{v}H^ 3(K_ v, {\mathbb{Q}}/{\mathbb{Z}}(2))\) is injective where v ranges over all places of F and \(K_ v\) denotes the composite field \(K\cdot F_ v\). The Hasse principles for \(H^ 3(K, {\mathbb{Q}}/{\mathbb{Z}}(2))\) imply the local-global principles for reduced norms of certain division algebras and for certain quadratic forms over K.
In the Appendix by Colliot-Thélène, it is shown that a sum of squares in F(X,Y) is written as a sum of 8 squares. two dimensional global fields; algebraic function field in one; variable over algebraic number field; Galois cohomology group; \(H^ 3\); Hasse principles; local-global principles; reduced norms; division algebras; quadratic forms; sum of squares K.~Kato, {A {H}asse principle for two dimensional global fields. With an appendix by {J}.-{L} {C}olliot-{T}hélène.}, J. Reine Angew. Math. {366} (1986), 142--180. DOI 10.1515/crll.1986.366.142; zbl 0576.12012; MR0833016 Galois cohomology, Brauer groups of schemes, Quadratic forms over global rings and fields, Galois cohomology, Quaternion and other division algebras: arithmetic, zeta functions, Waring's problem and variants, Arithmetic theory of algebraic function fields A Hasse principle for two dimensional global fields. Appendix by Jean-Louis Colliot-Thélène | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This is an exhaustive study of the action of the complex conjugation on complex algebraic curves that are defined by real polynomials. The complex conjugation defines an involution of these real curves. This involution defines an action on the symmetric powers of the curve and on the Picard scheme of the curve. The authors study that action and apply it to real theta-characteristics. They show as well how the topological invariants of a real curve X are determined by the action of the complex conjugation on the group \(H_ 1(X({\mathbb{C}}),{\mathbb{Z}}/2)\). This question was also considered by \textit{H. Jaffee} [Topology 19, 81-87 (1980; Zbl 0426.14013)].
Real hyperelliptic curves, real plane curves and real trigonal curves are considered as examples of the general theory. A topological argument leads to an interesting observation: entire components of the real moduli contain no hyperelliptic curves once the genus is at least 4. The paper ends with remarks on real moduli and with a real form of the Torelli theorem which was also proved independently by \textit{R. Silhol} [see e.g. Math. Z. 181, 345-364 (1982; Zbl 0492.14015)]. real theta-characteristics; real abelian varieties; complex conjugation on complex algebraic curves; real curves; Picard scheme; Real hyperelliptic curves; real plane curves; real trigonal curves; real moduli Gross, B. H.; Harris, J., Real algebraic curves, Ann. Sci. École Norm. Sup. (4), 14, 2, 157-182, (1981) Families, moduli of curves (algebraic), Arithmetic ground fields for curves, Real algebraic and real-analytic geometry, Jacobians, Prym varieties, Arithmetic ground fields for abelian varieties Real algebraic curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper is concerned with three local zeta functions: the Igusa zeta function and the topological and motivic zeta functions associated to an analytic function germ \(f\) in two variables.
It was known that each of these three zeta functions have at most a double pole, given by the log canonical threshold of the corresponding plane curve singularity. When the germ \(f\) is reduced, Loeser showed that such a double pole always creates a Jordan block of size 2 in the monodromy of \(f\).
The authors discuss this phenomenon in the case of a non-reduced germ \(f\) and obtain new information on the Bernstein-Sato polynomial in this situation, confirming a conjecture of Igusa, Denef and Loeser. Igusa and topological zeta function; Bernstein-Sato polynomial; monodromy; log canonical threshold; plane curve singularities Melle-Hernández, A.; Torrelli, T.; Veys, W.: Monodromy Jordan blocks, b-functions and poles of zeta functions for germs of plane curves. J. algebra 324, No. 6, 1364-1382 (2010) Singularities in algebraic geometry, Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Singularities of curves, local rings, Complex surface and hypersurface singularities Monodromy Jordan blocks, \(b\)-functions and poles of zeta functions for germs of plane curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Closed loop solitons in a plane, whose curvatures obey the modified Korteweg-de Vries equation, are investigated. It is shown that their tangential vectors are expressed by a ratio of Weierstrass sigma functions in the genus one case and a ratio of Baker's hyperelliptic sigma functions in the genus two case. This study is closely related to classical and quantized elastica problems. hyperelliptic function; space curve; elastica; modified Korteweg-de Vries equation; Weierstrass sigma functions; Baker's hyperelliptic sigma functions Matsutani, S.: Closed loop solitons and sigma functions: classical and quantized elasticas with genera one and two. J. geom. Phys. 39, 50-61 (2001) KdV equations (Korteweg-de Vries equations), Curves in Euclidean and related spaces, Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, Special algebraic curves and curves of low genus Closed loop solitons and sigma functions: Classical and quantized elasticas with genera one and two | 0 |
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