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2,200 | On the Selberg class of Dirichlet series: small weights | math.NT | In the study of Dirichlet series with arithmetic significance there has
appeared (through the study of known examples) certain expectations, namely (i)
if a functional equation and Euler product exists, then it is likely that a
type of Riemann hypothesis will hold, (ii) that if in addition the function has
a simple pol... | math |
2,201 | Algorithms in algebraic number theory | math.NT | In this paper we discuss the basic problems of algorithmic algebraic number
theory. The emphasis is on aspects that are of interest from a purely
mathematical point of view, and practical issues are largely disregarded. We
describe what has been done and, more importantly, what remains to be done in
the area. We hope t... | math |
2,202 | Additive functions on shifted primes | math.NT | Best possible bounds are obtained for the concentration function of an
additive arithmetic function on sequences of shifted primes. | math |
2,203 | Selberg's Conjectures and Artin $L$-functions | math.NT | The author reviews results and conjectures of Selberg on a class of Dirichlet
series functions which share properties with the Riemann zeta function, and he
relates this work to the theory of Artin L-functions. | math |
2,204 | A report on Wiles' Cambridge lectures | math.NT | In lectures at the Newton Institute in June of 1993, Andrew Wiles announced a
proof of a large part of the Taniyama-Shimura Conjecture and, as a consequence,
Fermat's Last Theorem. This report for nonexperts discusses the mathematics
involved in Wiles' lectures, including the necessary background and the
mathematical h... | math |
2,205 | Mean values of Dedekind sums | math.NT | For a positive integer k and an arbitrary integer h, the Dedekind sum s(h,k)
was first studied by Dedekind because of the prominent role it plays in the
transformation theory of the Dedekind eta-function, which is a modular form of
weight 1/2 for the full modular group SL_2(Z). There is an extensive literature
about th... | math |
2,206 | Zeta functions do not determine class numbers | math.NT | We show that two number fields with the same zeta function, and even with
isomorphic adele rings, do not necessarily have the same class number. | math |
2,207 | The Dimension of the Space of Cusp Forms of Weight One | math.NT | A new upper bound is given for the dimension of the space of holomorphic cusp
forms of weight one and prime level $q$: $$ \hbox{dim}\, S_1(q) << q^{11/12}
\log^4{q} $$ with an absolute implied constant. | math |
2,208 | On Calculations of Zeros of Various L-functions | math.NT | As we have shown several years ago [Y2], zeros of $L(s, \Delta )$ and
$L^(2)(s, \Delta )$ can be calculated quite efficiently by a certain
experimental method. Here $\Delta$ denotes the cusp form of weight 12 with
respect to SL$(2, Z)$ and $L(s, \Delta )$ (resp. $L^(2)(s, \Delta )$) denotes
the standard (resp. symmetri... | math |
2,209 | Conformal Characters and Theta Series | math.NT | We describe the construction of vector valued modular forms transforming
under a given congruence representation of the modular group SL$(\bold Z)$ in
terms of theta series. We apply this general setup to obtain closed and easily
computable formulas for conformal characters of rational models of
$W$-algebras. | math |
2,210 | Mean Values of the Logarithmic Derivative of the zeta Function and the GUE Hypothesis | math.NT | The GUE Hypothesis, which concerns the distribution of zeros of the Riemann
zeta-function, is used to evaluate some integrals involving the logarithmic
derivative of the zeta-function. Some connections are shown between the GUE
Hypothesis and other conjectures. | math |
2,211 | Generalized Kummer congruences and $p$-adic families of motives | math.NT | We describe some new general constructions of $p$-adic $L$-functions attached
to certain arithmetically defined complex $L$-functions coming from motives
over $\bold Q$ with coefficiens in a number field $T$, with $[T:\bold
Q]<\infty$. These constructions are equivalent to proving some generalized
Kummer congruences fo... | math |
2,212 | Galois representations and modular forms | math.NT | In this article, I discuss material which is related to the recent proof of
Fermat's Last Theorem: elliptic curves, modular forms, Galois representations
and their deformations, Frey's construction, and the conjectures of Serre and
of Taniyama--Shimura. | math |
2,213 | Families of curves and weight distributions of codes | math.NT | In this expository paper we show how one can, in a uniform way, calculate the
weight distributions of some well-known binary cyclic codes. The codes are
related to certain families of curves, and the weight distributions are related
to the distribution of the number of rational points on the curves. | math |
2,214 | Polynomials involving the floor function | math.NT | Some identities are presented that generalize the formula x^3 = 3x floor(x
floor(x)) - 3 floor(x) floor(x floor(x)) + floor(x)^3 + 3 frac(x) frac(x
floor(x)) + frac(x)^3 to a representation of the product x_0x_1 ... x_{n-1}. | math |
2,215 | On the Mumford-Tate conjecture for abelian varieties with reduction conditions | math.NT | We study monodromy action on abelian varieties satisfying certain bad
reduction conditions. These conditions allow us to get some control over the
Galois image. As a consequence we verify the Mumford--Tate conjecture for such
abelian varieties. | math |
2,216 | Torsion in Rank-1 Drinfeld Modules and the Uniform Boundedness Conjecture | math.NT | It is conjectured that for fixed $A$, $r \ge 1$, and $d \ge 1$, there is a
uniform bound on the size of the torsion submodule of a Drinfeld $A$-module of
rank $r$ over a degree $d$ extension $L$ of the fraction field $K$ of $A$. We
verify the conjecture for $r=1$, and more generally for Drinfeld modules having
potentia... | math |
2,217 | On the central critical value of the triple product L-function | math.NT | We compute the central critical value of the triple product $L$-function
associated to three cusp forms $f_1,f_2,f_3$ with trivial character for groups
$\Gamma_0(N_i)$ with square free levels $N_i$ not all of which are $1$ and
weights $k_i$ satisfying $k_1\ge k_2\ge k_3$ and $k_1<k_2+k_3$. This
generalizes work of Gros... | math |
2,218 | On the equation $a^p + 2^alpha b^p + c^p =0$ | math.NT | We discuss the equation $a^p + 2^\a b^p + c^p =0$ in which $a$, $b$, and $c$
are non-zero relatively prime integers, $p$ is an odd prime number, and $\a$ is
a positive integer. The technique used to prove Fermat's Last Theorem shows
that the equation has no solutions with $\a>1$ or $b$ even. When $\a=1$ and $b$
is odd,... | math |
2,219 | Fractional Power Series and Pairings on Drinfeld Modules | math.NT | Let $C$ be an algebraically closed field containing the finite field $F_q$
and complete with respect to an absolute value $|\;|$. We prove that under
suitable constraints on the coefficients, the series $f(z) = \sum_{n \in \Z}
a_n z^{q^n}$ converges to a surjective, open, continuous $F_q$-linear
homomorphism $C \righta... | math |
2,220 | Cycles of Quadratic Polynomials and Rational Points on a Genus-Two Curve | math.NT | It has been conjectured that for $N$ sufficiently large, there are no
quadratic polynomials in $\bold Q[z]$ with rational periodic points of period
$N$. Morton proved there were none with $N=4$, by showing that the genus~$2$
algebraic curve that classifies periodic points of period~4 is birational to
$X_1(16)$, whose r... | math |
2,221 | A note on the fourth power moment of the Riemann zeta-function | math.NT | We give explicit formulae for all of the terms in the asymptotic expansion of
the mean fourth power of the Riemann zeta-function on the critical line. | math |
2,222 | Siegel Modular Forms and Theta Series attached to quaternion algebras II | math.NT | We continue our study of Yoshida's lifting, which associates to a pair of
automorphic forms on the adelic multiplicative group of a quaternion algebra a
Siegel modular form of degree 2. We consider here the case that the automorphic
forms on the quaternion algebra correspond to modular forms of arbitrary even
weights a... | math |
2,223 | On vanishing sums for roots of unity | math.NT | Consider the $m$-th roots of unity in {\bf C}, where $m>0$ is an integer. We
address the following question: For what values of $n$ can one find $n$ such
$m$-th roots of unity (with repetitions allowed) adding up to zero? We prove
that the answer is exactly the set of linear combinations with non-negative
integer coeff... | math |
2,224 | A Truncated Integral of the Poisson Summation Formula | math.NT | Let $G$ be a reductive algebraic group defined over $\bQ$, with anisotropic
centre. Given a rational action of $G$ on a finite-dimensional vector space
$V$, we analyze the truncated integral of the theta series corresponding to a
Schwartz-Bruhat function on $V(\bA)$. The Poisson summation formula then yields
an identit... | math |
2,225 | Modular forms on GL(3) and Galois representations | math.NT | This paper gives an expository account of our experiments concerning
relations between modular forms for congruence subgroups of SL(3,Z) and three
dimensional Galois representations. The main new result presented here is a
calculation of the variations of the Hodge structure corresponding to the
motives we consider in ... | math |
2,226 | Nevanlinna Theory and Rational Points | math.NT | S. Lang conjectured in 1974 that a hyperbolic algebraic variety defined over
a number field has only finitely many rational points, and its analogue over
function fields. We discuss the Nevanlinna-Cartan theory over function fields
of arbitrary dimension and apply it for Diophantine property of hyperbolic
projective hy... | math |
2,227 | On Holomorphic Curves in Semi-Abelian Varieties | math.NT | The algebraic degeneracy of holomorphic curves in a semi-Abelian variety
omitting a divisor is proved (Lang's conjecture generalized to semi-Abelian
varieties) by making use of the {\it jet-projection method} and the logarithmic
Wronskian jet differential after Siu-Yeung. We also prove a structure theorem
for the locus... | math |
2,228 | On vanishing sums of $\,m\,$th roots of unity in finite fields | math.NT | In an earlier work, the authors have determined all possible weights $n$ for
which there exists a vanishing sum $\zeta_1+\cdots +\zeta_n=0$ of $m$th roots
of unity $\zeta_i$ in characteristic 0. In this paper, the same problem is
studied in finite fields of characteristic $p$. For given $m$ and $p$, results
are obtaine... | math |
2,229 | The distribution of the eigenvalues of Hecke operators | math.NT | For each prime $p$, we determine the distribution of the $p^{th}$ Fourier
coefficients of the Hecke eigenforms of large weight for the full modular
group. As $p\to\infty$, this distribution tends to the Sato--Tate distribution. | math |
2,230 | p-adic measures and square roots of triple product L-functions | math.NT | Let p be a prime number, and let f, g, and h be three modular forms of
weights $\kappa$, $\lambda$, and $\mu$ for $SL(2,\Bbb{Z})$. We suppose $\kappa
\geq \lambda + \mu$. In joint work with Kudla, one of the authors obtained a
formula for the normalized {\it square root} of the value at $s = {1/2}(\kappa
+ \lambda + \m... | math |
2,231 | Fourier coefficients of half-integral weight modular forms modulo ell | math.NT | For each prime $\ell$, let $|\cdot|_\ell$ be an extension to $\bar \Q$ of the
usual $\ell$-adic absolute value on $\Q$. Suppose $g(z) = \sum_{n=0}^\infty
c(n)q^n \in M_{k+\half}(N)$ is an eigenform whose Fourier coefficients are
algebraic integers. Under a mild condition, for all but finitely many primes
$\ell$ there a... | math |
2,232 | Small generators of number fields | math.NT | This is a revised version of ANT-0045.
If K is a number field of degree n with discriminant D, if K=Q(a) then
H(a)>c(n)|D|^(1/(2n-2)) where H(a) is the height of the minimal polynomial of
a. We ask if one can always find a generator a of K such that
d(n)|D|^(1/(2n-2))>H(a) holds. The answer is yes for real quadratic ... | math |
2,233 | On a Refined Stark Conjecture for Function Fields | math.NT | We prove that a refinement of Stark's Conjecture formulated by Rubin is true
up to primes dividing the order of the Galois group, for finite, abelian
extensions of function fields over finite fields. We also show that in the case
of constant field extensions a statement stronger than Rubin's holds true. | math |
2,234 | Gras-Type Conjectures for Function Fields | math.NT | Based on results obtained in a companion paper [MSRI preprint 1997-002], we
construct groups of special $S$--units for function fields of characteristic
$p>0$, and show that they satisfy Gras--type Conjectures. We use these results
in order to give a new proof of Chinburg's $\Omega_3$--Conjecture on the Galois
module s... | math |
2,235 | The Frobenius and monodromy operators for curves and abelian varieties | math.NT | In this paper, we give explicit descriptions of Hyodo and Kato's Frobenius
and Monodromy operators on the first $p$-adic de Rham cohomology groups of
curves and Abelian varieties with semi-stable reduction over local fields of
mixed characteristic. This paper was motivated by the first author's paper "A
$p$-adic Shimur... | math |
2,236 | On weights of $l$-adic representation | math.NT | Let $J$ be an abelian variety over a number field such that the center of its
endomorphism ring is equal to the ring of integers. If the endomorphism ring
splits at a prime number $l$, then the $l$-adic representation is defined by
the minuscule weights (microweights) of simple classical Lie algebras. | math |
2,237 | Characteristic p Galois representations that are produced by Drinfeld | math.NT | We examine which representations of the absolute Galois group of a field of
finite characteristic with image over a finite field of the same characteristic
may be constructed by the Galois group's action on the division points of an
appropriate Drinfeld module. | math |
2,238 | Explicit exponential maps for Hecke characters at ordinary primes | math.NT | In this paper we establish the explicit exponential map for the Galois
representation from a Hecke character at an ordinary prime. Such explicit maps
are important in verifying the Bloch-Kato conjecture for Hecke characters. | math |
2,239 | On Bloch-Kato's Tamagawa number conjecture for Hecke characters of imaginary quadratic number fields | math.NT | This is essentially the author's thesis submited to The University of Chicago
(May 1997).
I prove the validity of Tamagawa number conjecture of Bloch-Kato for certain
Hecke characters. I study the exponential map and local Tamagawa number for all
odd primes (both ordinary and supersingular), using Kato's explicit rec... | math |
2,240 | Sur le rang de J_0(q) | math.NT | In this paper, we prove an unconditionnal bound for the analytic rank (i.e
the order of vanishing at the critical point of the $L$ function) of the new
part $J^n_0(q)$, of the jacobian of the modular curve $X_0(q)$. Our main
resultis the following upper bound: for $q$ prime, one has
$$rank_a(J_0^n(q))\ll \dim J_0^n(q)$... | math |
2,241 | Sur les zeros des fonctions L automorphes | math.NT | We study, on average over f, zeros of the L-functions of primitive weight two
forms of level q (fixed). We prove, on the one hand, density theorems for the
zeros (similar to the results of Bombieri, Jutila, Motohashi, Selberg in the
case of characters), which are applied in \cite{KM} to obtain a sharp
unconditionnal es... | math |
2,242 | An estimate for the multiplicity of binary recurrences | math.NT | In this paper we improve drastically the estimate for the multiplicity of a
binary recurrence. The main contribution comes from an effective version of the
Faltings' Product Theorem. | math |
2,243 | Formal groups, elliptic curves, and some theorems of Couveignes | math.NT | The formal group law of an elliptic curve has seen recent applications to
computational algebraic geometry in the work of Couveignes to compute the order
of an elliptic curve over finite fields of small characteristic. The purpose of
this paper is to explain in an elementary way how to associate a formal group
law to a... | math |
2,244 | Elementary background in elliptic curves | math.NT | This paper gives additional background in algebraic geometry as an
accompaniment to the article, ``Formal Groups, Elliptic Curves, and some
Theorems of Couveignes'' [arXiv:math.NT/9708215]. Section 1 discusses the
addition law on elliptic curves, and Sections 2 and 3 explain about function
fields, uniformizers, and pow... | math |
2,245 | Variations of Hodge-de Rham structure and elliptic modular units | math.NT | This is a revised version of ANT-0049.
Given an elliptic curve E --> B over a base B with zero section i, we denote,
letting E':= E - i(B), by L(E) the Q-vector space with basis ({s}, s \in
E'(B)). Assume that B is smooth and separated over a field of characteristic 0.
On the lowest step, the weak version of the elli... | math |
2,246 | Elliptic curves of high rank over function fields | math.NT | By the Mordell-Weil theorem the group of Q(z)-rational points of an elliptic
curve is finitely generated. It is not known whether the rank of this group can
get arbitrary large as the curve varies. Mestre and Nagao have constructed
examples of elliptic curves E with rank at least 13.
In this paper a method is explain... | math |
2,247 | Separability, multi-valued operators, and zeroes of L-functions | math.NT | Let $\k$ be a global function field in 1-variable over a finite extension of
$\Fp$, $p$ prime, $\infty$ a fixed place of $\k$, and $\A$ the ring of
functions of $\k$ regular outside of $\infty$. Let $E$ be a Drinfeld module or
$T$-module. Then, as in \cite{go1}, one can construct associated characteristic
$p$ $L$-funct... | math |
2,248 | Remarks on normal bases | math.NT | We prove that any Galois extension of commutative rings with normal basis and
abelian Galois group of odd order has a self dual normal basis. Also we show
that if S/R is an unramified extension of number rings with Galois group of odd
order and $R$ is totally real then the normal basis does not exist for S/R. | math |
2,249 | Torsion points on y^2=x^6+1 | math.NT | Let C be the curve y^2=x^6+1 of genus 2 over a field of characteristic zero.
Consider C embedded in its Jacobian J by sending one of the points at infinity
on C to the origin of J. In this brief note we show that the points of C whose
image on J are torsion are precisely the two points at infinity and the six
points wi... | math |
2,250 | A double complex for computing the sign-cohomology of the universal ordinary distribution | math.NT | We give a new method for solving a problem originally solved about 20 years
ago by Sinnott and Kubert, namely that of computing the cohomology of the
universal ordinary distribution with respect to the action of the two-element
group generated by complex conjugation. We develop the method in sufficient
generality so as... | math |
2,251 | Computing all S-integral points on elliptic curves | math.NT | In this note we combine the advantages of the methods of Siegel-Baker-Coates
and of Lang-Zagier for the computation of S-integral points on elliptic curves
in Weierstrass normal form over the rationals. In this way we are able to
overcome the absence of an explicit lower bound for linear forms in q-adic
elliptic logari... | math |
2,252 | Formes modulaires p-adiques | math.NT | This is the text of a talk to the study week on \emph{Modular forms and
Galois representations} held in Luminy, 1997. We give a survey of $p$-adic
modular forms, as developped by Serre, Katz, Hida, Wiles, Coleman and others... | math |
2,253 | Quadratic minima and modular forms | math.NT | We give upper bounds on the size of the gap between the constant term and the
next non-zero Fourier coefficient of an entire modular form of given weight for
\Gamma_0(2). Numerical evidence indicates that a sharper bound holds for the
weights h \equiv 2 . We derive upper bounds for the minimum positive integer
represen... | math |
2,254 | Corestriction Principle in non-abelian Galois cohomology | math.NT | This is a revision of a McMaster University preprint, with extension.
In this paper we prove that over local or global fields of characteristic 0,
the Corestriction Principle holds for kernel and image of all maps which are
connecting maps in group cohomology and the groups of $R$-equivalences. Some
related questions... | math |
2,255 | The Riemann Hypothesis for the Goss zeta function for F_q[T] | math.NT | Let q be a power of a prime p. We prove an assertion of Carlitz which takes q
as parameter. Diaz-Vargas' proof of the Riemann Hypothesis for the Goss zeta
function for F_p[T] depends on his verification of Carlitz's assertion for the
specific case q = p. Our proof of the general case allows us to extend
Diaz-Vargas' pr... | math |
2,256 | Economical numbers | math.NT | A number $n$ is said to be economical if the prime power factorisation of $n$
can be written with no more digits than $n$ itself. We show that under a
plausible hypothesis, related to the twin prime conjecture, there are
arbitrarily long sequences of consecutive economial numbers, and exhibit such a
sequence of length ... | math |
2,257 | On Atkin-Lehner quotients of Shimura curves | math.NT | Poonen and Stoll have shown that the reduced Shafarevich-Tate group of a
principally polarized abelian variety over a global field can have order twice
a square (the odd case) as well as a square (the even case). For a curve over a
global field, they give a local diophantine criterion for its jacobian to be
even or odd... | math |
2,258 | The Carmichael numbers up to $10^{16}$ | math.NT | We extend our previous computations to show that there are 246683 Carmichael
numbers up to $10^{16}$. As before, the numbers were generated by a
back-tracking search for possible prime factorisations together with a ``large
prime variation''. We present further statistics on the distribution of
Carmichael numbers. | math |
2,259 | Torsion points of abelian varieties in abelian extensions | math.NT | Let A be an abelian variety defined over a number field K and let Kab be the
maximal abelian extension of K. We show that there only finitely many torsion
points of A which are defined over Kab iff A has no abelian subvariety with
complex multiplication over K. We use this to give another proof of Ribet's
result that A... | math |
2,260 | Selmer groups and the Eisenstein-Klingen ideal | math.NT | In this article, we set up a strategy to prove one divisibility towards the
main Iwasawa conjecture for the Selmer groups attached to the twisted adjoint
modular Galois representations associated to Hida families. This conjecture
asserts the equality of the p-adic L-function interpoling the critical values
of the symme... | math |
2,261 | The Smallest Solution of φ(30n+1)<φ(30n) is ... | math.NT | It is known that there are infinitely many solutions to the inequality
\phi(30n+1)<\phi(30n), where \phi is the familiar Euler totient function.
However, there are no solutions with n<20,000,000, and computing a solution
would seem to involve factoring integers with hundreds of digits. In this note,
we describe how to ... | math |
2,262 | Dense Egyptian Fractions | math.NT | Every positive rational number has representations as Egyptian fractions
(sums of reciprocals of distinct positive integers) with arbitrarily many terms
and with arbitrarily large denominators. However, such representations normally
use a very sparse subset of the positive integers up to the largest
demoninator. We sho... | math |
2,263 | Solubility of Systems of Quadratic Forms | math.NT | We derive an upper bound for the least number of variables needed to
guarantee that a system of t quadratic forms (t>=2) over a field F has a
nontrivial zero. In particular, if F is a local field, then 2t^2+3 variables
insure the existence of a nontrivial zero (2t^2+1 if t is even), while if F=Q_p
with p>=11, then 2t^2... | math |
2,264 | On Tate-Shafarevich groups of abelian varieties | math.NT | Let $K/F$ be a finite Galois extension of number fields with Galois group
$G$, let $A$ be an abelian variety defined over $F$, and let ${\cyr W}(A_{^{/
K}})$ and ${\cyr W}(A_{^{/ F}})$ denote, respectively, the Tate-Shafarevich
groups of $A$ over $K$ and of $A$ over $F$. Assuming that these groups are
finite, we derive... | math |
2,265 | Computation of a universal deformation ring | math.NT | We compute the universal deformation ring of an odd Galois two dimensional
representation of Gal$(M/Q)$ with an upper triangular image, where $M$ is the
maximal abelian pro-$p$-extension of $F_{\infty}$ unramified outside a finite
set of places S, $F_{\infty}$ being a free pro-$p$-extension of a subextension
$F$ of the... | math |
2,266 | Diophantine triples and construction of high-rank elliptic curves | math.NT | Using the theory of Diophantine m-tuples, i.e. sets with the property that
the product of its any two distinct elements increased by 1 is a perfect
square, we construct an elliptic curve over Q(t) of rank at least 4 with three
non-trivial torsion points. By specialization, we obtain an example of elliptic
curve over Q ... | math |
2,267 | A more general abc conjecture | math.NT | This note formulates a conjecture generalizing both the abc conjecture of
Masser-Oesterl\'e and the author's diophantine conjecture for algebraic points
of bounded degree. It also shows that the new conjecture is implied by the
earlier conjecture.
As with most of the author's conjectures, this new conjecture stems fr... | math |
2,268 | Abelian varieties with complex multiplication (for pedestrians) | math.NT | This is the text of an article that I wrote and disseminated in September
1981, except that I've updated the references, corrected a few misprints, and
added a table of contents, some footnotes, and an addendum.
The original article gave a simplified exposition of Deligne's extension of
the Main Theorem of Complex Mu... | math |
2,269 | On Néron models, divisors and modular curves | math.NT | Let $p$ be a prime number such that the modular curve $X_0(p)$ has genus at
least two. We show that the only points of the reduction mod $p$ of $X_0(p)$
with image in the reduction mod $p$ of $J_0(p)$ in the cuspidal group are the
two cusps. This answers a question of Robert Coleman. For the proof we give a
description... | math |
2,270 | $F_q$-Linear Calculus over Function Fields | math.NT | We define analogues of higher derivatives for $F_q$-linear functions over the
field of formal Laurent series with coefficients in $F_q$. This results in a
formula for Taylor coefficients of a $F_q$-linear holomorphic function, a
definition of classes of $F_q$-linear smooth functions which are characterized
in terms of ... | math |
2,271 | Lower Bounds for the Number of Smooth Values of a Polynomial | math.NT | We investigate the problem of showing that the values of a given polynomial
are smooth (i.e., have no large prime factors) a positive proportion of the
time. Although some results exist that bound the number of smooth values of a
polynomial from above, a corresponding lower bound of the correct order of
magnitude has h... | math |
2,272 | The Least Prime Primitive Root and the Shifted Sieve | math.NT | We derive, for all prime moduli p except those in a very thin set, an upper
bound for the least prime primitive root (mod p) of order of magnitude a
constant power of log p. The improvement over previous results, where the upper
bound was log p to an exponent tending to infinity with p, lies in the use of
the linear si... | math |
2,273 | Uniform Bounds for the Least Almost-Prime Primitive Root | math.NT | We investigate, using the weighted linear sieve, the distribution of
almost-primes among the residue classes (mod p) that generate the
multiplicative group of reduced residue classes. We are concerned with finding
an upper bound for the least prime or almost-prime primitive root (mod p) that
holds uniformly for all p, ... | math |
2,274 | Farmer Ted Goes Natural | math.NT | A traditional "Farmer Ted" calculus problem is to minimize the perimeter of a
rectangular chicken coop given the area N, so that as little as possible will
be spent on the fencing. But what if N is an integer, and we are only allowed
to consider rectangles with integer side lengths? Often it will be more
cost-effective... | math |
2,275 | A conjecture for the sixth power moment of the Riemann zeta-function | math.NT | The authors conjecture an asymptotic expression for the sixth power moment of
the Riemann zeta function. They establish related results on the asymptotics of
the zeta function that support the conjecture. | math |
2,276 | Integer solutions of a sequence of decomposable form inequalities | math.NT | In this paper, we prove the finiteness of the number of integer solutions of
the decomposable form inequalities. We also study the number of integer
solutions of a sequence of decomposable form inequalities. | math |
2,277 | A peculiar modular form of weight one | math.NT | In this paper we construct a modular form f of weight one attached to an
imaginary quadratic field K. This form, which is non-holomorphic and not a cusp
form, has several curious properties. Its negative Fourier coefficients are
non-zero precisely for neqative integers -n such that n >0 is a norm from K,
and these coef... | math |
2,278 | Group structures of elementary supersingular abelian varieties over finite fields | math.NT | Let A be a supersingular abelian variety over a finite field k. We give an
approximate description of the structure of the group A(k) of rational points
of A over k in terms of the characteristic polynomial f of the Frobenius
endomorphism of A relative to k. If f=g^e for a monic irreducible polynomial g
and a positive ... | math |
2,279 | Some cases of the Fontaine-Mazur conjecture, II | math.NT | This is an updated and extended version of ANT-0024.
We prove more special cases of the Fontaine-Mazur conjecture regarding p-adic
Galois representations unramified at p, and we present evidence for and
consequences of a generalization of it. | math |
2,280 | Counterexamples to a conjecture of Lemmermeyer | math.NT | We produce infinitely many finite 2-groups that do not embed with index 2 in
any group generated by involutions. This disproves a conjecture of Lemmermeyer
and restricts the possible Galois groups of unramified 2-extensions, Galois
over the rationals, of quadratic number fields. | math |
2,281 | Modular symbols for Q-rank one groups and Voronoi reduction | math.NT | We present an algorithm to compute the action of the Hecke operators on the
top dimensional integral cohomology of certain torsion-free arithmetic
subgroups of algebraic groups of Q-rank one. This generalizes the modular
symbol algorithm to a setting including Bianchi groups and Hilbert modular
groups. In addition, we ... | math |
2,282 | Symplectic modular symbols | math.NT | Let K be a number field with euclidean ring of integers O. Let G be a
finite-index torsion-free subgroup of Sp(2n, O). We exhibit a finite,
geometrically defined spanning set of the top dimensional integral cohomology
of G by generalizing the modular symbol algorithm of Ash and Rudolph. | math |
2,283 | The Explicit Formula and a Propagator | math.NT | I give a new derivation of the Explicit Formula for the general number field
K, which treats all primes in exactly the same way, whether they are discrete
or archimedean, and also ramified or not. In another token, I advance a
probabilistic interpretation of Weil's positivity criterion, as opposed to the
usual geometri... | math |
2,284 | Finiteness of minimal modular symbols for SL(n) | math.NT | Let K be a number field with ring of integers O, and let G be a finite-index
subgroup of SL(n,O). Using a classical construction from the geometry of
numbers and the theory of modular symbols, we exhibit a finite spanning set for
the highest nonvanishing rational cohomology group of G. | math |
2,285 | Difference subgroups of commutative algebraic groups over finite fields | math.NT | The work of Chatzidakis and Hrushovski on the model theory of difference
fields in characteristic zero showed that groups defined by difference
equations have a very restricted structure. Recent work of Chatzidakis,
Hrushovski and Peterzil [CHP] extends the class of difference fields for which
this sort of result is kn... | math |
2,286 | Iwasawa theory for elliptic curves | math.NT | We study this subject by first proving that the p-primary subgroup of the
classical Selmer group for an elliptic curve with good, ordinary reduction at a
prime p has a very simple and elegant description which involves just the
Galois module of p-power torsion points. We then prove theorems of Mazur,
Schneider, and Per... | math |
2,287 | Explicit classification for torsion subgroups of rational points of elliptic curves | math.NT | The classification of elliptic curves E over the rationals Q is studied
according to their torsion subgroups E_{tors}(Q) of rational points. Explicit
criteria for the classification are given when E_{tors}(Q) are cyclic groups
with even orders. The generator points P of E_{tors}(Q) are also explicitly
presented in each... | math |
2,288 | Steinitz class of Mordell groups of elliptic curves with complex multiplication | math.NT | Let E be an elliptic curve having Complex Multiplication by the full ring O_K
of integers of K=Q(\sqrt{-D}), let H=K(j(E)) be the Hilbert class field of K.
Then the Mordell-Weil group E(H) is an O_K-module, and its structure denpends
on its Steinitz class St(E), which is studied here. In partucular, when D is a
prime n... | math |
2,289 | On relations between Jacobians of certain modular curves | math.NT | The topic of this paper concerns a certain relation between the jacobians of
various quotients of the modular curve $X(p)$, which relates the jacobian of
the quotient of $X(p)$ by the normaliser of a non-split Cartan subgroup of
$GL_2(F_p)$ to the jacobians of more standard modular curves. In this paper, we
confirm a c... | math |
2,290 | Large torsion subgroups of split Jacobians of curves of genus two or three | math.NT | We construct examples of families of curves of genus 2 or 3 over Q whose
Jacobians split completely and have various large rational torsion subgroups.
For example, the rational points on a certain elliptic surface over P^1 of
positive rank parameterize a family of genus-2 curves over Q whose Jacobians
each have 128 rat... | math |
2,291 | Safarevic's theorem on solvable groups as Galois groups | math.NT | In this paper we present a complete proof of I.R.Safarevic's famous theorem
that every finite solvable group occurs as a Galois group over Q. | math |
2,292 | Congruences between Selmer groups | math.NT | In this paper we study when two congruent $l$-adic Galois representations
have congruent Selmer groups. We obtain results for representations from
cyclotomic characters, Hecke characters and adjoints of modular forms. | math |
2,293 | The Explicit Formula in simple terms | math.NT | This is a semi-expository paper on the easier aspects of the Explicit Formula
for the Riemann Zeta Function. The topics reviewed here include: Weil's
criterion for the Riemann Hypothesis and its probabilistic interpretation,
various formulations of the contribution corresponding to the real place,
Haran's version of th... | math |
2,294 | The cubic moment of central values of automorphic L-functions | math.NT | The authors study the central values of L-functions in certain families; in
particular they bound the sum of the cubes of these values.Contents: | math |
2,295 | Bounding the torsion in CM elliptic curves | math.NT | Merel has shown that the order of torsion subgroup of an elliptic curve over
a number field can be bounded in terms of only the degree of the number field.
The purpose of this note is to investigate what could be the `right bound'.
In this paper we use the result of Deuring on the supersingular primes for a
CM ellipt... | math |
2,296 | Explicit upper bound for the rank of J_0(q) | math.NT | We provide, on the Birch and Swinnerton-Dyer conjecture, an explicit upper
bound for the rank of the Mordell-Weil group of the Jacobian of the modular
curve X_0(q) for q prime large enough, namely rank J_0(q)< 6.5 dim J_0(q).
The file j0q-pari.dvi contains the .dvi version of the commented listing of
the Pari/GP prog... | math |
2,297 | Number fields with discriminant +-2^a 3^b: Examples from three point covers | math.NT | In this paper we construct number fields ramified at 2 and 3 only, with
various moderate-sized non-solvable Galois groups. We construct these fields by
specializing three point covers, some from the literature and some new here.
The specialization points come from solutions to a generalized Fermat equation.
We conclude... | math |
2,298 | Spectral Analysis of the local Conductor Operator | math.NT | The conductor operator acts on a function through multiplying it with the
logarithm of the norm of the variable both in position and in momentum space
and adding the outcomes. It makes sense at each completion of an arbitrary
number field and arose in previous papers by the author where it was shown to
be intimately co... | math |
2,299 | On the nonasymptotic prime number distribution | math.NT | The objective of this paper is to introduce an approach to the study of the
nonasymptotic distribution of prime numbers. The natural numbers are
represented by theorem 1 in the matrix form ^2N. The first column of the
infinite matrix ^2N starts with the unit and contains all composite numbers in
ascending order.The inf... | math |
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