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Theorem 9.5.6 Let \( \Gamma \) be an arithmetic Kleinian group which contains a non-elementary Fuchsian subgroup. Then \( \Gamma \) contains infinitely many commensurability classes of arithmetic Fuchsian subgroups. | Proof: By Corollary 9.5.3, \( \Gamma \) contains an arithmetic Fuchsian group \( G \) where \( \left\lbrack {k : \ell }\right\rbrack = 2 \) with \( k = {k\Gamma },\ell = {kG} = k \cap \mathbb{R} \) and \( {A\Gamma } \cong {AG}{ \otimes }_{\ell }k \) . Thus \( {\operatorname{Ram}}_{f}\left( {A\Gamma }\right) = \left\{ {... | Yes |
Theorem 9.5.7 Let \( L \) be an alternating link or closed 3-braid such that \( {S}^{3} \smallsetminus L \) is hyperbolic. If \( \sum \) is a closed embedded incompressible surface in \( {S}^{3} \smallsetminus L \), then \( \sum \) cannot be totally geodesic. | Proof: This relies on the Meridian Lemma which asserts that with \( L \) and \( \sum \) as in the statement, \( \sum \) contains a circle isotopic in \( {S}^{3} \smallsetminus L \) to a meridian. This meridian would then arise from a parabolic element, which, if \( \sum \) were totally geodesic, would lie in a cocompac... | Yes |
Lemma 9.6.1 Let \( F \) be a non-elementary Fuchsian subgroup of the Bianchi group \( {\Gamma }_{d} \). Then \( F \) preserves a circle or straight-line in \( \mathbb{C} \cup \infty \)\n\n\[ a{\left| z\right| }^{2} + {Bz} + \overline{Bz} + c = 0, \]\n\nwhere \( a, c \in \mathbb{Z} \) and \( B \in {O}_{d} \). | Proof: Since \( F \) is a non-elementary Fuchsian subgroup, it does preserve a circle or straight-line \( \mathcal{C} \) in \( \mathbb{C} \cup \infty \). Assume this has equation \( a{\left| z\right| }^{2} + {Bz} + \overline{Bz} + c = 0 \) with \( a \) and \( c \) real numbers and \( B \) complex. By conjugating in \( ... | No |
Theorem 9.6.3 Every non-elementary Fuchsian subgroup of \( {\Gamma }_{d} \) is conjugate in \( \operatorname{PSL}\left( {2,\mathbb{C}}\right) \) to a subgroup of an arithmetic Fuchsian group arising from a quaternion algebra \( A\left( {d, D}\right) = \left( \frac{-d, D}{\mathbb{Q}}\right) \) for some positive \( D \in... | Proof: Let \( \mathcal{O} \) denote the order \( \mathbb{Z}\left\lbrack {1, i, j,{ij}}\right\rbrack \) in the algebra \( A\left( {d, D}\right) \) . It is clear that \( \mathbb{Q}\left( \sqrt{-d}\right) \) splits \( A\left( {d, D}\right) \) and a particular embedding is given by\n\n\[ \rho \left( {{a}_{0} + {a}_{1}i + {... | No |
Theorem 9.6.6 Let \( M \) be a finite volume hyperbolic 3-manifold with cusps \( {C}_{i} \), for \( i = 1,\ldots, n \) . Let \( {T}_{i} \) be a choice of horospherical cusp torus for \( i = 1,\ldots, n \) . If \( {\alpha }_{i} \) is an essential simple closed curve on \( {T}_{i} \) whose length (as measured on \( {T}_{... | In fact, one can say something more precise. The negative curvature metric referred to in this theorem is constructed from the hyperbolic metric on the 3-manifold \( M \) together with a particular choice of negatively curved metric on a solid torus. Briefly, after truncating via a choice of cusp tori as described abov... | Yes |
Theorem 9.6.7 Let \( M \) be a finite-volume hyperbolic 3-manifold with cusps \( {C}_{i} \), containing a closed immersed totally geodesic surface \( S \). Then \( S \) remains incompressible in all but a finite number of surgeries on any cusp \( {C}_{i} \). | Sketch Proof: For simplicity and because it carries all the important ideas, we assume \( M \) has a single cusp. Let \( M = {\mathbf{H}}^{3}/\Gamma \) and arrange a lift of the cusp \( C \) to be at infinity in \( {\mathbf{H}}^{3} \), so that the peripheral subgroup of \( \Gamma \) consists of translations. Since \( S... | No |
Corollary 9.6.8 Let \( M \) denote the complement of the figure 8 knot in \( {S}^{3} \). Then all but a finite number of Dehn surgeries on \( M \) contain a closed incompressible surface. | This follows directly from Theorem 9.6.7. However, one can use the description of cocompact Fuchsian subgroups of \( \operatorname{PSL}\left( {2,{O}_{3}}\right) \) to obtain sharper estimates on the number of excluded surgeries. Looking at the proof of Theorem 9.6.7, one sees that it is the choice of horoball and the s... | No |
Corollary 9.6.9 The figure 8 knot complement contains a closed totally geodesic surface that remains incompressible in all except possibly the following surgeries: | Proof: Let \( K \) denote the figure 8 knot. We will consider the representation of \( {\pi }_{1}\left( {{S}^{3} \smallsetminus K}\right) \) into \( \operatorname{PSL}\left( {2,{O}_{3}}\right) \) given in \( §{1.4.3} \) in which a meridian and longitude are represented by the matrices\n\n\[ \mu = \left( \begin{array}{l... | Yes |
Theorem 9.7.1 Let \( k \mid F \) be a finite extension of number fields and let A be a quaternion division algebra over \( k \) . The following statements are equivalent:\n\n1. \( A \cong \left( \frac{a, b}{k}\right) \) for some \( a \in k, b \in F \) .\n\n2. For every finite place \( w \) of \( F \) which is ramified ... | Proof: Suppose that condition 1 holds and \( v \in {\operatorname{Ram}}_{f}\left( A\right) \) . Then \( A{ \otimes }_{k}{k}_{v} \cong \) \( \left( \frac{a, b}{{k}_{v}}\right) \) is a division algebra. Thus \( {k}_{v}\left( \sqrt{b}\right) \) is a quadratic extension of \( {k}_{v} \) (see Lemma 2.1.6). Hence \( {F}_{w}\... | Yes |
Lemma 9.8.2 Let \( \Gamma \) be a Kleinian group of finite covolume which contains one of the Euclidean triangle groups \( \left( {3,3,3}\right) ,\left( {2,3,6}\right) \) or \( \left( {2,4,4}\right) \) as a subgroup. Then \( {k\Gamma } \) contains \( \mathbb{Q}\left( \sqrt{-1}\right) \) or \( \mathbb{Q}\left( \sqrt{-3}... | Proof: Clearly \( \Gamma \) has a cusp so that \( {A\Gamma } \cong {M}_{2}\left( {\mathbb{Q}\left( \sqrt{-d}\right) }\right) \) for some \( d \) . If \( \Delta \) is a torsion-free subgroup of \( \Gamma \) of finite index, then a cusp is a flat torus isometric to \( \mathbb{C}/\Lambda \) for some lattice \( \Lambda \) ... | Yes |
Theorem 10.2.2 With notation as above, the following sequence is exact for \( A \in \mathcal{A}\left( k\right) \) :\n\n\[ 1 \rightarrow {k}^{ * } \rightarrow {A}_{k}^{ * }\overset{\Phi }{ \rightarrow }\operatorname{SO}\left( {{V}_{\rho }, n}\right) \rightarrow 1 \] | Let \( \mathcal{L} \) be an order in \( B \) so that \( \mathcal{O} = \mathcal{L}{ \otimes }_{{R}_{k}}{R}_{k\left( z\right) } \) is an order in \( A \) (see Exercise 6.3 No. 3). By construction, \( \rho \left( \mathcal{O}\right) = \mathcal{O} \) . If \( L = \mathcal{O} \cap {V}_{\rho } \), then \( L \) is an \( {R}_{k}... | No |
\[ {H}_{\mathbb{Q}} = \left\{ {{\widehat{S}}^{-1}\left( {g,{\sigma }_{2}\left( g\right) ,\ldots ,{\sigma }_{d}\left( g\right) }\right) \widehat{S} \mid g \in {G}_{k}}\right\} \] \[ {H}_{\mathbb{Z}} = \left\{ {{\widehat{S}}^{-1}\left( {g,{\sigma }_{2}\left( g\right) ,\ldots ,{\sigma }_{d}\left( g\right) }\right) \wideha... | Note that, up to conjugation, \( H \) has the form \( G \times {}^{{\sigma }_{2}}G \times \cdots \times {}^{{\sigma }_{d}}G \) so that there is a morphism \( p : H \rightarrow G \) obtained by mapping \( A \) onto the first factor of \( \widehat{S}A{\widehat{S}}^{-1} \) . This, then, is the restriction of scalars const... | Yes |
Theorem 10.3.7 Every \( k \) -form of \( {\mathrm{{PGL}}}_{2} \) is isomorphic over \( k \) to a quotient \( {A}^{ * }/Z{\left( A\right) }^{ * } \), where \( A \) is a quaternion algebra over \( k \) . | For the proof of Theorem 10.3.7, recall that every quaternion algebra \( A \) splits over some quadratic extension \( L \) of \( k \) so that over \( L,{A}^{ * }/Z{\left( A\right) }^{ * } \) is isomorphic to \( {\mathrm{{PGL}}}_{2} \) . For the converse, we invoke some results from non-abelian Galois cohomology. | No |
Lemma 10.3.9\n\n\[ \n{H}^{1}\left( {\mathcal{G},\mathrm{{GL}}\left( {V}_{K}\right) }\right) = 0. \n\] | Proof: Let \( a \) be a 1-cocycle. For \( c \in {M}_{n}\left( K\right) \), define \( b = \mathop{\sum }\limits_{{g \in \mathcal{G}}}{a}_{g}{}^{g}c \) . Since the automorphisms \( \{ g\} \) form a set of algebraically independent mappings, the equation \( \det \left( b\right) = 0 \) can have only finitely many solutions... | Yes |
Theorem 10.3.11 (Vinberg) There is a least field of definition of \( \Gamma \) which is an invariant of the commensurability class of \( \Gamma \) . Furthermore, it is the field \( \mathbb{Q} \) (tr Ad \( \gamma : \gamma \in \Gamma \) ), where Ad is the adjoint representation of \( G \) . | In the special case where \( G = {\mathrm{{PGL}}}_{2} \) and \( \Gamma \) is discrete of finite covolume, then \( \Gamma \) is Zariski dense by Borel’s density theorem. Furthermore, \( {k\Gamma } = \) \( \mathbb{Q}\left( {\operatorname{tr}\text{Ad}\gamma : \gamma \in \Gamma }\right) \) (see Exercise 3.3, No. 4). Using ... | No |
Lemma 10.4.4 Let \( G \) be a finite-covolume Kleinian group normalised by an orientation-reversing involution. Then \( \left\lbrack {{kG} : {kG} \cap \mathbb{R}}\right\rbrack = 2 \) . | Proof: Let \( r \) be the involution, which, by conjugacy if necessary, can be taken to be the extension of complex conjugation to \( {\mathbf{H}}^{3} \) . Choose a subgroup \( {G}_{0} \) of finite index in \( G \) for which \( \mathbb{Q}\left( {\operatorname{tr}{G}_{0}}\right) = {kG} \) and \( {G}_{0} \) is normalised... | No |
Theorem 11.1.1 Let \( k \) be a totally real number field, \( A \) be a quaternion algebra over \( k \) which is ramified at all real places except one and \( \mathcal{O} \) be a maximal order in \( A \) . Then the hyperbolic covolume of the Fuchsian group \( {P\rho }\left( {\mathcal{O}}^{1}\right) \) is | \[ \frac{{8\pi }{\Delta }_{k}^{3/2}{\zeta }_{k}\left( 2\right) \mathop{\prod }\limits_{{\mathcal{P} \mid \Delta \left( A\right) }}\left( {N\left( \mathcal{P}\right) - 1}\right) }{{\left( 4{\pi }^{2}\right) }^{\left\lbrack k : \mathbb{Q}\right\rbrack }}. \] | Yes |
Theorem 11.1.3 Let \( k \) be a number field with exactly one complex place, A be a quaternion algebra over \( k \) ramified at all real places and \( \mathcal{O} \) be a maximal order in \( A \) . Then if \( \rho \) is a \( k \) -representation of \( A \) to \( {M}_{2}\left( \mathbb{C}\right) \) then | \[ \text{Hyperbolic Vol}\left( {{\mathbf{H}}^{3}/{P\rho }\left( {\mathcal{O}}^{1}\right) }\right) = \frac{4{\pi }^{2}{\left| {\Delta }_{k}\right| }^{3/2}{\zeta }_{k}\left( 2\right) \mathop{\prod }\limits_{{\mathcal{P} \mid \Delta \left( A\right) }}\left( {N\left( \mathcal{P}\right) - 1}\right) }{{\left( 4{\pi }^{2}\rig... | No |
Theorem 11.3.1 Let \( K > 0 \) . There are only finitely many conjugacy classes of arithmetic Fuchsian groups \( \Gamma \) such that \( \operatorname{Vol}\left( {{\mathbf{H}}^{2}/\Gamma }\right) < K \) . | Proof: For arithmetic Fuchsian groups, the same argument as that employed in Theorem 11.2.1 holds, except for the first part. For any finite covolume Fuchsian group \( \Gamma ,\operatorname{Vol}\left( {{\mathbf{H}}^{2}/\Gamma }\right) = {2\pi }\left| {\chi \left( \Gamma \right) }\right| \), where \( \chi \left( \Gamma ... | Yes |
Lemma 11.3.5 If \( \\left( {N, M}\\right) > 2 \), then \( \\mathbb{Q}\\left( {\\cos {2\\pi }/N,\\cos {2\\pi }/M}\\right) = {R}_{\\left\\lbrack N, M\\right\\rbrack } \) . | Proof: Note that, \( \\operatorname{Gal}\\left( {{C}_{\\left\\lbrack N, M\\right\\rbrack } \\mid \\mathbb{Q}}\\right) \\cong {\\mathbb{Z}}_{\\left\\lbrack N, M\\right\\rbrack }^{ * } \) and \( {R}_{\\left\\lbrack N, M\\right\\rbrack } \) is the fixed field of the subgroup \( \\{ \\pm 1\\} \) . The field \( \\mathbb{Q}\... | Yes |
Theorem 11.3.6 There are finitely many arithmetic Fuchsian triangle groups \( \left( {{e}_{1},{e}_{2},{e}_{3}}\right) \) and all \( {e}_{i} \leq {134} \) . | We now make use of the other values of \( a \) to reduce the bound on \( N \), so that a search for all arithmetic triangle groups can be made. The values \( a = 3,4,5 \) employed in (11.20) imply, respectively, that \( N \leq {44},{33},{26} \) . Combined with the restrictions for which (11.17) holds, this yields that ... | Yes |
Lemma 11.4.1 Let \( \mathcal{D} \) be any order in the quaternion algebra \( A \) . Then,\n\n\[ N\left( \mathcal{D}\right) = \left\{ {x \in {A}^{ * } \mid x \in N\left( {\mathcal{D}}_{\mathcal{P}}\right) \forall \mathcal{P} \in {\Omega }_{f}}\right\} .\n\] | Proof: It is clear that for each \( x \in N\left( \mathcal{D}\right), x \in N\left( {\mathcal{D}}_{\mathcal{P}}\right) \) for all \( \mathcal{P} \) . Now suppose that \( x \in {A}^{ * } \) is such that \( x \in N\left( {\mathcal{D}}_{\mathcal{P}}\right) \) for all \( \mathcal{P} \) . Let \( x\mathcal{D}{x}^{-1} = {\mat... | Yes |
Theorem 11.4.3 Let \( \Gamma \in \mathcal{C}\left( A\right) \) . Let \( S\left( \Gamma \right) \) be the set of primes \( \mathcal{P} \) such that \( \Gamma \) has an element which is odd at \( \mathcal{P} \) . Then there exists a maximal order \( \mathcal{O} \) such that \( \Gamma \) is conjugate to a subgroup of \( {... | Proof: Let \( \mathcal{D} \) be a maximal order so that the closure of \( \Gamma \) in \( P\left( {A}_{\mathcal{P}}^{ * }\right) \) is a compact open subgroup which is equal to \( P\left( {N\left( {\mathcal{D}}_{\mathcal{P}}\right) }\right) \) for all \( \mathcal{P} \in {\Omega }_{f} \smallsetminus T \) , where \( T \)... | Yes |
Theorem 11.5.1 For \( \mathcal{O} \) a maximal order in \( A \) , \[ \left\lbrack {{\Gamma }_{\varnothing ,\mathcal{O}} : {\Gamma }_{S,\mathcal{O}}}\right\rbrack = {2}^{-m}\mathop{\prod }\limits_{{\mathcal{P} \in S}}\left( {N\left( \mathcal{P}\right) + 1}\right) \] for some \( 0 \leq m \leq \left| S\right| \) . Also, i... | Proof: Let \( \mathcal{O} \) and \( {\mathcal{O}}^{\prime } \) be maximal orders of \( A \) such that \( \mathcal{O} \cap {\mathcal{O}}^{\prime } \) is an Eichler order of level \( \prod \mathcal{P} \) for \( \mathcal{P} \in S \) . Now \( {\Gamma }_{1} = {\Gamma }_{\varnothing ,\mathcal{O}} \cap {\Gamma }_{S,\mathcal{O... | Yes |
Theorem 11.5.2 Let \( e \) be the number of primes in \( k \) dividing 2 and not contained in \( {\operatorname{Ram}}_{f}\left( A\right) \) . Let \( \mathcal{O} \) be a maximal order in \( A \) and let \( \Gamma \in \mathcal{C}\left( A\right) \) . Then the covolume of \( \Gamma \) is an integral multiple of \( {2}^{-e}... | Proof: By Theorem 11.4.3, Covol \( \left( \Gamma \right) \) is an integral multiple of Covol \( \left( {\Gamma }_{S,\mathcal{O}}\right) \) for some maximal order \( \mathcal{O} \) . The right-hand side of (11.21) is a multiple of \( \mathop{\prod }\limits_{{\mathcal{P} \in S}}\frac{N\left( \mathcal{P}\right) + 1}{2} \)... | Yes |
Theorem 11.6.1 (Eichler)\n\n\[ n\left( {\mathcal{O}}^{ * }\right) = {R}_{k,\infty }^{ * }.\] | Proof: Clearly \( n\left( {\mathcal{O}}^{ * }\right) \subset {R}_{k,\infty }^{ * } \), so suppose that \( t \in {R}_{k,\infty }^{ * } \) . By the Norm Theorem 7.4.1, there exists \( \alpha \in {A}^{ * } \) such that \( n\left( \alpha \right) = t \) . For all but a finite set \( S \) of primes, \( \alpha ,{\alpha }^{-1}... | Yes |
2. \( {\Gamma }_{{R}_{f}}/{\Gamma }_{{\mathcal{O}}^{1}} \cong {R}_{f,\infty }^{ * }/{\left( {R}_{f}^{ * }\right) }^{2} \) | Proof: For Part 2, let \( \bar{n} : {\Gamma }_{{R}_{f}} \rightarrow {R}_{f,\infty }^{ * }/{\left( {R}_{f}^{ * }\right) }^{2} \) be defined by \( \bar{n}\left( {P\left( \alpha \right) }\right) = \) \( n\left( \alpha \right) {\left( {R}_{f}^{ * }\right) }^{2} \) . This is then a well-defined homomorphism. By a similar ar... | No |
Corollary 11.6.4\n\n\[ \left\lbrack {{\Gamma }_{{R}_{f}} : {\Gamma }_{{\mathcal{O}}^{1}}}\right\rbrack \leq {2}^{{r}_{1} + {r}_{2} + {r}_{f}} \] | The Dirichlet Unit Theorem (see Theorem 0.4.2) can be extended to cover groups of \( S \) -units from which the above corollary follows. Thus this divisor of the index \( \left\lbrack {{\Gamma }_{\mathcal{O}} : {\Gamma }_{{\mathcal{O}}^{1}}}\right\rbrack \) can be calculated starting from a knowledge of the group of un... | No |
Theorem 11.6.5 With the notation as given in this section,\n\n\[ \left\lbrack {{\Gamma }_{\mathcal{O}} : {\Gamma }_{{R}_{f}}}\right\rbrack = \left\lbrack {{}_{2}{J}_{1} : {J}_{2}}\right\rbrack \]\n\nIf \( k \) has class number 1, then \( {\Gamma }_{\mathcal{O}} = {\Gamma }_{{R}_{f}} \) . | Proof: When \( \alpha \in N\left( \mathcal{O}\right) \), then \( \alpha \in {\mathcal{O}}_{\mathcal{P}}^{ * } \) for all but a finite set \( S \) of primes. For \( \mathcal{P} \in S \smallsetminus {\operatorname{Ram}}_{f}\left( A\right) ,\alpha \in {t}_{\mathcal{P}}{\mathcal{O}}_{\mathcal{P}}^{ * } \) for some \( {t}_{... | Yes |
The smallest covolume arithmetic Kleinian group which can be defined over a quadratic field is \( \operatorname{PGL}\left( {2,{O}_{3}}\right) \) . The smallest covolume cocompact arithmetic Kleinian group defined over a quadratic field is the group which is the extension of the tetrahedral group described in Example 11... | The group \( \operatorname{PGL}\left( {2,{O}_{3}}\right) \) is an extension of \( \operatorname{PSL}\left( {2,{O}_{3}}\right) \) and so, for example, from calculations using Corollary 11.2.4, has covolume \( {\mu }_{0} = \) 0.084578 approximately. The group described in Example 11.6.7 has the form \( {P\rho }\left( {N\... | Yes |
Lemma 12.2.1 Let \( \Gamma \) be a non-elementary group and assume that \( {k\Gamma } = \) \( \mathbb{Q}\left( {\operatorname{tr}\Gamma }\right) \) and is a number field. For all non-trivial \( \gamma \in \Gamma ,{k\Gamma }\left( {\lambda }_{\gamma }\right) \) embeds isomorphically as a subfield of \( {A}_{0}\Gamma = \... | Proof: Consider the characteristic polynomial \( {p}_{\gamma }\left( x\right) \) . If this splits over \( {k\Gamma } \), then \( {\lambda }_{\gamma } \in {k\Gamma } \) and so trivially embeds in \( {A}_{0}\Gamma \) . Thus assume \( {p}_{\gamma }\left( x\right) \) is irreducible over \( {k\Gamma } \) . Let \( B \) be th... | Yes |
Corollary 12.2.4 With \( \Gamma \) as in Lemma 12.2.1, \( {k\Gamma }\left( {\lambda }_{\gamma }\right) \) splits \( {A}_{0}\Gamma \) . | Proof: If \( {A}_{0}\left( \Gamma \right) \) is a division algebra, then \( {k\Gamma }\left( {\lambda }_{\gamma }\right) \) is a quadratic extension and the result follows from Theorem 12.2.3. If \( {A}_{0}\Gamma \) is not a division algebra, it is already split. | Yes |
Lemma 12.2.5 Let \( \Gamma \) be derived from a quaternion algebra and let \( \gamma \) be loxodromic. Then \( \left\lbrack {k\left( {\lambda }_{\gamma }\right) : k}\right\rbrack = 2 \) . | Proof: As noted earlier, this holds unless \( A \) splits, in which case \( A \cong \) \( {M}_{2}\left( {\mathbb{Q}\left( \sqrt{-d}\right) }\right) \) for some \( d \) . Since \( \operatorname{tr}\gamma \) is an algebraic integer, \( {\lambda }_{\gamma } \) is a unit (see Exercise 12.1, No. 1) and it will lie in \( k \... | No |
Theorem 12.2.6 Let \( A \) be a quaternion algebra defined over \( k \), where \( k \) has exactly one complex place and \( A \) is ramified at all real places. Let \( L \) be a quadratic extension of \( k \) . Then \( L \) embeds in \( A \) if and only if there is an order \( \mathcal{O} \) in \( A \) and an element \... | Proof: This is immediate if such a \( \gamma \) exists. For the converse, suppose that \( L \) embeds in \( A \) . Since \( k \) has one complex place, it follows from Dirichlet’s Unit Theorem 0.4.2 that the \( \mathbb{Z} \) -rank of \( {R}_{L}^{ * } \) is strictly greater than that of \( {R}_{k}^{ * } \) . Thus there ... | Yes |
Corollary 12.2.7 Let \( \Gamma \) be a Kleinian group derived from a quaternion algebra \( A \) over \( k \) . Let \( L \) be a quadratic extension of \( k \) . Then \( L \) embeds in \( A \) if and only if \( \Gamma \) contains an element \( \gamma \) of infinite order with \( L = k\left( {\lambda }_{\gamma }\right) \... | Proof: Since \( \Gamma \) is commensurable with \( {P\rho }\left( {\mathcal{O}}^{1}\right) \) as in Theorem 12.2.6, for some integer \( m,{\gamma }^{m} \in \Gamma \), where \( \gamma \) is as in the theorem. However, then \( L = k\left( {u}^{m}\right) = k\left( {\lambda }_{{\gamma }^{m}}\right) . \) | No |
Theorem 12.2.8 Suppose that \( \Gamma \) is a Kleinian group derived from a quaternion algebra and \( \gamma \in \Gamma \) is loxodromic. Then there is a group \( {\Gamma }_{0} \) commensurable with \( \Gamma \) containing an element \( {\gamma }_{0} \) with \( {\gamma }_{0}^{2} = \gamma \) in \( \operatorname{PSL}\lef... | Proof: Let us assume that \( \Gamma = {P\rho }\left( {\mathcal{O}}^{1}\right) \) for some maximal order \( \mathcal{O} \) in \( A = {A\Gamma } \) . Let \( \gamma = {P\rho }\left( u\right) \) for some \( u \in {\mathcal{O}}^{1} \) . Now \( 1 + u \in \mathcal{O} \) and \( {\left( 1 + u\right) }^{2} = \) \( u\left( {2 + \... | Yes |
Corollary 12.2.9 If \( L \) is any real quadratic extension of \( \mathbb{Q} \), there is a hyperbolic element \( \gamma \left( L\right) \) in \( \operatorname{PSL}\left( {2,\mathbb{Z}}\right) \) such that \( L \) embeds in \( {M}_{2}\left( \mathbb{Q}\right) \) as \( \mathbb{Q}\left( {\gamma \left( L\right) }\right) \)... | Proof: Since \( {M}_{2}\left( \mathbb{Q}\right) \) has no ramification, \( L \) embeds in \( {M}_{2}\left( \mathbb{Q}\right) \) by Theorem 12.2.3 and the result is completed by Theorem 12.2.6 | No |
Corollary 12.2.10 Let \( k \) be a number field which is either totally real or has exactly one complex place. Let \( \alpha = u + {u}^{-1} \in {R}_{k} \) where \( u \) is quadratic over \( k \) and satisfies the following hypotheses:\n\n(a) \( u \) is not a root of unity.\n\n(b) If \( k \) is totally real, then \( {\a... | Proof: We deal with the case where \( k \) has one complex place, the totally real case being handled similarily. Also assume that \( \left\lbrack {k : \mathbb{Q}}\right\rbrack \) is even, the odd degree case being a slight variation (see Exercise 12.2, No. 3). Let \( {v}_{1},{v}_{2},\ldots ,{v}_{n} \) denote the real ... | No |
Lemma 12.3.2 Suppose that \( u \) is an algebraic integer such that \( \left| u\right| > 1 \) , \( {u}^{-1} \) is a conjugate of \( u \), and \( u \) satisfies one of the conditions (a) or (b) of Lemma 12.3.1. Then in case (a) (resp. (b)), there is a Kleinian (resp. Fuchsian) group \( \Gamma \) derived from a quaternio... | Proof: We only sketch this proof, as the details are similar to the proofs of Theorem 12.2.6 and Corollary 12.2.10.\n\nSuppose \( u \) is not real, so that the conjugates not on the unit circle are \( u,{u}^{-1},\bar{u} \) and \( {\bar{u}}^{-1} \) . Thus the only non-real conjugates of \( \theta = u + {u}^{-1} \) are \... | No |
Lemma 12.3.3 With \( \gamma \) as described in Lemma 12.3.1, \( {\ell }_{0}\left( \gamma \right) = \ln M\left( P\right) \) or \( 2\ln M\left( P\right) \) according to whether \( \gamma \) is loxodromic or hyperbolic, where \( P\left( x\right) \) is the minimum polynomial of \( u \) . | Proof: In the loxodromic case, the conjugates \( u,{u}^{-1},\bar{u} \) and \( {\bar{u}}^{-1} \) lie off the unit circle so that \( M\left( P\right) = {\left| u\right| }^{2} \) . Since \( {\ell }_{0}\left( \gamma \right) = 2\ln \left| u\right| \), the result follows. The hyperbolic case is similar. | Yes |
Lemma 12.3.7 If \( \left\lbrack {k : \mathbb{Q}}\right\rbrack \) is odd, there is a minimal Salem number in \( S\left( k\right) \) . | Proof: Let \( u \in S\left( k\right) \) so that \( u \) determines a real place of \( k \) . Let \( A \) be a quaternion algebra over \( k \) which is ramified at all real places except the designated one and has no finite ramification. By the Classification Theorem 7.3.6, there are a finite number \( \left( { \leq \le... | Yes |
Lemma 12.4.3 \( {\Phi }_{\mathcal{P}} \subset N\left( {m, n, s}\right) \) if and only if \( s{\pi }^{-m}\left( {{\alpha }_{1} - {\alpha }_{2}}\right) \in {R}_{\mathcal{P}} \) . | Proof: Since \( N\left( {m, n, s}\right) \) is an \( {R}_{\mathcal{P}} \) -order containing \( 1,{\Phi }_{\mathcal{P}} \subset N\left( {m, n, s}\right) \) if and only if \( \left( \begin{matrix} {\alpha }_{1} & 0 \\ 0 & {\alpha }_{2} \end{matrix}\right) \in N\left( {m, n, s}\right) \) . This occurs if and only if\n\n\[... | Yes |
Lemma 12.4.4 Let \( {\mathcal{O}}_{\mathcal{P}} \) be a maximal order in \( {M}_{2}\left( {k}_{\mathcal{P}}\right) \) which contains \( {\Phi }_{\mathcal{P}} \) . Then in the tree \( {\mathcal{T}}_{\mathcal{P}} \) of maximal orders in \( {M}_{2}\left( {k}_{\mathcal{P}}\right) \), either one, two or all neighbours of \(... | Proof: Suppose \( {\Phi }_{\mathcal{P}} \subset N\left( {m, n, s}\right) \) . Now suppose that \( {\alpha }_{1} - {\alpha }_{2} \in {R}_{\mathcal{P}}^{ * } \) . Then since \( s{\pi }^{-m}\left( {{\alpha }_{1} - {\alpha }_{2}}\right) \in {R}_{\mathcal{P}} \) by Lemma 12.4.3, we must have \( m = 0 \) and \( s = 0 \) . In... | No |
Theorem 12.4.6 Let \( {\mathcal{O}}_{1} \) and \( {\mathcal{O}}_{2} \) be maximal orders in a quaternion division algebra \( A \) over the number field \( k \) such that \( {P\rho }\left( {\mathcal{O}}_{1}^{1}\right) \) and \( {P\rho }\left( {\mathcal{O}}_{2}^{1}\right) \) are arithmetic Fuchsian groups. Then \( {P\rho... | Proof: As the groups are cocompact, the isomorphism class is determined by the signature which is determined by the covolume and the number of conjugacy classes of primitive elements of finite order. For maximal orders, the covolumes are equal by Theorem 11.1.1. If either group contains an element of order \( n \), the... | Yes |
Lemma 12.4.7 Let \( {\mathcal{O}}_{1} \) and \( {\mathcal{O}}_{2} \) be maximal orders in a quaternion algebra A over a number field \( k \) and \( \rho \) be a \( k \) -representation in \( {M}_{2}\left( \mathbb{R}\right) \) (respectively \( \left. {{M}_{2}\left( \mathbb{C}\right) }\right) \) such that \( {P\rho }\lef... | Proof: Let \( \gamma = P\left( c\right) \), where \( c \in \mathrm{{GL}}\left( {2,\mathbb{R}}\right) \) or \( \mathrm{{GL}}\left( {2,\mathbb{C}}\right) \) . Now \( \rho \left( A\right) = \) \( A\left( {\rho \left( {\mathcal{O}}_{1}^{1}\right) }\right) = A\left( {\rho \left( {\mathcal{O}}_{2}^{1}\right) }\right) \) and ... | Yes |
For the two-dimensional case, let \( k = \mathbb{Q}\left( \sqrt{10}\right) \), which has class number 2. Let \( A \) be defined over \( k \) such that it is ramified at the real place corresponding to \( - \sqrt{10} \) . Note that a fundamental unit in \( {R}_{k}^{ * } \) is \( 3 + \sqrt{10} \) . Thus \( {h}_{\infty } ... | As noted in Theorem 12.4.6, these groups will be isomorphic. As they are cocompact and torsion free, their isomorphism class is determined by their genus, which can be deduced from a computation of the volume formula in Theorem 11.1.1, to be 19. | Yes |
Theorem 12.4.9 For any integer \( n \geq 2 \), there are \( n \) isospectral non-isometric hyperbolic 3-manifolds. | Proof: For each \( t \), choose a quadratic imaginary number field \( k \) such that \( {\Delta }_{k} \) has at least \( {4t} \) distinct prime divisors. Then by the Dirichlet density theorem (Theorem 0.3.12), choose a prime ideal \( \mathcal{P} \) of \( k \) such that \( \mathcal{P} \) splits completely in \( k\left( ... | Yes |
Lemma 12.4.10 Let \( {A}_{1} \) and \( {A}_{2} \) be quaternion algebras defined over the totally real field \( k \) such that \( {A}_{1} \) and \( {A}_{2} \) are ramified at exactly the same set of real places. Let \( {\mathcal{L}}_{1} \) and \( {\mathcal{L}}_{2} \) be as defined above. Then \( {A}_{1} \cong {A}_{2} \... | Proof: If \( {A}_{1} \cong {A}_{2} \) and \( L \in {\mathcal{L}}_{1} \), then \( L{ \otimes }_{v}{k}_{v} \) is a field for every \( v \in \) \( \operatorname{Ram}\left( {A}_{1}\right) = \operatorname{Ram}\left( {A}_{2}\right) \) and so \( L \in {\mathcal{L}}_{2} \) . Thus \( {\mathcal{L}}_{1} = {\mathcal{L}}_{2} \) .\n... | Yes |
Theorem 12.4.11 If \( {M}_{1} \) and \( {M}_{2} \) are a pair of isospectral arithmetic hyperbolic 2-manifolds, then \( {M}_{1} \) and \( {M}_{2} \) are commensurable. | Proof: Let \( {M}_{i} = {\mathbf{H}}^{2}/{\Gamma }_{i}, i = 1,2 \) . As noted before Lemma 12.4.10, the defining fields are equal: \( {k}_{1} = {k}_{2} = k \) . Now let \( L \in {\mathcal{L}}_{1} \) . Then by Theorem 12.2.6 (see comments following Theorem 12.2.8), there is an order \( \mathcal{O} \) in \( {A}_{1} \) an... | Yes |
Theorem 12.4.12 Let \( {M}_{1} = {\mathbf{H}}^{3}/{\Gamma }_{1} \) and \( {M}_{2} = {\mathbf{H}}^{3}/{\Gamma }_{2} \) be isospectral arithmetic hyperbolic 3-manifolds. Then they are commensurable. | Proof : As in the Fuchsian case, from the earlier discussion before Lemma 12.4.10, we can assume that \( {\Gamma }_{1} \) and \( {\Gamma }_{2} \) share a common invariant trace field \( k \) say. Let \( {A}_{1} \) and \( {A}_{2} \) be the invariant quaternion algebras of \( {\Gamma }_{1} \) and \( {\Gamma }_{2} \) , re... | Yes |
Lemma 12.5.1 Let \( K \) be a \( \mathcal{P} \) -adic field with ring of integers \( {R}_{K} \) . Then \( u \in \mathrm{{GL}}\left( {2, K}\right) \) fixes an edge or a vertex of the tree \( \mathcal{T} \) if and only if the element \( \operatorname{disc}\left( u\right) /\det \left( u\right) \in {R}_{K} \), where \( \op... | Proof: The element \( P\left( u\right) \) fixes an edge or a vertex if and only if it is contained in a compact subgroup of \( \operatorname{PGL}\left( {2, K}\right) \) . This will be true if and only if it is true when \( K \) is replaced by a finite extension. Thus we can normalise so that \( u = \left( \begin{matrix... | Yes |
Lemma 12.5.2 Let \( K \) be as in the preceding lemma. Let \( u \in \mathrm{{GL}}\left( {2, K}\right) \) be such that \( \operatorname{tr}u \in {R}_{K},\det \left( u\right) \in {R}_{K}^{ * } \) and \( P\left( u\right) \) is non-trivial. Then \( P\left( u\right) \) fixes an edge of \( \mathcal{T} \) if and only if eithe... | Proof: By Lemma 12.5.1, \( P\left( u\right) \) must fix an edge or a vertex. If \( K\left( u\right) \) is not a field, then \( u \) is conjugate in \( \mathrm{{GL}}\left( {2, K}\right) \) to a scalar multiple of \( \left( \begin{matrix} \lambda & {\pi }^{r}{t}^{\prime } \\ 0 & 1 \end{matrix}\right) \) , where \( \lambd... | Yes |
Theorem 12.5.3 (Chinburg and Friedman) Let \( k \) be a number field and \( A \) a quaternion division algebra over \( k \) which satisfies the Eichler condition. Suppose that \( u \in {A}^{ * } \) is such that \( \operatorname{tr}u \in {R}_{k}, n\left( u\right) \in {R}_{k}^{ * } \) and \( u \notin {k}^{ * } \) . Let \... | Proof: Recall that \( A = {La} + {Lb} \) for some elements \( a, b \in A \), so that \( {R}_{k}\left\lbrack u\right\rbrack \) is contained in the order \( {\mathcal{O}}_{\ell }\left( I\right) \), where \( I = {R}_{L}a + {R}_{L}b \) . Thus let \( \mathcal{O} \) be a fixed maximal order such that \( \Omega = {R}_{k}\left... | Yes |
Theorem 12.5.4 Let \( A \) be a quaternion division algebra over a number field \( k \) such that \( \mathcal{C}\left( A\right) \) is a class of arithmetic Kleinian or Fuchsian groups.\n\nThe group \( P\left( {A}^{1}\right) \) contains an element of order \( n \)\n\n\( \Leftrightarrow {\xi }_{2n} + {\xi }_{2n}^{-1} \in... | Now assume that \( P\left( {A}^{1}\right) \) contains an element of order \( n \) . If the members of \( \mathcal{C}\left( A\right) \) are Fuchsian groups, then, for every maximal order \( \mathcal{O} \) in \( A,{\Gamma }_{{\mathcal{O}}^{1}} \) contains an element of order \( n \) . If the members of \( \mathcal{C}\lef... | Yes |
Corollary 12.5.5 For every \( n \), there are infinitely many arithmetic Fuchsian or Kleinian groups which contain an element of order \( n \), and which are pairwise non-commensurable. | Proof: We give the proof in the Kleinian case, leaving the necessary alterations for the Fuchsian case as an exercise (see Exercise 12.5, No. 1). Let \( {k}_{0} = \mathbb{Q}\left( {\cos \pi /n}\right) \) and denote the distinct real embeddings of \( {k}_{0} \) by \( {\sigma }_{1} = \) Id, \( {\sigma }_{2},\ldots ,{\sig... | No |
Lemma 12.5.6 Let \( n > 2 \) and let \( {\xi }_{n} \) be a primitive \( n \) th root of unity. Let \( A \) be a quaternion division algebra over \( k \) . Then \( P\left( {A}^{ * }\right) \) contains an element of order \( n \) if and only if \( {\xi }_{n} + {\xi }_{n}^{-1} \in k,{\xi }_{n} \notin k \), and \( k\left( ... | Proof: Suppose that \( P\left( u\right) \) has order \( n \) so that \( {u}^{n} \in {k}^{ * } \) but \( {u}^{n/d} \notin k \) for any divisor of \( d \neq 1 \) . Let \( \sigma \) be the non-trivial automorphism of \( k\left( u\right) \mid k \) and let \( \xi = \sigma \left( u\right) /u \) . Then \( \xi \) is a primitiv... | Yes |
Lemma 12.5.7 Let \( A \) be a quaternion algebra over \( k \) such that \( \mathcal{C}\left( A\right) \) consists of arithmetic Fuchsian or Kleinian groups, and let \( u \in {A}^{ * } \smallsetminus {k}^{ * } \) . Then \( P\left( u\right) \) belongs to a maximal discrete group if and only if \( \operatorname{disc}\left... | Proof: Let \( \rho : A \rightarrow {M}_{2}\left( \mathbb{C}\right) \) be a \( k \) -representation. Let \( \rho \left( u\right) = \left( \begin{array}{ll} a & b \\ c & d \end{array}\right) \) so that \( {P\rho }\left( u\right) = P\left( \begin{matrix} a/\sqrt{n\left( u\right) } & b/\sqrt{n\left( u\right) } \\ c/\sqrt{n... | No |
Theorem 12.5.9 Let \( A \) be a quaternion division algebra over a number field \( k \) such that \( \mathcal{C}\left( A\right) \) is a class of arithmetic Kleinian groups. Let \( n > 2 \) . Then \( P\left( {A}^{ * }\right) \) contains an element of order \( n \) | Proof: The equivalent statements concerning elements of order \( n \) in \( P\left( {A}^{ * }\right) \) follow from Lemma 12.5.6 and Theorem 7.3.3.\n\nFor the computations below, we note that \( n\left( {1 + {\xi }_{n}}\right) = 2 + {\xi }_{n} + {\xi }_{n}^{-1} \) , \( \operatorname{disc}\left( {\xi }_{n}\right) = {\le... | Yes |
Example 12.6.2 Let \( k \) denote the field \( \mathbb{Q}\left( x\right) \), where \( {x}^{5} + {x}^{4} - 3{x}^{3} - 2{x}^{2} + x - 1 = 0 \) . Then \( {\Delta }_{k} = - {9759} \) and \( \left\{ {1, x,{x}^{2},{x}^{3},{x}^{4}}\right\} \) is an integral basis. We will show that the minimal-volume manifold derived from any... | Using Kummer's Theorem, one can identify the primes of small norm and thus obtain an approximation \( {\zeta }_{k}\left( 2\right) \approx {1.149} \) . Let \( \mathcal{O} \) be a maximal order in a quaternion algebra \( A \) defined over \( k \) so that \[ \operatorname{Vol}\left( {{\mathbf{H}}^{3}/{\Gamma }_{{\mathcal{... | Yes |
Proposition 1.24. Let \( \mathcal{D} \) be a finite-dimensional Hilbert space, and let \( V \in \) \( {C}_{\text{loc }}^{2}\left( {\mathcal{D} \rightarrow \mathbf{R}}\right) \) be such that such that \( V\left( 0\right) = 0,\nabla V\left( 0\right) = 0 \), and \( {\nabla }^{2}V\left( 0\right) \) is strictly positive def... | Proof. Applying the Picard theorem (converting the second-order ODE into a first-order ODE in the usual manner) we see that there is a maximal interval of existence \( I = \left( {{T}_{ - },{T}_{ + }}\right) \) containing 0, which supports a unique classical solution \( u \in {C}_{\mathrm{{loc}}}^{2}\left( {I \rightarr... | Yes |
Proposition 1.35 (Duhamel’s formula). Let \( I \) be a time interval, let \( {t}_{0} \) be a time in \( I \), and let \( L \in \operatorname{End}\left( \mathcal{D}\right), u \in {C}^{1}\left( {I \rightarrow \mathcal{D}}\right), f \in {C}^{0}\left( {I \rightarrow \mathcal{D}}\right) \) . Then we have\n\n(1.47)\n\n\[ \n{... | Proof. If we make the ansatz \( {}^{17}u\left( t\right) = {e}^{tL}v\left( t\right) \) for some \( v : I \rightarrow \mathcal{D} \) in (1.47), then (1.47) is equivalent to\n\n\[ \n{\partial }_{t}v\left( t\right) = {e}^{-{tL}}f\left( t\right) \n\]\n\nwhich by the fundamental theorem of calculus is equivalent to\n\n\[ \nv... | Yes |
Proposition 1.38 (Abstract iteration argument). Let \( \mathcal{N},\mathcal{S} \) be two Banach spaces. Suppose we are given a linear operator \( D : \mathcal{N} \rightarrow \mathcal{S} \) with the bound\n\n(1.51)\n\n\[ \parallel {DF}{\parallel }_{\mathcal{S}} \leq {C}_{0}\parallel F{\parallel }_{\mathcal{N}} \]\n\nfor... | This proposition is established by the arguments used to prove the contraction mapping principle, and is left as an exercise. | No |
Proposition 1.41 (Linear stability implies nonlinear stability). Let \( \mathcal{D} \) be a finite-dimensional real Hilbert space, and let \( L \in \mathrm{{End}}\left( \mathcal{D}\right) \) be a linear operator which is linearly stable in the sense that \( {}^{19} \) there exists \( \sigma > 0 \) such that \( \langle ... | Proof. The uniqueness of \( u \) follows from the Picard uniqueness theorem, so it suffices to establish existence, as well as the estimate (1.54). A simple Gronwall argument (see Exercise 1.54) gives the dissipative estimate (1.55) \[ {\begin{Vmatrix}{e}^{tL}{u}_{0}\end{Vmatrix}}_{\mathcal{D}} \leq {e}^{-{\sigma t}}{\... | No |
Proposition 1.46. Suppose that an ODE (1.57) is endowed with a Lax pair \( L : \mathcal{D} \rightarrow \operatorname{End}\left( H\right), P : \mathcal{D} \rightarrow \operatorname{End}\left( H\right) \) . Then for any non-negative integer \( k \), the moment \( \operatorname{tr}\left( {L}^{k}\right) \) is preserved by ... | Proof. We begin with the moments \( \operatorname{tr}\left( {L}^{k}\right) \) . Let \( u : I \rightarrow \mathcal{D} \) solve (1.57). From the Leibnitz rule and the first trace identity\n\n(1.60)\n\n\[ \operatorname{tr}\left( {AB}\right) = \operatorname{tr}\left( {BA}\right) \]\n\nwe have\n\n\[ {\partial }_{t}\mathrm{{... | Yes |
Proposition 3.2 (Uniqueness for classical NLS solutions). Let \( I \) be a time interval containing \( {t}_{0} \), and let \( u,{u}^{\prime } \in {C}_{t, x}^{2}\left( {I \times {\mathbf{R}}^{d} \rightarrow \mathbf{C}}\right) \) be two classical solutions to (3.1) with the same initial datum \( {u}_{0} \) for some fixed... | Proof. By time translation symmetry we can take \( {t}_{0} = 0 \) . By time reversal symmetry we may assume that \( I \) lies in the upper time axis \( \lbrack 0, + \infty ) \) . Let us write \( {u}^{\prime } = u + v \) . Then \( v \in {C}_{t, x}^{2}\left( {I \times {\mathbf{R}}^{d} \rightarrow \mathbf{C}}\right), v\le... | Yes |
Proposition 3.8 (Classical NLS solutions). Let \( p > 1 \) be an odd integer. let \( k > d/2 \) be an integer, and let \( \mu = \pm 1 \) . Then the NLS (3.1) is unconditionally locally wellposed in \( {H}_{x}^{k, k}\left( {\mathbf{R}}^{d}\right) \) in the subcritical sense. More specifically, for any \( R > \) 0 there ... | Proof. The key observations \( {}^{20} \) are Exercise 2.52, and the fact that the space \( {H}_{x}^{k, k}\left( {\mathbf{R}}^{d}\right) \) is a Banach space algebra:\n\n(3.24)\n\n\[ \parallel {fg}{\parallel }_{{H}_{x}^{k, k}\left( {\mathbf{R}}^{d}\right) }{ \lesssim }_{k, d}\parallel f{\parallel }_{{H}_{x}^{k, k}\left... | No |
Proposition 3.11 (Persistence of regularity). Let \( I \) be a time interval containing \( {t}_{0} = 0 \), let \( s \geq 0 \), and let \( u \in {C}_{t}^{0}{H}_{x}^{s}\left( {I \times {\mathbf{R}}^{d}}\right) \) be a strong \( {H}_{x}^{s} \) solution to an algebraic NLS equation. If the quantity \( \parallel u{\parallel... | Proof. We use the energy method. By time reversal symmetry we may take \( I = \left\lbrack {0, T}\right\rbrack \) for some \( T > 0 \) . From the Duhamel formula\n\n\[ u\left( t\right) = {e}^{\mathrm{i}{t\Delta }/2}u\left( 0\right) - {i\mu }{\int }_{0}^{t}{e}^{i\left( {t - {t}^{\prime }}\right) \Delta /2}{\left| u\left... | Yes |
Proposition 3.15 (Subcritical \( {L}_{x}^{2} \) NLS solutions). [Tsu] Let \( p \) be an \( {L}_{x}^{2} \) - subcritical exponent (so \( 1 < p < 1 + \frac{4}{d} \) ) and let \( \mu = \pm 1 \) . Then the NLS (3.1) is locally wellposed in \( {L}_{x}^{2}\left( {\mathbf{R}}^{d}\right) \) in the subcritical sense. More speci... | Proof. We modify the proof of Proposition 3.8. Again we fix \( R \) and choose \( T > 0 \) later. We will apply Proposition 1.38 for a suitable choice of norms \( \mathcal{S},\mathcal{N} \) and some \( \varepsilon > 0 \) ; a specific instance of our scheme in the case \( d = 1, p = 3 \) is described in Figure 2. One su... | Yes |
Proposition 3.19 \( \left( {{H}_{x}^{1}\left( {\mathbf{R}}^{3}\right) }\right. \) subcritical NLS solutions \( ) \) . Let \( \mu = \pm 1 \) . If \( 2 \leq \) \( p < 5 \), then the NLS (3.1) is locally wellposed in \( {H}_{x}^{1}\left( {\mathbf{R}}^{3}\right) \) in the subcritical sense. | Proof. (Sketch) We apply Proposition 1.38 with \( \mathcal{S} = {S}^{1}\left( {\left\lbrack {-T, T}\right\rbrack \times {\mathbf{R}}^{3}}\right) ,\mathcal{N} = \) \( {N}^{1}\left( {\left\lbrack {-T, T}\right\rbrack \times {\mathbf{R}}^{3}}\right) \) . By arguing as in Proposition 3.15, we will be done as soon as we sho... | No |
Proposition 3.21 \( \left( {{\dot{H}}_{x}^{1}\left( {\mathbf{R}}^{3}\right) }\right. \) critical NLS solutions). Let \( \mu = \pm 1 \) and \( p = 5 \) . Then the NLS (3.1) is locally wellposed in \( {\dot{H}}_{x}^{1}\left( {\mathbf{R}}^{3}\right) \) in the critical sense. More precisely, given any \( R > 0 \) there exi... | We leave the proof of this to the exercises. Note that the \( {L}_{t, x}^{10} \) norm is controlled (via Sobolev embedding) by the \( {L}_{t}^{10}{\dot{W}}_{x}^{1,{30}/{13}} \) norm, which in turn is controlled by the \( {\dot{S}}^{1} \) norm. From Strichartz estimates we thus conclude that | No |
Proposition 3.28 (Existence of wave operators). Let \( d = 3, p = 3 \), and \( \mu = + 1 \) . Then the wave operator \( {\Omega }_{ + } : {H}_{x}^{1} \rightarrow {H}_{x}^{1} \) exists and is continuous. | Proof. (Sketch) To construct the wave operator \( {\Omega }_{ + } \), we need to evolve a state at \( t = + \infty \) to \( t = 0 \) . We shall factor this problem into two sub-problems; first we shall solve the \ | No |
Proposition 3.30 (Spacetime bound implies asymptotic completeness). Let \( d = 3, p = 3 \), and \( \mu = + 1 \) . Suppose that there exists a bound of the form\n\n(3.49)\n\n\[ \parallel u{\parallel }_{{S}^{1}\left( {\mathbf{R} \times {\mathbf{R}}^{3}}\right) }{ \lesssim }_{{\begin{Vmatrix}{u}_{0}\end{Vmatrix}}_{{H}_{x}... | Proof. We shall demonstrate the surjectivity here, and leave the continuity to an exercise. We need to show that for any \( {u}_{0} \in {H}_{x}^{1} \), the global \( {H}_{x}^{1} \) -wellposed solution \( u \) to (3.1) scatters in \( {H}_{x}^{1} \) ; by the preceding discussion, this is equivalent to the conditional con... | No |
Proposition 3.32. Consider the two-dimensional defocusing cubic NLS (thus \( d = 2, p = 3,\mu = + 1 \), and the equation is \( {L}_{x}^{2} \) -critical). Let \( {u}_{0} \in {H}_{x}^{0,1} \) . Then there exists a global \( {L}_{x}^{2} \) -wellposed solution to (3.1), and furthermore the \( {L}_{t, x}^{4}\left( {\mathbf{... | Proof. We shall use an argument from [BC]. By time reversal symmetry and gluing arguments we may restrict attention to the time interval \( \lbrack 0, + \infty ) \) . Since \( {u}_{0} \) lies in \( {H}_{x}^{0,1} \), it also lies in \( {L}_{x}^{2} \) . Applying the \( {L}_{x}^{2} \) wellposedness theory (Proposition 3.1... | Yes |
Proposition 3.34. Let \( \psi \in {\mathcal{S}}_{x}\left( \mathbf{R}\right) \) and \( 0 < \varepsilon \ll 1 \) . If \( \varepsilon \) is sufficiently small depending on \( \psi \), then we have a solution \( v \) to (3.54) on the slab \( \left( {0,1}\right) \times \mathbf{R} \) obeying the bounds\n\n\[ \parallel v\left... | Proof. To construct \( v \), we use the ansatz \( v = \widetilde{v} + w \) . Subtracting (3.55) from (3.54), we see that \( w \) needs to solve the equation\n\n\[ i{\partial }_{t}w + \frac{1}{2}{\partial }_{xx}w = \frac{1}{{t}^{\left( {5 - p}\right) /2}}\left( {F\left( {\widetilde{v} + w}\right) - F\left( \widetilde{v}... | Yes |
Proposition 3.38 (Almost conservation law). Let \( s > 1/2 \) . Suppose \( t \) is a time such that \( E\left\lbrack {{Iu}\left( t\right) }\right\rbrack \lesssim 1 \) . Then \( \left| {{\partial }_{t}E\left\lbrack {{Iu}\left( t\right) }\right\rbrack }\right| { \lesssim }_{s}{N}^{-1/2} \) . | Proof. For a general classical field \( v \), we have the identity\n\n\[ \n{\partial }_{t}E\left\lbrack {v\left( t\right) }\right\rbrack = - 2\operatorname{Re}{\int }_{\mathbf{R}}\overline{{\partial }_{t}v}\left( {i{\partial }_{t}v + \frac{1}{2}{\partial }_{xx}v - {\left| v\right| }^{4}v}\right) {dx} \n\]\n\nwhich can ... | No |
Proposition 4.1 (Maximal function estimates). [KPV2] Let \( u \) be an \( {H}_{x}^{s} \) solution to the inhomogeneous Airy equation \( {\partial }_{t}u + {\partial }_{xxx}u = F \) on \( \left\lbrack {0, T}\right\rbrack \times \mathbf{R} \). Then we have\n\n(4.13)\n\n\[ \parallel u{\parallel }_{{L}_{t}^{2}{L}_{x}^{\inf... | These estimates are proven by a \( T{T}^{ * } \) argument similar to those used to prove \( \textbf{Strichartz estimates, but with the roles of space and time reversed; see [KPV2].} \) | No |
Corollary 4.3. Let \( {\mathbf{C}}^{n} \) be a standard complex phase space. let \( H \in {C}_{\mathrm{{loc}}}^{2}\left( {{\mathbf{C}}^{n} \rightarrow }\right. \) R) be a Hamiltonian of quadratic growth (to ensure global solutions), and let \( u \) be a classical solution to the Hamiltonian flow (1.28). Then for any \(... | Thus the Hamiltonian flow cannot have any sort of \ | No |
Proposition 4.4 (Uniform weak approximation by finite-dimensional flow). [Bou2] Let \( N \gg 1 \) and \( T > 0 \) . Let \( S\left( T\right) : {l}^{2}\left( \mathbf{Z}\right) \rightarrow {l}^{2}\left( \mathbf{Z}\right) \) denote the evolution map associated to the NLS (4.29) for time \( T \), and let \( {S}_{N}\left( T\... | It is important here that this approximation property hold for arbitrarily large data and long times, that the bound is uniform for all data \( {u}_{0} \) of a given size, and that the bound goes to zero as \( N \rightarrow \infty \) . It asserts that the flow \( {S}_{N}\left( T\right) \) becomes an increasingly good a... | No |
Proposition 5.5 (Exterior energy decay). Let \( u \) be as in Proposition 5.3. Then we have\n\n\[ \mathop{\inf }\limits_{{\sigma > 0}}\mathop{\limsup }\limits_{{t \rightarrow {T}_{ * }^{ - }}}{E}_{B\left( {x,{T}_{ * } - t + \sigma }\right) \smallsetminus B\left( {x,{T}_{ * } - t}\right) }\left\lbrack {u\left\lbrack t\r... | This Proposition follows easily from the energy flux machinery that we develop later in this section and is deferred to an exercise. | No |
Proposition 5.7 (Blowup implies potential energy concentration). If \( u \) is an \( {H}_{x}^{1} \times {L}_{x}^{2} \) solution with energy at most \( {E}_{0} \) with a maximal time of existence \( 0 < {T}_{ * } < \infty \), then there exists \( x \in {\mathbf{R}}^{3} \) such that \[ \mathop{\limsup }\limits_{{t \right... | Proof. Suppose for contradiction that for every \( x \in {\mathbf{R}}^{3} \) that we have \[ \mathop{\limsup }\limits_{{t \rightarrow {T}_{ * }^{ - }}}{\int }_{B\left( {x,{T}_{ * } - t}\right) }{\left| u\left( t, y\right) \right| }^{6}{dy} < {\epsilon }_{2} \] where \( {\epsilon }_{2} = {\epsilon }_{2}\left( {E}_{0}\ri... | No |
Proposition 5.9 (Non-concentration of potential energy). [Stru2], [SStru2], [Gri2] If \( u \) is an \( {H}_{x}^{1} \times {L}_{x}^{2} \) solution with energy at most \( {E}_{0} \) with a maximal time of existence \( 0 < {T}_{ * } < \infty \), then\n\n\[{\int }_{B\left( {0,{T}_{ * } - t}\right) }{\left| u\left( t, x\rig... | To prove Proposition 5.9 we need to introduce some more non-perturbative methods. To justify certain formal computations (such as energy identities) let us assume for the moment that \( u \) is in fact smooth on \( \left\lbrack {0,{T}_{ * }}\right) \times {\mathbf{R}}^{3} \) ; this hypothesis can be removed once the co... | No |
Proposition 5.15 (Frequency delocalisation implies spacetime bound). Let \( u : {I}_{ * } \times {\mathbf{R}}^{3} \rightarrow \mathbf{C} \) be a classical solution to (5.2) of energy at most \( {E}_{\text{crit }} \) . Let \( \eta > 0 \) , and suppose there exists a dyadic frequency \( {N}_{\text{lo }} > 0 \) and a time... | Proof. (Sketch) We introduce a small parameter \( \varepsilon > 0 \) depending on \( \eta \) . The first task is to find a safe location in frequency space with which to truncate the solution into noninteracting components. If \( K\left( \eta \right) \) is sufficiently large depending on \( \varepsilon \), we can use P... | Yes |
Corollary 5.16 (Frequency localisation of energy at each time). [CKSTT11] Let \( u \) be a minimal energy blowup solution. Then for every time \( t \in {I}_{ * } \) there exists a dyadic frequency \( N\left( t\right) \in {2}^{\mathbb{Z}} \) such that for every \( {\eta }_{5} \leq \eta \leq {\eta }_{0} \) we have small ... | Here \( 0 < \mathrm{c}\left( \eta \right) \ll 1 \ll C\left( \eta \right) < \infty \) are quantities depending on \( \eta \). (5.33) \[ {\begin{Vmatrix}{P}_{ \leq c\left( \eta \right) N\left( t\right) }u\left( t\right) \end{Vmatrix}}_{{\dot{H}}_{x}^{1}} \leq \eta , \] (5.34) \[ {\begin{Vmatrix}{P}_{ \geq C\left( \eta \r... | Yes |
Proposition 5.17 (Physical space localisation of energy at each time). Let \( u,{I}_{0}, N\left( \right) \) be as above. Then for every \( t \in {I}_{0} \), there exists an \( x\left( t\right) \in {\mathbf{R}}^{3} \) such that\n\n(5.36)\n\n\[{\int }_{\left| {x - x\left( t\right) }\right| \leq C\left( {\eta }_{1}\right)... | The properties (5.36), (5.37), (5.38) were already essentially obtained in Proposition 5.12, modulo some technical details such as exceptional intervals, and the fact that the estimates there only held for a significant fraction of each interval \( {I}_{1} \) rather than being universal for all \( t \) in the middle in... | Yes |
Proposition 5.18 (Reverse Sobolev inequality). Let \( u \) be a minimal blowup solution. Then for every \( {t}_{0} \in {I}_{0} \), any \( {x}_{0} \in {\mathbf{R}}^{3} \), and any \( R \geq 0 \), \[ {\int }_{B\left( {{x}_{0}, R}\right) }{\left| \nabla u\left( {t}_{0}, x\right) \right| }^{2}{dx} \lesssim {\eta }_{1} + {O... | The deduction of this proposition from Proposition 5.17 is rather easy and is left as an exercise. This should be compared with (5.16), which gives a bound of \( O\left( 1\right) \) for the left-hand side of (5.40). Thus, we can assume the kinetic energy is small, except on those regions of space on which the potential... | No |
Proposition 5.19 (Frequency-localised interaction Morawetz estimate). Let \( u \) be a minimal blowup solution. Then for all \( {N}_{ * } < \mathrm{c}\left( {\eta }_{3}\right) {N}_{\min } \)\n\n\[{\int }_{{I}_{0}}\int {\left| {P}_{ \geq {N}_{ * }}u\left( t, x\right) \right| }^{4}{dxdt} \lesssim {\eta }_{1}{N}_{ * }^{-3... | We now briefly discuss the proof of this proposition. We rescale \( {N}_{ * } = 1 \) and define \( {u}_{\mathrm{{hi}}},{u}_{\mathrm{{lo}}} \) as before. The hypothesis \( {N}_{ * } < \mathrm{c}\left( {\eta }_{3}\right) {N}_{\min } \) ensures that \( {u}_{\mathrm{{lo}}} \) has very small energy, and also that \( {u}_{\t... | Yes |
Lemma 3.9. Con is a closure operation on \( P\left( X\right) \), that is, it has the properties\n\n(i) \( A \subseteq \operatorname{Con}\left( A\right) \) ,\n\n(ii) If \( {A}_{1} \subseteq {A}_{2} \), then \( \operatorname{Con}\left( {A}_{1}\right) \subseteq \operatorname{Con}\left( {A}_{2}\right) \) ,\n\n(iii) \( \ope... | Proof:\n\n(i) Trivial.\n\n(ii) Suppose \( q \in \operatorname{Con}\left( {A}_{1}\right) \) . Let \( v \) be any valuation such that \( v\left( {A}_{2}\right) \subseteq \) \( \{ 1\} \) . Then \( v\left( {A}_{1}\right) \subseteq \{ 1\} \) and so \( v\left( q\right) = 1 \) since \( q \in \operatorname{Con}\left( {A}_{1}\r... | Yes |
Lemma 4.4. (i) If \( q \in \operatorname{Ded}\left( A\right) \), then \( q \in \operatorname{Ded}\left( {A}^{\prime }\right) \) for some finite subset \( {A}^{\prime } \) of \( A \) . | Proof: (i) This holds because a proof of \( q \) from \( A \), being a finite sequence of elements of \( P\left( X\right) \), can contain only finitely many members of \( A \) . | Yes |
Theorem 4.11. (The Substitution Theorem). Let \( X, Y \) be any two sets, and let \( \varphi : P\left( X\right) \rightarrow P\left( Y\right) \) be a homomorphism of the (free) proposition algebra on \( X \) into the (free) proposition algebra on \( Y \) . Let \( w = w\left( {{x}_{1},\ldots ,{x}_{n}}\right) \) be any el... | Proof: (a) Suppose \( {p}_{1},\ldots ,{p}_{r} \) is a proof of \( w \) from \( A \) . If \( {p}_{i} \in A \), then trivially \( \varphi \left( {p}_{i}\right) \in \varphi \left( A\right) \) . Since \( \varphi \) is a homomorphism, it follows that if \( {p}_{i} \) is an axiom of the propositional calculus on \( X \), the... | Yes |
Theorem 2.1. (The Soundness Theorem) Let \( A \subseteq P\left( X\right), p \in P\left( X\right) \) . If \( A \vdash p \), then \( A \vDash p \) . | Proof: Suppose there exists a proof \( {p}_{1},\ldots ,{p}_{n} \) of \( p \) from \( A \) . We have to show \( p \) is a consequence of \( A \) .\n\nLet \( v : P\left( X\right) \rightarrow {\mathbb{Z}}_{2} \) be a valuation for which \( v\left( A\right) \subseteq \{ 1\} \) . We shall use induction over the length \( n ... | Yes |
Corollary 2.2. (The Consistency Theorem) \( F \) is not a theorem of \( \operatorname{Prop}\left( X\right) \) . | Proof: If \( \vdash F \), then \( \vDash F \) by the Soundness Theorem. Since axioms are tautologies, \( v\left( F\right) = 1 \) for every valuation \( v \), contradicting the definition of valuation. This implies that there are no valuations. But \( P\left( X\right) \) is free and every map of \( X \) into \( {\mathbb... | Yes |
Theorem 2.4. (The Deduction Theorem) Let \( A \subseteq P\left( X\right) \), and let \( p \) , \( q \in P\left( X\right) \). Then \( A \vdash p \Rightarrow q \) if and only if \( A \cup \{ p\} \vdash q \). | Proof: (a) Suppose \( A \vdash p \Rightarrow q \). Let \( {p}_{1},\ldots ,{p}_{n} \) be a proof of \( {p}_{n} = p \Rightarrow q \) from \( A \). Then \( {p}_{1},\ldots ,{p}_{n}, p, q \) is a proof of \( q \) from \( A \cup \{ p\} \).\n\n(b) Suppose \( A \cup \{ p\} \vdash q \). Then we have a proof \( {p}_{1},\ldots ,{... | Yes |
We show \( \{ p \Rightarrow q, q \Rightarrow r\} \vdash p \Rightarrow r \) . | First we show \( \{ p, p \Rightarrow q, q \Rightarrow r\} \vdash r \), and a proof of this is \( p, p \Rightarrow q, q, q \Rightarrow r, r \). It follows from the Deduction Theorem that \( \{ p \Rightarrow q, q \Rightarrow r\} \vdash p \Rightarrow r \) . | Yes |
Lemma 2.10. The subset \( A \subseteq P\left( X\right) \) is maximal consistent if and only if\n\n(i) \( F \notin A \), and\n\n(ii) \( A = \operatorname{Ded}\left( A\right) \), and\n\n(iii) for all \( p \in P\left( X\right) \), either \( p \in A \) or \( \sim p \in A \) . | Proof: (a) Let \( A \) be maximal consistent. Since \( A \) is consistent, \( F \notin \) \( \operatorname{Ded}\left( A\right) \) and therefore \( F \notin A \) . Since \( \operatorname{Ded}\left( {\operatorname{Ded}\left( A\right) }\right) = \operatorname{Ded}\left( A\right) ,\operatorname{Ded}\left( A\right) \) is co... | Yes |
Lemma 2.11. Let \( A \) be a consistent subset of \( P\left( X\right) \) . Then \( A \) is contained in a maximal consistent subset. | Proof: Let \( \sum = \{ T \subseteq P\left( X\right) \mid T \supseteq A, F \notin \operatorname{Ded}\left( T\right) \} \) . Since \( A \in \sum ,\sum \neq \varnothing \) . Suppose \( \left\{ {T}_{\alpha }\right\} \) is a totally ordered family of members of \( \sum \), and put \( T = \mathop{\bigcup }\limits_{\alpha }{... | Yes |
Theorem 2.12. (The Satisfiability Theorem) Let \( A \) be a consistent subset of \( P\left( X\right) \). Then there exists a valuation \( v : P\left( X\right) \rightarrow {\mathbb{Z}}_{2} \), such that \( v\left( A\right) \subseteq \{ 1\} \). | Proof: Let \( M \) be a maximal consistent subset containing \( A \). For \( p \in P\left( X\right) \), put \( v\left( p\right) = 1 \) if \( p \in M \) and \( v\left( p\right) = 0 \) if \( p \notin M \). We now prove \( v \) is a valuation.\n\nCertainly \( v\left( F\right) = 0 \), because \( F \notin M \). It remains t... | Yes |
Theorem 2.13. (The Adequacy Theorem) Let \( A \subseteq P\left( X\right), p \in P\left( X\right) \). If \( A \vDash p \) in \( \operatorname{Prop}\left( X\right) \), then \( A \vdash p \) in \( \operatorname{Prop}\left( X\right) \). | Proof: Suppose \( A \vDash p \), so that \( v\left( A\right) \subseteq \{ 1\} \) implies \( v\left( p\right) = 1 \) for every valuation \( v \). If \( A \cup \{ \sim p\} \) is consistent, it follows from the Satisfiability Theorem that there is a valuation \( v \) such that \( v\left( {A\cup \{ \sim p\} }\right) \subse... | Yes |
Theorem 3.3. \( L\left( {X}_{n}\right) \) is the set of all truth functions \( f : {\mathbb{Z}}_{2}^{n} \rightarrow {\mathbb{Z}}_{2} \) . | Proof: The constant functions \( 0,1 \in L\left( {X}_{n}\right) \) since \( 0 = \bar{F} \) and \( 1 = \left( \overline{F \Rightarrow F}\right) \) . Thus the result holds for \( n = 0 \) .\n\nIf \( f, g \) are truth functions \( {\mathbb{Z}}_{2}^{n} \rightarrow {\mathbb{Z}}_{2} \), we define the truth function \( f \Rig... | Yes |
Lemma 3.4. Let \( w = w\left( {{x}_{1},\ldots ,{x}_{n}}\right) \in P\left( X\right) \). Then \( \vDash w \) if and only if its associated truth function \( \bar{w} : {\mathbb{Z}}_{2}^{n} \rightarrow {\mathbb{Z}}_{2} \) is the constant 1. | Proof: Suppose \( \bar{w} = 1 \). Let \( v : P\left( X\right) \rightarrow {\mathbb{Z}}_{2} \) be any valuation of \( P\left( X\right) \). Put \( {a}_{i} = v\left( {x}_{i}\right) \). Then the restriction of \( v \) to \( P\left( {X}_{n}\right) \) is a valuation of \( P\left( {X}_{n}\right) \), and \( v\left( w\right) = ... | Yes |
Corollary 3.6. \( \operatorname{Prop}\left( X\right) \) is decidable for provability. | Proof. An element \( p \in P\left( X\right) \) is a theorem if and only if it is valid. | No |
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