Q stringlengths 4 3.96k | A stringlengths 1 3k | Result stringclasses 4
values |
|---|---|---|
Lemma 3.5.1 Let \( \gamma \in \Gamma \) . Then \( \operatorname{tr}\gamma \) is an integer polynomial in \( \{ \operatorname{tr}\delta \mid \delta \in P\} \) . | Proof: We proceed by induction on the length of \( \gamma \) . From (3.13) and (3.14), the result is clearly true if \( \ell \left( \gamma \right) = 1 \) or 2 . So suppose \( \ell \left( \gamma \right) \geq 3 \) and the result holds for all elements of length less than \( \ell \left( \gamma \right) \) . If \( \gamma \n... | Yes |
Lemma 3.5.2 Let \( \gamma \in \Gamma \) . Then \( \operatorname{tr}\gamma \) is an integer polynomial in \( \{ \operatorname{tr}\delta \mid \delta \in Q\} \) . | Proof: For each permutation \( \tau \) of \( {S}_{n} \), define\n\n\[ \n{\tau }^{ * }\left( Q\right) = \left\{ {{\gamma }_{\tau \left( {i}_{1}\right) }\ldots {\gamma }_{\tau \left( {i}_{r}\right) } \mid 1 \leq {i}_{1} < \cdots < {i}_{r} \leq n}\right\} \n\]\n\nso that \( P = { \cup }_{\tau \in {S}_{n}}{\tau }^{ * }\lef... | Yes |
Lemma 3.5.3 Let \( \gamma \in \Gamma \) . Then \( \operatorname{tr}\gamma \) is a rational polynomial in \( \{ \operatorname{tr}\delta \mid \) \( \delta \in R\} \) . | Proof: This follows immediately from Lemma 3.5.2 and the identity (3.20). \( ▱ \) | No |
Lemma 3.5.5 With \( \Gamma \) as above and \( \operatorname{tr}{\gamma }_{i} \neq 0 \) for \( i = 1,2,\ldots, n \), then \( {k\Gamma } = \mathbb{Q}\left( {\operatorname{tr}{\Gamma }^{SQ}}\right) \) | Proof: Clearly \( {\Gamma }^{SQ} \subset {\Gamma }^{\left( 2\right) } \) so that \( \mathbb{Q}\left( {\operatorname{tr}{\Gamma }^{SQ}}\right) \subset {k\Gamma } \) . Now from (3.12), if \( \operatorname{tr}\gamma \neq 0 \), then \( \gamma = {\left( \operatorname{tr}\gamma \right) }^{-1}\left( {{\gamma }^{2} + I}\right)... | Yes |
Lemma 3.5.6 Let \( \Gamma \) be a finitely generated non-elementary subgroup of \( \mathrm{{SL}}\left( {2,\mathbb{C}}\right) \) . Let \( k = \mathbb{Q}\left( \left\{ {\operatorname{tr}{\gamma }^{2} : \gamma \in \Gamma }\right\} \right) \) . Then \( k = {k\Gamma } \) . | Proof: Note that \( \operatorname{tr}{\gamma }^{2} = {\operatorname{tr}}^{2}\gamma - 2 \) so that \( k \subset {k\Gamma } \) . Now choose a set of generators \( {\gamma }_{1},\ldots ,{\gamma }_{n} \) of \( \Gamma \) such that \( \operatorname{tr}{\gamma }_{i} \neq 0,\operatorname{tr}{\gamma }_{i}^{2}{\gamma }_{j}^{2} \... | Yes |
Lemma 3.5.8 Let \( \Gamma = \langle g, h\rangle \), with \( \operatorname{tr}h = 0 \), be a non-elementary subgroup of \( \mathrm{{SL}}\left( {2,\mathbb{C}}\right) \). Then\n\n\[ \n{k\Gamma } = \mathbb{Q}\left( {{\operatorname{tr}}^{2}g,\operatorname{tr}\left\lbrack {g, h}\right\rbrack }\right) \n\] | It is of interest to note how these relate to the invariant trace field when \( \Gamma \) is non-elementary. In the case where \( \operatorname{tr}h = 0 \), it is immediate from Lemma 3.5.8 that\n\n\[ \n{k\Gamma } = \mathbb{Q}\left( {\gamma \left( {g, h}\right) ,\beta \left( g\right) }\right) \n\] | No |
Lemma 3.5.9 Let \( \Gamma = \left\langle {{\gamma }_{1},{\gamma }_{2},{\gamma }_{3}}\right\rangle \), with \( \operatorname{tr}{\gamma }_{i} \neq 0 \) for \( i = 1,2 \) and 3 . Then \( {k\Gamma }\;{is}\;{generated}\;{over}\;\mathbb{Q}\;{by}\;\{ \operatorname{tr}{}^{2}{\gamma }_{i},1 \leq i \leq 3;\;\operatorname{tr}{\g... | Proof: From Lemma 3.5.5 and (3.26), \( {k\Gamma } \) is generated over \( \mathbb{Q} \) by the traces of seven elements. Then using (3.28) and \[ {\gamma }_{1}^{2}{\gamma }_{2}^{2}{\gamma }_{3}^{2} = \mathop{\prod }\limits_{{i = 1}}^{3}\left( {\left( {\operatorname{tr}{\gamma }_{i}}\right) {\gamma }_{i} - I}\right) \] ... | Yes |
Theorem 3.6.2 If \( g \) and \( h \) are elements of the non-elementary group \( \Gamma \) such that \( \langle g, h\rangle \) is irreducible, \( g \) and \( h \) do not have order 2 in \( \operatorname{PSL}\left( {2,\mathbb{C}}\right) \) and \( g \) is not parabolic, then\n\n\[ \n{A\Gamma } = \left( \frac{{\operatorna... | Proof: The elements \( {g}^{2} \) and \( {h}^{2} \) satisfy the conditions stated in the previous theorem so we can apply the method used in the proof of that theorem. Thus in (3.35), replacing \( {t}_{0} \) by \( {t}_{0}^{2} - 2,{t}_{1} \) by \( {t}_{1}^{2} - 2 \) and \( {t}_{2} \) by \( {t}_{0}{t}_{1}{t}_{2} - {t}_{0... | Yes |
Theorem 4.2.1 Let \( M = {\mathbf{H}}^{3}/\Gamma \) be a hyperbolic manifold such that the cokernel of the map \( \left( {{H}_{1}\left( {\partial \bar{M},\mathbb{Z}}\right) \rightarrow {H}_{1}\left( {\bar{M},\mathbb{Z}}\right) }\right) \) is finite of odd order. Then \( {k\Gamma } = \mathbb{Q}\left( {\operatorname{tr}\... | Proof: Let \( P \) denote the subgroup of \( \Gamma \) which is generated by parabolic elements. Then \( \Gamma /P \) is isomorphic to the \( \operatorname{Coker}\left( {{H}_{1}\left( {\partial \bar{M},\mathbb{Z}}\right) \rightarrow {H}_{1}\left( {\bar{M},\mathbb{Z}}\right) }\right) \) . Now \( \Gamma /{\Gamma }^{\left... | Yes |
Corollary 4.2.2 If \( M = {\mathbf{H}}^{3}/\Gamma \) is the complement of a link in a \( \mathbb{Z}/2 \) - homology sphere, then \( {k\Gamma } = \mathbb{Q}\left( {\operatorname{tr}\Gamma }\right) \) and \( {A\Gamma } = {M}_{2}\left( {\mathbb{Q}\left( {\operatorname{tr}\Gamma }\right) }\right) \) . | Proof: The first part follows from the theorem and the second from the fact that \( M \) is non-compact (see Theorem 3.3.8). | No |
Theorem 4.2.3 If \( \Gamma \) is any Kleinian group of finite covolume which is non-cocompact, then \( \Gamma \) will have a faithful discrete representation in the \( \operatorname{group}\operatorname{PSL}\left( {2,\mathbb{Q}\left( {\operatorname{tr}\Gamma }\right) }\right) \) . | Proof: Choose a lift of a cusp of \( \Gamma \) to be at \( \infty \) and normalise so that the parabolic element \( g = \left( \begin{array}{ll} 1 & 1 \\ 0 & 1 \end{array}\right) \) lies in \( \Gamma \) . With further normalisation, let \( f \in \Gamma \) be such that \( f\left( \infty \right) = 0 \) . Thus \( \Gamma \... | Yes |
Theorem 4.3.1 If \( \Delta \) is a finitely generated non-elementary normal subgroup of the finitely generated Kleinian group \( \Gamma \), then \( {k\Delta } = {k\Gamma } \) and also \( {A\Delta } = {A\Gamma } \). | Proof: The group \( {\Delta }^{\left( 2\right) } \) is characteristic in \( \Delta \) and thus normal in \( \Gamma \). Also\n\n\[ \n{k\Delta } = \mathbb{Q}\left( {\operatorname{tr}{\Delta }^{\left( 2\right) }}\right) \subset \mathbb{Q}\left( {\operatorname{tr}{\Gamma }^{\left( 2\right) }}\right) = {k\Gamma }.\n\]\n\nCh... | Yes |
Corollary 4.3.3 If \( \Gamma \) is the covering group of a hyperbolic fibre bundle as at (4.1) and \( {F}_{1} \) is a subgroup of finite index in \( F \), which lies in \( {\Gamma }^{\left( 2\right) } \), then \( {kF} = \mathbb{Q}\left( {\operatorname{tr}{F}_{1}}\right) = {k\Gamma } \) and \( {AF} = {A}_{0}{F}_{1} = {A... | Proof: Since \( {F}_{1} \subset {\Gamma }^{\left( 2\right) } \), it follows as in the proof of the theorem that \( {A\Gamma } = {A}_{0}{F}_{1} \cdot {k\Gamma } \). Furthermore,\n\n\[ \n{kF} = \mathbb{Q}\left( {\operatorname{tr}{F}_{1}^{\left( 2\right) }}\right) \subset \mathbb{Q}\left( {\operatorname{tr}{F}_{1}}\right)... | Yes |
Theorem 4.8.1 There exist infinitely many commensurability classes of compact hyperbolic 3-manifolds. | For \( n = 2 \), solving (4.10) and (4.11) gives \( a = \left( \frac{3 + \sqrt{17}}{2}\right) \left( \frac{1 + \sqrt{(4 - \sqrt{17}}}{2}\right) \) . Thus \( k{\Gamma }_{2} = \mathbb{Q}\left( a\right) \) has degree 4 over \( \mathbb{Q} \), one complex and two real places. By a direct calculation on determining when \( x... | No |
Theorem 5.1.2 Let \( \Gamma \) be a finitely generated subgroup of \( \operatorname{PSL}\left( {2,\mathbb{C}}\right) \) such that the following three conditions all hold.\n\n1. \( {\Gamma }^{\left( 2\right) } \) is irreducible.\n\n2. \( \operatorname{tr}\left( \Gamma \right) \) consists of algebraic integers.\n\n3. For... | Proof: Note that since \( \Gamma \) is finitely generated, so is \( {\Gamma }^{\left( 2\right) } \) and so from \( §{3.5} \) , all traces in \( {\Gamma }^{\left( 2\right) } \) are obtained from integral polynomials in a finite number of traces. Thus \( {k\Gamma } \) is a finite extension of \( \mathbb{Q} \) .\n\nIt suf... | Yes |
Lemma 5.1.3 With \( \Gamma \) as described in Theorem 5.1.2 satisfying conditions 1 and 2, condition 3 is equivalent to the following requirement:\n\n\( {3}^{\prime } \) . All embeddings \( \sigma \), apart from the identity and \( \mathbf{c} \), complex conjugation, are real and \( {A\Gamma } \) is ramified at all rea... | Proof: If condition \( {\mathcal{3}}^{\prime } \) holds and \( \sigma : {k\Gamma } \rightarrow \mathbb{R} \), then there exists \( \tau : {A\Gamma } \rightarrow \) \( \mathcal{H} \), Hamilton’s quaternions, such that \( \sigma \left( {\operatorname{tr}f}\right) = \operatorname{tr}\left( {\tau \left( f\right) }\right) \... | Yes |
Lemma 5.2.4 Let \( \Gamma \) be a finitely generated non-elementary subgroup of \( \mathrm{{SL}}\left( {2,\mathbb{C}}\right) \). Then \( \Gamma \) has integral traces if and only if \( \Gamma \) is conjugate in \( \mathrm{{SL}}\left( {2,\mathbb{C}}\right) \) to a subgroup of \( \mathrm{{SL}}\left( {2,\mathbb{A}}\right)... | Proof: One way is obvious, so we assume that \( \Gamma \) has integral traces. Since \( \Gamma \) is finitely generated, the trace field of \( \Gamma \) is a finite extension \( k \) of \( \mathbb{Q} \). Let \( {A}_{0}\Gamma \) be the quaternion algebra generated over \( k \) by elements of \( \Gamma \) and \( \mathcal... | Yes |
Theorem 5.2.7 Let \( G \) be a subgroup of \( \mathrm{{SL}}\left( {2, K}\right) \) which is not virtually solvable and contains an element \( g \) for which \( v\left( {\operatorname{tr}g}\right) < 0 \) . Then \( G \) has a non-trivial splitting as the fundamental group of a graph of groups. | ## 5.2.2 Non-integral Traces\n\nThe proof of Theorem 5.2.2 can now be completed. The trace field, \( k \), of \( \Gamma \) is a finite extension of \( \mathbb{Q} \) . By Corollary 3.2.4, we can assume that \( \widehat{\Gamma } \) is a subgroup of \( \mathrm{{SL}}\left( {2, L}\right) \), where \( \left\lbrack {L : k}\ri... | Yes |
Theorem 5.3.1 Let \( \Gamma \) be a Kleinian group of finite covolume which satisfies the following conditions:\n\n(a) \( {k\Gamma } \) contains no proper subfield other than \( \mathbb{Q } \).\n\n(b) \( {A\Gamma } \) is ramified at at least one infinite place of \( {k\Gamma } \).\n\nThen \( \Gamma \) contains no hyper... | Proof: Note that \( \Gamma \) contains a hyperbolic element if and only if \( {\Gamma }^{\left( 2\right) } \) contains a hyperbolic element. Let us suppose that \( \gamma \in {\Gamma }^{\left( 2\right) } \) is hyperbolic, and let \( t = \operatorname{tr}\left( \gamma \right) \) . By assumption, \( t \in {k\Gamma } \cap... | Yes |
Theorem 5.3.4 Let \( M \) be a closed hyperbolic 3-manifold containing a totally geodesic immersion of a closed surface. Then there is a finite covering of \( M \) which contains an embedded closed orientable totally geodesic surface. | To prove this theorem, we first recall the notion of subgroup separability.\n\nDefinition 5.3.5 Let \( G \) be a group and \( H \) a finitely generated subgroup. Then \( G \) is said to be H-subgroup separable if given any element \( g \in G \smallsetminus H \) , there is a finite index subgroup \( K \) of \( G \) with... | No |
Lemma 5.3.6 Let \( \mathcal{C} \) be a circle or straight line in \( \mathbb{C} \cup \infty \) and \( M = {\mathbf{H}}^{3}/\Gamma \) , a closed hyperbolic 3-manifold. Let \[ \operatorname{Stab}\left( {\mathcal{C},\Gamma }\right) = \{ \gamma \in \Gamma : \gamma \mathcal{C} = \mathcal{C}\} \] Then \( \operatorname{Stab}\... | Proof: Let \( H \) denote \( \operatorname{Stab}\left( {\mathcal{C},\Gamma }\right) \) . We may assume without loss of generality that \( H \neq 1 \) -because 1 is separable since \( \Gamma \) is residually finite. Note that \( H \) is either a Fuchsian group or a \( {\mathbb{Z}}_{2} \) -extension of a Fuchsian group. ... | Yes |
Lemma 5.3.7 Let \( M = {\mathbf{H}}^{3}/\Gamma \) be a finite-volume hyperbolic 3-manifold and \( f : S \hookrightarrow M \) be an incompressible immersion of a closed surface. Let \( H = {f}_{ * }\left( {{\pi }_{1}\left( S\right) }\right) \subset \Gamma \) . If \( \Gamma \) is H-subgroup separable, there is a finite c... | Proof: Let \( p \) denote the cover \( {\mathbf{H}}^{3} \rightarrow M \) . Since \( S \) is compact, standard covering space arguments imply that there is a compact set \( D \subset {\mathbf{H}}^{3} \) with \( p\left( D\right) = f\left( S\right) \) . Since \( \Gamma \) acts discontinuously on \( {\mathbf{H}}^{3} \), th... | Yes |
Example 5.3.9 Twist Knots: Certain twist knots as shown in Figure 5.2 furnish examples which satisfy the conditions of Theorem 5.3.8 (see Theorem 1.5.6). These twist knots are two-bridge knots of the form \( \left( {p/p - 2}\right) \) (see \( §{4.5} \) ). If we choose \( p \) to be of the form \( {4m} + 3 \), then we o... | In the cases \( m = \) \( 1,2 \), we obtain, respectively, the polynomials\n\n\[ 1 + {2z} - 3{z}^{2} + {z}^{3},\;1 + {3z} - {13}{z}^{2} + {16}{z}^{3} - 7{z}^{4} + {z}^{5}, \]\n\nwhich are irreducible over \( \mathbb{Q} \) . | Yes |
Lemma 5.3.11 The points \( {a}_{1},{a}_{2},{b}_{1} \) and \( {b}_{2} \) lie on a circle in \( \mathbf{C} \cup \infty \) . The cross-ratio \( \left\lbrack {{a}_{1},{a}_{2},{b}_{1},{b}_{2}}\right\rbrack \) is a real number lying in the interval \( \left( {0,1}\right) \) . | Proof: By an element of \( \operatorname{PSL}\left( {2,\mathbb{C}}\right) \), we can map \( {a}_{1} \rightarrow 0,{b}_{1} \rightarrow 1 \) and \( {b}_{2} \rightarrow \infty \) . Assume that \( {a}_{2} \) maps to \( w \) . Because \( {\delta A} \neq A \) and \( {\delta A} \cap A \neq \varnothing \) , \( w \) must be a r... | Yes |
Lemma 5.3.12 With notation as above, \( {t}^{2},{rt} \in {k\Gamma } \) and, hence, so does \( t/r = - {\left\lbrack {a}_{1},{a}_{2},{b}_{1},{b}_{2}\right\rbrack }^{-1}. \) | Proof: Since \( {t}^{2} = {\operatorname{tr}}^{2}\gamma - 4 = {\operatorname{tr}}^{2}\eta - 4,{t}^{2} \in {k\Gamma } \). Also, the element \( \gamma {\eta }^{-1} \) is a commutator in \( \Gamma \) and so lies in \( {\Gamma }^{\left( 2\right) } \). Thus \( {rt} = 2 - \operatorname{tr}\left( {\gamma {\eta }^{-1}}\right) ... | Yes |
Theorem 5.3.13 If \( M \) has a non-simple closed geodesic, then \( {A\Gamma } \cong \left( \frac{a, b}{k\Gamma }\right) \) for some \( a \in {k\Gamma } \) and \( b \in {k\Gamma } \cap \mathbb{R} \) . | Proof: Assume that \( M \) has a non-simple geodesic \( g \) . We shall compute the expression for \( {A\Gamma } \) using the elements \( \eta \) and \( {\gamma }^{-1} \) described above, which generate an irreducible subgroup. Thus \( {A\Gamma } \cong \left( \frac{a,{b}^{\prime }}{k}\right) \), where \( a = \operatorn... | Yes |
Corollary 5.3.14 With the notation of Theorem 5.3.13, suppose that there are no elements \( a \in {k\Gamma } \) and \( b \in {k\Gamma } \cap \mathbb{R} \) such that \( {A\Gamma } \) is isomorphic over \( {k\Gamma } \) to the quaternion algebra \( \left( \frac{a, b}{k\Gamma }\right) \) . Then all of the closed geodesics... | It will be shown in \( §{9.7} \), that there exist number fields \( k \) with exactly one complex place and quaternion algebras over \( k \) such that there are no elements \( a \in k, b \in k \cap \mathbb{R} \) as described in this corollary. The arithmetic groups \( \Gamma \) which arise from these, furnish examples ... | No |
Lemma 5.4.1 Let \( \Gamma \) be a Kleinian group of finite covolume with invariant quaternion algebra \( A \) and number field \( k \) . If \( \Gamma \) contains a subgroup isomorphic to \( {A}_{4} \), then\n\n\[ A \cong \left( \frac{-1, - 1}{k}\right) \]\n\nIn particular, the only finite primes at which \( A \) can be... | Proof: Suppose that \( \Gamma \) contains a subgroup isomorphic to \( {A}_{4} \) . Then since \( {A}_{4} \) is generated by two elements of order \( 3,{A}_{4} = {A}_{4}^{\left( 2\right) } \subset {\Gamma }^{\left( 2\right) } \) . Thus by conjugation, we can assume that \( \sigma \left( {B{A}_{4}}\right) \subset \mathca... | Yes |
Lemma 5.4.2 Let \( \Gamma \) be a finite-covolume Kleinian group which contains a subgroup isomorphic to \( {A}_{5} \) . If, furthermore, \( \left\lbrack {{k\Gamma } : \mathbb{Q}}\right\rbrack = 4 \), then \( {A\Gamma } \) has no finite ramification. | Proof: As above, let \( k = {k\Gamma } \) and \( A = {A\Gamma } \) . Now \( A \) can, at worst, have dyadic finite ramification. Also, by Lemma 5.4.1, \( A \) is ramified at all real places of which there are either 0 or 2 . Since \( \Gamma \) must contain an element of order \( 5,\mathbb{Q}\left( \sqrt{5}\right) \subs... | Yes |
Theorem 5.5.1 \( {k}_{\Delta }\Gamma = {k\Gamma } \) . | Proof: Denote \( {k}_{\Delta }\Gamma \) by \( {k}_{\Delta } \) for short. If we lift the triangulation of \( M \) to \( {\mathbf{H}}^{3} \), we get a tesselation of \( {\mathbf{H}}^{3} \) by ideal tetrahedra. Let \( V \) be the set of vertices of these tetrahedra in the sphere at infinity. Let \( {k}_{1} \) be the fiel... | No |
Lemma 5.5.2 \( {k}_{1} = {k}_{2} = {k}_{\Delta } \) . | Proof: \( {k}_{1} \subseteq {k}_{2} \) since \( {k}_{1} \) is generated by cross-ratios of elements of \( {k}_{2} \) while \( {k}_{2} \subseteq {k}_{1} \) because the cross-ratio of \( 0,1,\infty \), and \( z \) is just \( z \) . The inclusion \( {k}_{\Delta } \subseteq {k}_{1} \) is straightforward (see Exercise 5.5, ... | No |
Theorem 5.6.4 Let \( K = \mathbb{Q}\left( {\sqrt{-{d}_{1}},\ldots ,\sqrt{-{d}_{r}}}\right) \), where the positive integers \( {d}_{1},\ldots {d}_{r} \) are square-free. Then \( K \) is the invariant trace field of a finite-volume hyperbolic 3-manifold. | The proof of this relies on the fact that a twice-punctured disc in a hyperbolic 3-manifold has a unique hyperbolic structure. More precisely, we have (recall our convention that all immersions map boundary to boundary) the following lemma: | No |
Lemma 5.6.5 Let \( M = {\mathbf{H}}^{3}/\Gamma \) be a hyperbolic 3-manifold and \( f : D \hookrightarrow \) \( M \) an incompressible twice-punctured disc in \( M \) . Then, \( {f}_{ * }\left( {{\pi }_{1}\left( D\right) }\right) \subset \Gamma \) is conjugate in \( \operatorname{PSL}\left( {2,\mathbb{C}}\right) \) to ... | Proof: Since \( f\left( D\right) \) is a hyperbolic twice-punctured disc in \( M \), the fundamental group viewed as a subgroup \( F \) of \( \operatorname{PSL}\left( {2,\mathbb{C}}\right) \) is generated by a pair of parabolic elements, \( a \) and \( b \) say, whose product is also parabolic. Now by conjugating in \(... | Yes |
Lemma 5.6.6 For all \( d,\operatorname{PSL}\left( {2,{O}_{d}}\right) \) contains a torsion-free subgroup \( {G}_{d} \) such that \( {\mathbf{H}}^{3}/{G}_{d} \) contains an embedded totally geodesic twice-punctured disc. | Proof: The groups of the complements of the Whitehead link and the chain link with four components are subgroups in the cases \( d = 1 \) and \( d = 3 \), respectively and as seen in Figure 5.10, these complements contain obvious twice-punctured discs. Being totally geodesic follows from Lemma 5.6.5. Let these manifold... | No |
Lemma 6.2.2 Let \( V \) be a finite-dimensional space over \( k \) and let \( L \) be an \( R \) -lattice in \( V \) . Then \( L = \bigcap R\left( {v}_{\mathcal{P}}\right) L \), where the intersection is over all prime ideals of \( R \) . | Proof: Clearly \( L \) is contained in the intersection. Let \( {x}_{1},{x}_{2},\ldots ,{x}_{r} \) be a generating set for \( L \) over \( R \) ; thus it will also be a generating set for \( R\left( {v}_{\mathcal{P}}\right) L \) . Suppose that \( x \) lies in the intersection. Define the ideal \( J \) by\n\n\[ J = \{ y... | Yes |
Lemma 6.2.3 Let \( R \) be a Dedekind domain and let \( I \) be an ideal in the quaternion algebra \( A \) over \( k \) . For each prime ideal \( \mathcal{P} \) in \( R \), let \( I\left( {v}_{\mathcal{P}}\right) \) be an \( R\left( {v}_{\mathcal{P}}\right) \) -ideal in \( A \) such that \( I\left( {v}_{\mathcal{P}}\ri... | Proof: Let \( {x}_{1},{x}_{2},{x}_{3},{x}_{4} \in I \) be linearly independent over \( k \) and let \( L = \) \( R\left\lbrack {{x}_{1},{x}_{2},{x}_{3},{x}_{4}}\right\rbrack \) . Then \( L \) is an \( R \) -ideal and \( L \subset I \) and so \( \exists r \in R \) such that \( {rI} \subset L \) . It follows that, for al... | Yes |
Lemma 6.2.4 Let \( \mathcal{O} \) be an \( R \) -order in a quaternion algebra \( A \) over \( k \) . Then \( \mathcal{O} \) is maximal if and only if \( R\left( {v}_{\mathcal{P}}\right) { \otimes }_{R}\mathcal{O} \) is a maximal \( R\left( {v}_{\mathcal{P}}\right) \) -order for each prime ideal \( \mathcal{P} \) of \(... | Proof: Suppose that \( \mathcal{O} \) is maximal and \( i \) is the mapping identifying \( \mathcal{O} \) with its image in \( R\left( {v}_{\mathcal{P}}\right) { \otimes }_{R}\mathcal{O} \), via \( i\left( x\right) = 1 \otimes x \) . Suppose \( R\left( {v}_{\mathcal{P}}\right) { \otimes }_{R}\mathcal{O} \) is contained... | No |
Lemma 6.2.5 There is a bijection between \( R\left( {v}_{\mathcal{P}}\right) \) -ideals (resp. orders) in a quaternion algebra \( A \) over \( k \) and the \( {R}_{\mathcal{P}} \) -ideals (resp. orders) in the quaternion algebra \( {k}_{\mathcal{P}}{ \otimes }_{k}A \) over \( {k}_{\mathcal{P}} \) given by the mapping \... | Proof: Since \( R\left( {v}_{\mathcal{P}}\right) \) is a principal ideal domain, \( I \) will have a free basis \( \left\{ {{x}_{1},{x}_{2},{x}_{3},{x}_{4}}\right\} \) . Then in \( {k}_{\mathcal{P}}{ \otimes }_{k}A,\left( {{R}_{\mathcal{P}}{ \otimes }_{R\left( {v}_{\mathcal{P}}\right) }I}\right) \cap A \) consists of t... | Yes |
Lemma 6.2.7 Let \( A \) be a quaternion algebra over a number field \( k \), which has ring of integers \( R \) . Let \( I \) be an \( R \) -ideal in \( A \) . There is a bijection between \( R \) -ideals \( J \) of \( A \) and sequences of ideals \( \left\{ {\left( {L}_{\mathcal{P}}\right) : \mathcal{P} \in {\Omega }_... | Proof: If \( J \) is an ideal in \( A \), then there exists \( a, b \in {k}^{ * } \) such that \( {aJ} \subset I \subset \) \( {bJ} \) . For almost all \( \mathcal{P}, a \) and \( b \) are units in \( {R}_{\mathcal{P}} \) so that \( {J}_{\mathcal{P}} = {I}_{\mathcal{P}} \) for almost all \( \mathcal{P} \) . Now suppose... | Yes |
Example 6.2.9 Let \( A \) be the quaternion algebra \( \left( \frac{-1, - 1}{\mathbb{Q}}\right) \) and \( \mathcal{O} \) the order \( \mathbb{Z}\left\lbrack {1, i, j,{ij}}\right\rbrack \) . Recall that \( A \) splits at \( {\mathbb{Q}}_{p} \) for all odd primes \( p \) (see Theorem 2.6.6) so that \( {A}_{p} \cong {M}_{... | \[ i \mapsto \left( \begin{matrix} {x}_{1} & {y}_{1} \\ {y}_{1} & - {x}_{1} \end{matrix}\right) ,\;j \mapsto \left( \begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}\right) \] (where \( {y}_{1} = 0 \) if \( p \equiv 1\left( {\;\operatorname{mod}\;4}\right) \) ) provides a splitting of \( {A}_{p} \) . Under this mapping, the... | Yes |
Theorem 6.3.2 If \( \mathcal{O} \) has a free \( R \) -basis \( \left\{ {{u}_{1},{u}_{2},{u}_{3},{u}_{4}}\right\} \), then \( d\left( \mathcal{O}\right) \) is the principal ideal \( \det \left( {\operatorname{tr}\left( {{u}_{i}{u}_{j}}\right) }\right) R \) . | Proof: Clearly \( \det \left( {\operatorname{tr}\left( {{u}_{i}{u}_{j}}\right) }\right) R \subset d\left( \mathcal{O}\right) \) . Now let \( {x}_{1},{x}_{2},{x}_{3},{x}_{4} \in \mathcal{O} \) so that \( {x}_{i} = \mathop{\sum }\limits_{{k = 1}}^{4}{a}_{ik}{u}_{k},{a}_{ik} \in R \) . Thus\n\n\[ \det \left( {\operatornam... | Yes |
Theorem 6.3.4 Let \( A \) be a quaternion algebra over a field \( k \) . Let \( {\mathcal{O}}_{1} \) and \( {\mathcal{O}}_{2} \) be orders in \( A \) with \( {\mathcal{O}}_{1} \subset {\mathcal{O}}_{2} \) . Then \( d\left( {\mathcal{O}}_{2}\right) \mid d\left( {\mathcal{O}}_{1}\right) \) and \( d\left( {\mathcal{O}}_{1... | Now the ideal \( d\left( \mathcal{O}\right) \) is a finitely generated \( R \) -module and each generator is a finite linear combination of elements of the form \( \det \left( {\operatorname{tr}\left( {{x}_{i}{x}_{j}}\right) }\right) \), where \( {x}_{1},{x}_{2},{x}_{3},{x}_{4} \in \mathcal{O} \) . Thus there is a fini... | Yes |
Lemma 6.4.2 There exists a filtration of \( {\mathcal{O}}^{ * } \) :\n\n\[ \n{\mathcal{O}}^{ * } \supset 1 + \mathcal{Q} \supset 1 + {\mathcal{Q}}^{2} \supset 1 + {\mathcal{Q}}^{3}\cdots \n\] \n\nwhere \( {\mathcal{O}}^{ * }/1 + \mathcal{Q} \cong {\bar{F}}^{ * } \) and \( 1 + {\mathcal{Q}}^{i}/1 + {\mathcal{Q}}^{i + 1}... | Proof: Here \( \bar{F} \) is the residue class field \( {R}_{F}/\pi {R}_{F} \) which has order \( N{\left( \mathcal{P}\right) }^{2} \) . Elements of \( \mathcal{O} \) have the form \( \alpha = x + {yj}, x, y \in {R}_{F} \), and for \( \alpha \in {\mathcal{O}}^{ * }, x \in \) \( {R}_{F}^{ * } \) . The first isomorphism ... | Yes |
Theorem 6.4.3 Suppose that \( A \) is ramified at the finite prime \( \mathcal{P} \) and let \( p \) be the rational prime which \( \mathcal{P} \) divides, so that \( N\left( \mathcal{P}\right) = {p}^{t} \) for some \( t \) . Then \( \Gamma \) has a normal subgroup \( \Delta \) with finite cyclic quotient of order divi... | Proof: The group \( \Gamma \) has a faithful representation in \( {A}^{1} \) and, hence, in \( {A}_{\mathcal{P}}^{1} \) . Thus the image lies in \( {\mathcal{O}}_{\mathcal{P}}^{1} \) and so we let \( \Delta = \Gamma \cap \left( {1 + \mathcal{Q}}\right) \cap {\mathcal{O}}_{\mathcal{P}}^{1} \) . Then \( \Gamma /\Delta \)... | Yes |
Lemma 6.5.2 Let \( K \) be a \( \mathcal{P} \) -adic field with ring of integers \( R \) and uniformiser \( \pi \) . Let \( L \) and \( M \) be complete \( R \) -lattices in \( V \) such that \( M \subset L \) . Then there exists an \( R \) -basis \( \{ v, w\} \) of \( L \) and integers \( a \) and \( b \) such that \(... | Proof: Since there is an \( x \in R \) such that \( {xL} \subset M \), for each \( z \in L \), let \( {n}_{z} \) be such that \( {\pi }^{{n}_{z}}R = \{ c \in R \mid {cz} \in L\} \) . Over all generators of \( L \) (i.e., elements \( v \in L \) such that there is a \( w \) in \( L \) such that \( L = {Rv} + {Rw} \) ), c... | Yes |
Theorem 6.5.4 Let \( \mathcal{O} \) be a maximal order. The maximal orders at distance \( n \) from \( \mathcal{O} \) are also at distance \( n \) from \( \mathcal{O} \) in the graph measured by path length. In particular, the graph is a tree. | Proof: Let \( {\mathcal{O}}^{\prime } \) be a maximal order such that \( d\left( {\mathcal{O},{\mathcal{O}}^{\prime }}\right) = n \) . By a suitable choice of basis, \( \mathcal{O} = \operatorname{End}\left( {{e}_{1}R + {e}_{2}R}\right) \) and \( {\mathcal{O}}^{\prime } = \operatorname{End}\left( {{e}_{1}R + {e}_{2}{\p... | Yes |
Theorem 6.6.1 Let \( \Delta \left( A\right) \) be the discriminant of a quaternion algebra \( A \) over a number field \( k \) and let \( \mathcal{O} \) be an order in \( A \) . Then \( \mathcal{O} \) is a maximal order if and only if \( d\left( \mathcal{O}\right) = \Delta {\left( A\right) }^{2} \) . In particular, all... | Proof: By Corollary 6.2.9, \( \mathcal{O} \) is a maximal order if and only if \( {\mathcal{O}}_{\mathcal{P}} \) is maximal for every prime ideal \( \mathcal{P} \) . By Theorems 6.4.1 and 6.5.3, the discriminant of a maximal order in \( {A}_{\mathcal{P}} \) is either \( {\left( \mathcal{P}{R}_{\mathcal{P}}\right) }^{2}... | Yes |
Consider again the Example 6.2.8 where \( A = \left( \frac{-1, - 1}{\mathbb{Q}}\right) \). | Then \( {A}_{p} \) splits for all odd primes \( p \) but \( {A}_{2} \) is a division algebra. Thus \( \Delta \left( A\right) = 2\mathbb{Z} \). The discriminant of the order \( {\mathcal{O}}^{\prime } = \mathbb{Z}\left\lbrack {1, i, j,1/2\left( {1 + i + j + {ij}}\right) }\right\rbrack \) is easily shown via Theorem 6.3.... | Yes |
Lemma 6.6.3 Let \( I \) be an ideal in \( A \) such that \( {\mathcal{O}}_{r}\left( I\right) = \mathcal{O} \) is a maximal order. Then \( {I}_{\mathcal{P}} = {x}_{\mathcal{P}}{\mathcal{O}}_{\mathcal{P}} \) for some \( {x}_{\mathcal{P}} \in {A}_{\mathcal{P}}^{ * } \) . | Proof: Recall that \( {\mathcal{O}}_{\mathcal{P}} = {\mathcal{O}}_{r}\left( {I}_{\mathcal{P}}\right) \) (see Exercise 6.2, No. 4) and that each \( {\mathcal{O}}_{\mathcal{P}} \) is maximal.\n\nIf \( A \) is ramified at \( \mathcal{P} \), then \( {\mathcal{O}}_{\mathcal{P}} \) is the unique maximal order in \( {A}_{\mat... | No |
Corollary 6.6.4 Let \( I \) be an ideal in \( A \) . Then \( {\mathcal{O}}_{\ell }\left( I\right) \) is maximal if and only if \( {\mathcal{O}}_{r}\left( I\right) \) is maximal. | Proof: From the lemma, it is immediate that if \( {\mathcal{O}}_{r}\left( I\right) \) is maximal, then \( {\mathcal{O}}_{\ell }\left( I\right) \) is maximal. Now suppose that \( {\mathcal{O}}_{\ell }\left( I\right) \) is maximal. Now \( {I}^{-1} \) is an ideal (see Exercise 6.1, No. 1) and \( {\mathcal{O}}_{\ell }\left... | No |
Theorem 6.6.9 If \( \mathcal{O} \) is a maximal order in a quaternion algebra \( A \) over a number field \( k \), there are infinitely many principal congruence subgroups \( {\mathcal{O}}^{1}\left( I\right) \) which are torsion free. | The proof of this is left as an exercise (see Exercise 6.6, No. 7). | No |
Lemma 6.7.2 If \( \mathcal{C} \) denotes the set of conjugacy classes of maximal orders in \( A \), there is a bijection from \( \mathcal{C} \) to \( \mathcal{L}\mathcal{R}\left( \mathcal{O}\right) \smallsetminus (\mathcal{L}\left( \mathcal{O}\right) / \sim \), where \( \mathcal{O} \) is a fixed maximal order. | Proof: Denote equivalence classes of ideals by square brackets and conjugacy classes of orders also by square brackets. If \( I \in \mathcal{L}\left( \mathcal{O}\right) \), let \( {\mathcal{O}}^{\prime } = {\mathcal{O}}_{r}\left( I\right) \) , which is maximal. Define \( \theta : \mathcal{L}\left( \mathcal{O}\right) / ... | No |
Lemma 6.7.3 Let \( A \) be a quaternion algebra over a number field \( k \) such that there is at least one infinite place of \( k \) at which \( A \) is unramified. Then \[ \bar{n} : \mathcal{L}\left( \mathcal{O}\right) / \sim \rightarrow {I}_{k}/{P}_{k,\infty } \] is injective. | Proof: Let \( {I}_{1},{I}_{2} \in \mathcal{L}\left( \mathcal{O}\right) \) be such that \( n\left( {I}_{1}\right) = n\left( {I}_{2}\right) x \) for some \( x \in {k}_{\infty }^{ * } \) . Then \( {I}_{2}^{-1}{I}_{1} \) is a normal ideal and \( n\left( {{I}_{2}^{-1}{I}_{1}}\right) = n{\left( {I}_{2}\right) }^{-1}n\left( {... | Yes |
Lemma 6.7.4 If \( A \) is a quaternion algebra over a \( \mathcal{P} \) -adic field \( K \), then \( n : {A}^{ * } \rightarrow {K}^{ * } \) is surjective. | Proof: The result is clear if \( A = {M}_{2}\left( K\right) \) . Thus assume that \( A \cong \left( \frac{u,\pi }{K}\right) \) , where \( F = K\left( \sqrt{u}\right) \) is the unramified quadratic extension of \( K \) (see Theorem 2.6.3). Now \( {\left. n\right| }_{F} = {N}_{F \mid K} \) and \( {K}^{ * }/N\left( {F}^{ ... | Yes |
Lemma 6.7.5 Let \( \mathcal{D} \) denote the subgroup of \( {I}_{k} \) generated by all ideals in \( {\operatorname{Ram}}_{f}\left( A\right) \) and \( {I}_{k}^{2} \) the subgroup generated by the squares of all ideals in \( {I}_{k} \) . Then\n\n\[ n : \mathcal{L}\mathcal{R}\left( \mathcal{O}\right) \rightarrow \mathcal... | Proof: If \( X \) is a two-sided ideal of \( \mathcal{O} \), then \( {X}_{\mathcal{P}} \) is a two-sided ideal of \( {\mathcal{O}}_{\mathcal{P}} \) . If \( \mathcal{P} \) is ramified in \( A \), then in the standard representation of \( {A}_{\mathcal{P}} \) given in \( §{6.4},{X}_{\mathcal{P}} = {\mathcal{O}}_{\mathcal... | No |
Corollary 6.7.7 The type number is a power of 2. | Proof: This is immediate from (6.12), as it is the order of a finite factor group of the abelian group \( {I}_{k}/{I}_{k}^{2} \) of exponent \( 2 \) . | Yes |
Corollary 6.7.8 Let \( \mathcal{O} \) be a fixed maximal order in \( A \), where \( A \) is as given in Theorem. 6.7.6. Then every conjugacy class of maximal orders has a representative order \( {\mathcal{O}}^{\prime } \) such that there is a finite set \( S \) of primes, disjoint from those in \( {\operatorname{Ram}}_... | Proof: Let \( I \) be an \( \mathcal{O} \) -ideal such that \( {\mathcal{O}}_{\ell }\left( I\right) = \mathcal{O} \) and \( I \) represents the conjugacy class of \( {\mathcal{O}}^{\prime } = {\mathcal{O}}_{r}\left( I\right) \), as in Lemma 6.7.2. Now for all but a finite set \( {S}^{\prime } \) of primes, \( {\mathcal... | Yes |
Theorem 7.1.2 The additive adèle groups as described above are topologically isomorphic. | Proof: Note that \( {\left( L \mid k\right) }_{v} = L{ \otimes }_{k}{k}_{v} \cong \prod {L}_{w} \), where this is the finite product over the places \( w \) of \( L \) such that \( w \mid v \) (see \( §{0.8} \) ). Denote this isomorphism by \( {\Phi }_{v} \) . Now choose \( \mathcal{E} \subset L \) such that \( {R}_{L}... | Yes |
Theorem 7.1.3 Let \( k \) be a number field and \( E \) a vector space of finite dimension over \( k \) . Then \( E \) is discrete in \( {E}_{\mathcal{A}} \) and \( {E}_{\mathcal{A}}/E \) is compact. | Proof: If \( \left\lbrack {E : k}\right\rbrack = n \), then it suffices to prove the result for \( E = k \) since \( {E}_{\mathcal{A}} \cong {k}_{\mathcal{A}}^{n} \) . However, by Theorem 7.1.2, we need only show that \( \mathbb{Q} \) is discrete in \( {\mathbb{Q}}_{\mathcal{A}} \) and that \( {\mathbb{Q}}_{\mathcal{A}... | Yes |
Theorem 7.2.3 Let \( H \) be a finite extension of \( {\mathbb{Q}}_{p} \) or \( \mathbb{R} \) or a quaternion algebra over such a finite extension. Then \( a \mapsto {\psi }_{a} \), where \( \psi \) is the canonical character, defines a topological isomorphism \( H \rightarrow \widehat{H} \) . Furthermore, \( {\psi }_{... | Proof: We give the proof in the \( \mathcal{P} \) -adic case only, the real case being similar. Note that \( {\psi }_{p}\left( {\mathbb{Z}}_{p}\right) = 1 \) . Also, using a \( {\mathbb{Z}}_{p} \) -basis of \( {R}_{\mathcal{P}} \) or \( {\mathcal{O}}_{\mathcal{P}} \), the trace mapping restricted to \( {R}_{\mathcal{P}... | Yes |
Theorem 7.2.4 Let \( X \) denote a number field \( k \) or a quaternion algebra over a number field and let \( \psi = {\psi }_{\mathcal{A}} \) be the character on \( {X}_{\mathcal{A}} \) defined above. Then \( a \mapsto {\psi }_{a} \) is a topological isomorphism \( {X}_{\mathcal{A}} \rightarrow \widehat{{X}_{\mathcal{... | Proof: Let us use \( {C}_{v} \) to denote the compact subrings \( {R}_{v},{\mathcal{O}}_{v} \) in \( {X}_{v} \) when \( v \) is a finite place of \( k \) . Let \( \chi \in \widehat{{X}_{\mathcal{A}}} \) and \( x = \left( {x}_{v}\right) \in {X}_{\mathcal{A}} \) . Then \( \chi \left( x\right) = \prod {\chi }_{v}\left( {x... | Yes |
Corollary 7.2.5 \( {X}_{k} \) is the dual of \( {X}_{\mathcal{A}}/{X}_{k} \) . | Proof: The above proof shows that \( {X}_{\mathcal{A}}/{X}_{k} \) is topologically isomorphic to \( \widehat{{X}_{\mathcal{A}}}/{X}_{k * } \), and by duality, the dual of this space is \( {X}_{k} \) . \( ▱ \) | Yes |
Corollary 7.2.6 (Approximation Theorem) For every place \( v,{X}_{k} + \) \( {X}_{v} \) is dense in \( {X}_{\mathcal{A}} \) . | Proof: Let \( \chi \in \widehat{{X}_{\mathcal{A}}} \) and suppose \( \chi \) is trivial on \( {X}_{k} \) . Thus \( \chi \in {X}_{k * } \) and so \( \chi = {\psi }_{a} \), where \( a \in {X}_{k} \) by the theorem. However, if \( \chi \) is also trivial on \( {X}_{v} \) , then \( {\psi }_{v}\left( {a{x}_{v}}\right) = 1 \... | No |
Theorem 7.3.2 Let \( L \mid k \) be a quadratic extension of number fields and let \( N : {L}_{\mathcal{A}}^{ * } \rightarrow {k}_{\mathcal{A}}^{ * } \) denote the extension of the norm. Then \( \left\lbrack {{k}_{\mathcal{A}}^{ * } : {k}^{ * }N\left( {L}_{\mathcal{A}}^{ * }\right) }\right\rbrack = 2 \) . | Proof: Let \( \chi \) be a character on \( {k}_{\mathcal{A}}^{ * } \) which is trivial on \( {k}^{ * }N\left( {L}_{\mathcal{A}}^{ * }\right) \) . Recall that \( \chi = \left( {\chi }_{v}\right) \) and for each \( v,{\chi }_{v}^{2} = 1 \) (see Exercise 7.1, No. 4 and Theorem 0.7.13). Thus \( {\chi }^{2} = 1 \) . Now \( ... | No |
Theorem 7.3.3 Let \( A \) be a quaternion algebra over a number field \( k \) and let \( L \mid k \) be a quadratic extension. The following are equivalent:\n\n1. L embeds in \( A \) .\n\n2. A splits over \( L \) .\n\n3. \( L{ \otimes }_{k}{k}_{v} \) is a field for each \( v \in \operatorname{Ram}\left( A\right) \) . | Proof: \( 1 \Rightarrow 2 \) is Corollary 2.1.9.\n\n\( 2 \Rightarrow 3 \) . If \( L{ \otimes }_{k}A \cong {M}_{2}\left( L\right) \), then \( {L}_{w}{ \otimes }_{L}\left( {L{ \otimes }_{k}A}\right) \cong {M}_{2}\left( {L}_{w}\right) \) for every \( w \in \Omega \left( L\right) \) . Let \( w \mid v \), where \( v \in \Om... | Yes |
Lemma 7.3.4 Let \( K \) be a local field and let \( L = K\left( t\right) \) a separable quadratic extension so that \( t \) satisfies the minimum polynomial \( {X}^{2} - \operatorname{tr}\left( t\right) X + N\left( t\right) \) . If \( a \) and \( b \) are close enough to \( \operatorname{tr}\left( t\right) \) and \( N\... | Proof: If \( K = \mathbb{R} \), the discriminant \( {\operatorname{tr}}^{2}\left( t\right) - {4N}\left( t\right) < 0 \) and, hence, \( {a}^{2} - {4b} \) will also be \( < 0 \), and the result follows.\n\nNow suppose \( K = {k}_{\mathcal{P}} \), some \( \mathcal{P} \) -adic field, and denote the valuation of \( x \) in ... | Yes |
Theorem 7.3.5 Let \( k \) be a number field and let \( S \) be a finite set of places of \( k \) such that, for each \( v \in S \), there is a quadratic field extension \( {L}_{v} \) of \( {k}_{v} \). Then there exists a quadratic field extension \( L \) of \( k \) such that \( L{ \otimes }_{k}{k}_{v} = {L}_{v} \) for ... | Proof: Let \( w \) be a place of \( k, w \notin S \). Then by the Approximation Theorem (Corollary 7.2.6), \( k + {k}_{w} \) is dense in \( {k}_{\mathcal{A}} \). For each \( v \in S \), let \( {L}_{v} = {k}_{v}\left( {t}_{v}\right) \). Then we can find \( a, b \in k \) close to \( \operatorname{tr}\left( {t}_{v}\right)... | Yes |
Theorem 7.3.6 Let \( A \) be a quaternion algebra over the number field \( k \) and let \( \operatorname{Ram}\left( A\right) \) denote the set of places at which \( A \) is ramified. Then the following hold:\n\n1. \( \operatorname{Ram}\left( A\right) \) is finite of even cardinality.\n\n2. Let \( {A}_{1},{A}_{2} \) be ... | Proof: Parts 1 and 2 were established in \( §{2.7} \), so it remains to prove Part 3. Let \( S = \left\{ {{v}_{1},{v}_{2},\ldots ,{v}_{2r}}\right\} \) be a set of places as described in the statement. Then each such \( {k}_{{v}_{i}} \) admits a quadratic extension field \( {L}_{{v}_{i}} \) . By Theorem 7.3.5, there exi... | No |
Theorem 7.4.1 (Theorem on Norms) Let \( A \) be a quaternion algebra over the number field \( k \) . Then \( {k}_{\infty }^{ * } = n\left( {A}^{ * }\right) \) . | Proof: Let \( \alpha \in {A}^{ * } \) and \( v = \sigma \in {\operatorname{Ram}}_{\infty }\left( A\right) \) . Then \( v\left( {n\left( \alpha \right) }\right) = {n}_{\mathcal{H}}\left( {{i}_{v}\left( \alpha \right) }\right) > 0 \) where \( {i}_{v} : A \rightarrow {A}_{v} \cong \mathcal{H} \) .\n\nConversely, let \( x ... | Yes |
Lemma 7.5.3 When \( \mathbb{R} ⊄ H \), the volume of \( {\mathcal{B}}^{ * } \) with respect to the multiplicative Haar measure is given by the following formulas:\n\n(a) \( \operatorname{Vol}\left( {\mathcal{B}}^{ * }\right) = \operatorname{Vol}\left( {R}^{ * }\right) = 1 \) if \( H = K \) .\n\n(b) \( \operatorname{Vol... | Proof: When \( H = K \), for the additive measure,\n\n\[ \operatorname{Vol}\left( {R}^{ * }\right) = \operatorname{Vol}\left( R\right) - \operatorname{Vol}\left( {\pi R}\right) = 1 - \parallel \pi \parallel = 1 - N{\left( \pi \right) }^{-1} = 1 - {q}^{-1}. \]\n\nWhen \( H = A \), a quaternion division algebra,\n\n\[ \o... | Yes |
Lemma 7.5.7 \( \operatorname{Vol}\left( {\mathcal{H}}^{1}\right) = \operatorname{Vol}\left\{ {x \in {\mathcal{H}}^{ * } \mid n\left( x\right) = 1}\right\} = 4{\pi }^{2} \) . | Proof: For the usual measures on \( {\mathbb{R}}^{4} \), the volume of a ball of radius \( r \) is \( {\pi }^{2}{r}^{4}/2 \) . Thus, for \( x = {x}_{1} + {x}_{2}i + {x}_{3}j + {x}_{4}{ij}, n\left( x\right) = {x}_{1}^{2} + {x}_{2}^{2} + {x}_{3}^{2} + {x}_{4}^{2} \), so that \( \parallel x\parallel = n{\left( x\right) }^... | Yes |
Lemma 7.5.8 Let \( \mathcal{O} \) be a maximal order in the quaternion algebra \( A \) over the \( \mathcal{P} \) -adic field \( K \) . Let \( {D}_{K} \) denote the discriminant of \( K \) and \( q = \left| {R/{\pi R}}\right| \) . Then\n\n\[ \operatorname{Vol}\left( {\mathcal{O}}^{1}\right) = {D}_{K}^{-3/2}\left( {1 - ... | Proof: Note that the reduced norm \( n \) maps \( {\mathcal{O}}^{ * } \) onto \( {R}^{ * } \) (see Exercise 6.7, No. 1) so there is an exact sequence\n\n\[ 1 \rightarrow {\mathcal{O}}^{1}\overset{i}{ \rightarrow }{\mathcal{O}}^{ * }\overset{n}{ \rightarrow }{R}^{ * } \rightarrow 1 \]\n\nThus for the volume of \( {\math... | No |
Lemma 7.6.2 \( {X}_{k}^{ * } \) is discrete in \( {X}_{\mathcal{A}}^{ * } \) and \( {A}_{k}^{1} \) is discrete in \( {A}_{\mathcal{A}}^{1} \) . | Now define \( {X}_{\mathcal{A},1} \) to be the kernel of the module map on \( {X}_{\mathcal{A}}^{ * } \) so that the exact sequence\n\n\[ 1 \rightarrow \frac{{X}_{\mathcal{A},1}}{{X}_{k}^{ * }} \rightarrow \frac{{X}_{\mathcal{A}}^{ * }}{{X}_{k}^{ * }} \rightarrow \begin{Vmatrix}{X}_{\mathcal{A}}^{ * }\end{Vmatrix} \rig... | No |
Theorem 7.6.3 With respect to the measures obtained above | \[ \operatorname{Vol}\left( \frac{{X}_{\mathcal{A}}}{{X}_{k}}\right) = 1,\;\operatorname{Vol}\left( \frac{{X}_{\mathcal{A},1}}{{X}_{k}^{ * }}\right) = 1,\;\operatorname{Vol}\left( \frac{{A}_{\mathcal{A}}^{1}}{{A}_{k}^{1}}\right) = 1. \] | Yes |
Lemma 7.7.1 For each \( x \in {X}_{k}^{ * } \subset {X}_{\mathcal{A}},\parallel x\parallel = 1 \) . | Proof: Recall that \( {X}_{k} \) is discrete in \( {X}_{\mathcal{A}} \) . Let \( {dt} \) denote the measure on \( {X}_{\mathcal{A}}/{X}_{k} \) compatible with the exact sequence at (7.6), the Tamagawa measure \( {dz} \) on \( {X}_{\mathcal{A}} \) and the counting measure on \( {X}_{k} \) . Let \( Y \) be a measurable s... | Yes |
Lemma 7.7.2 Let \( X = k \) or \( A \), where \( A \) is a quaternion division algebra over \( k \). For \( m, M \in {\mathbb{R}}^{ + } \), define\n\n\[ Y = \left\{ {y \in {X}_{\mathcal{A}}^{ * } \mid 0 < m \leq \parallel y{\parallel }_{\mathcal{A}} \leq M}\right\} \]\n\nThen the image of \( Y \) in \( {X}_{\mathcal{A}... | Proof: Recall (§0.8 and Exercise 7.1, No. 2) that a compact set in \( {X}_{\mathcal{A}}^{ * } \) has the form \( \left\{ {x \in {X}_{\mathcal{A}}^{ * } \mid \left( {x,{x}^{-1}}\right) \in C \times {C}^{\prime }}\right\} \), where \( C \) and \( {C}^{\prime } \) are compact sets in \( {X}_{\mathcal{A}} \). Thus we need ... | Yes |
Lemma 7.7.4 For any \( v \in {\Omega }_{\infty } \), there exists a compact set \( C \) in \( {X}_{\mathcal{A}}^{ * } \) such that \( {X}_{\mathcal{A}}^{ * } = {X}_{k}^{ * }{X}_{v}^{ * }C \) . | Proof: When \( X \) has no divisors of zero, this follows from Lemma 7.7.2. For in that case, \( \begin{Vmatrix}{X}_{v}^{ * }\end{Vmatrix} = {\mathbb{R}}^{ + } = \begin{Vmatrix}{X}_{\mathcal{A}}^{ * }\end{Vmatrix} \). Now suppose \( X = {M}_{2}\left( k\right) \) so that \( {X}_{\mathcal{A}}^{ * } \) is the restricted p... | Yes |
Theorem 7.7.7 (Eichler) Let \( A \) be a quaternion algebra over a number field \( k \) where \( A \) satisfies the Eichler condition. Let \( \mathcal{O} \) be a maximal order and let \( I \) be an ideal such that \( {\mathcal{O}}_{\ell }\left( I\right) = \mathcal{O} \) . Then \( I \) is principal; that is, \( I = \mat... | Proof: Clearly, if \( I = \mathcal{O}\alpha \), then \( n\left( I\right) = {R}_{k}n\left( \alpha \right) \) and \( n\left( \alpha \right) \in {k}_{\infty }^{ * } \) . Now suppose that \( n\left( I\right) = {R}_{k}x \), where \( x \in {k}_{\infty }^{ * } \) . By the Norm Theorem 7.4.1, there exists \( \alpha \in {A}^{ *... | Yes |
Lemma 8.1.3 Let \( Z \) be the direct product of a locally compact group \( X \) and a compact group \( Y \) . Let \( W \) be a subgroup of \( Z \) whose projection on \( X \) is the subgroup \( V \) . Then the following hold:\n\n1. If \( W \) is discrete in \( Z \), then \( V \) is discrete in \( X \) . Furthermore, \... | Proof: Let \( p : Z \rightarrow X \) denote the projection. Let \( D \) be a compact neighbourhood of the identity in \( X \) . Then \( V \cap D = p\left( {W \cap {p}^{-1}\left( D\right) }\right) \) . Now \( {p}^{-1}\left( D\right) = D \times Y \) is a compact neighbourhood of the identity in \( Z \) . Thus, since \( W... | No |
Theorem 8.2.3 Let \( \Gamma \) be an arithmetic Kleinian group commensurable with \( {P\rho }\left( {\mathcal{O}}^{1}\right) \), where \( \mathcal{O} \) is an order in a quaternion algebra \( A \) over \( k \) . The following are equivalent:\n\n1. \( \Gamma \) is non-cocompact.\n\n2. \( k = \mathbb{Q}\left( \sqrt{-d}\r... | Proof: If \( \Gamma \) is non-cocompact, then so is \( {P\rho }\left( {\mathcal{O}}^{1}\right) \), and so \( A \) cannot be a division algebra. So \( A \cong {M}_{2}\left( k\right) \) (see Theorem 2.1.7). If \( \left\lbrack {k : \mathbb{Q}}\right\rbrack \geq 3 \), then \( k \) has at least one place at which \( A \) wi... | Yes |
Theorem 8.3.1 If \( \Gamma \) is an arithmetic Kleinian group which is commensurable with \( \rho \left( {\mathcal{O}}^{1}\right) \), where \( \mathcal{O} \) is an order in a quaternion algebra \( A \) over the field \( k \) and \( \rho \) is a \( k \) -embedding, then \( {k\Gamma } = k \) and \( {A\Gamma } = \rho \lef... | Note that this result already imposes two necessary conditions on \( \Gamma \) if it is to be arithmetic; namely, that \( {k\Gamma } \) has exactly one complex place and that \( {A\Gamma } \) is ramified at all real places. In Chapter 3, a variety of methods were given to calculate \( {k\Gamma } \) and \( {A\Gamma } \)... | No |
Corollary 8.3.5 Let \( \Gamma \) be a finite-covolume Kleinian group. Then \( \Gamma \) is arithmetic if and only if \( {\Gamma }^{\left( 2\right) } \) is derived from a quaternion algebra. | The following deduction is immediate from the proof of Theorem 8.3.2. | No |
Theorem 8.3.11 (Takeuchi) The \( \\left( {\\ell, m, n}\\right) \) Fuchsian triangle group \( \\Delta \) is arithmetic if and only if for every \( \\sigma \\in \\operatorname{Gal}\\left( {{k\\Delta } \\mid \\mathbb{Q}}\\right) ,\\sigma \\neq \\mathrm{{Id}} \), then \( \\sigma \\left( {\\lambda \\left( {\\ell, m, n}\\rig... | If \( n \) is large enough, there always exists an element \( \\sigma \) in the Galois group such that the inequality in this theorem fails. This was established by Takeuchi. It follows that there can be only finitely many arithmetic Fuchsian triangle groups. We will show that there are finitely many arithmetic Fuchsia... | No |
Theorem 8.4.1 Let \( {\Gamma }_{1} \) and \( {\Gamma }_{2} \) be subgroups of \( \operatorname{PSL}\left( {2,\mathbb{C}}\right) \) which are arithmetic Kleinian groups. Then \( {\Gamma }_{1} \) and \( {\Gamma }_{2} \) are commensurable in the wide sense in \( \operatorname{PSL}\left( {2,\mathbb{C}}\right) \) if and onl... | Proof: Let \( g \in \mathrm{{SL}}\left( {2,\mathbb{C}}\right) \) be such that \( g{\Gamma }_{1}{g}^{-1} \) and \( {\Gamma }_{2} \) are commensurable. Then \( k{\Gamma }_{1} = k{\Gamma }_{2} \) and the mapping \( \phi : A{\Gamma }_{1} \rightarrow A{\Gamma }_{2} \) given by \( \phi \left( {\sum {a}_{i}{\gamma }_{i}}\righ... | Yes |
Corollary 8.4.2 There exist infinitely many commensurability classes of compact hyperbolic 3-manifolds \( {\mathbf{H}}^{3}/\Gamma \) such that the groups \( \Gamma \) have the same trace field. | Proof: Let \( k \) be a number field with exactly one complex place. There are infinitely many isomorphism classes of quaternion algebras \( A \) over \( k \) such that \( \mathrm{{Ram}}\left( A\right) \neq \varnothing , \) which are ramified at all real places of \( k \) by Theorem 7.3.6 and for each one, a representa... | Yes |
Consider again the tetrahedral group \( {\Gamma }_{1} \) whose Coxeter symbol is shown in Figure 8.3 (see §4.7.2) and the Fibonacci group \( {\Gamma }_{2} = {F}_{10} \) (see §4.8.2). It was shown in \( §{8.3} \) that \( {\Gamma }_{1} \) is arithmetic with defining field \( \mathbb{Q}\left( t\right) \) of degree 4 over ... | \[ \langle x, y, z \mid {x}^{2} = {y}^{2} = {z}^{3} = {\left( yz\right) }^{2} = {\left( zx\right) }^{5} = {\left( yx\right) }^{3} = 1\rangle \] is commensurable with the group with presentation \[ \left\langle {{x}_{1},{x}_{2},\ldots ,{x}_{10} \mid {x}_{i}{x}_{i + 1} = {x}_{i + 2}\text{ for all }i\left( {\;\operatornam... | Yes |
Theorem 8.4.4 Let \( \Gamma \subset \operatorname{PSL}\left( {2,\mathbb{C}}\right) \) be an arithmetic Kleinian group. Then \( \operatorname{Comm}\left( \Gamma \right) = P\left( {A{\Gamma }^{ * }}\right) . | Proof: Clearly, if \( {\Gamma }_{1} \) and \( {\Gamma }_{2} \) are commensurable, \( \operatorname{Comm}\left( {\Gamma }_{1}\right) = \operatorname{Comm}\left( {\Gamma }_{2}\right) \) . Thus if \( \mathcal{O} \) is an order in \( {A\Gamma } \) and \( x \in A{\Gamma }^{ * }, x\mathcal{O}{x}^{-1} \) is also an order and ... | Yes |
Theorem 8.4.6 Let \( {\Gamma }_{1} \) and \( {\Gamma }_{2} \) be subgroups of \( \operatorname{PSL}\left( {2,\mathbb{R}}\right) \) which are arithmetic Fuchsian groups. Then \( {\Gamma }_{1} \) and \( {\Gamma }_{2} \) are commensurable in the wide sense if and only if \( k{\Gamma }_{1} = k{\Gamma }_{2} \) and there exi... | The proof of this is identical to that given for Theorem 8.4.1. Note further, that we can take the wide commensurability class in the Fuchsian case to mean up to conjugacy in Isom \( {\mathbf{H}}^{2} = \mathrm{{PGL}}\left( {2,\mathbb{R}}\right) \) since such conjugacy leaves the traces invariant. | No |
Lemma 8.5.2 If \( \Gamma = \left\langle {{\gamma }_{1},{\gamma }_{2},\ldots ,{\gamma }_{n}}\right\rangle \) and the set \( \left\{ {\operatorname{tr}{\gamma }_{i},\operatorname{tr}{\gamma }_{i}{\gamma }_{j} : i, j = }\right. \) \( 1,2,\ldots, n\} \) consists of algebraic integers, then \( \{ \operatorname{tr}\gamma : \... | Proof: Since \( x = \operatorname{tr}{XYZ} \) satisfies a monic polynomial whose coefficients are algebraic integers in the traces of \( X, Y \) and \( Z \) and their products in pairs (see (3.24)), the result follows for \( \operatorname{tr}{\gamma }_{i}{\gamma }_{j}{\gamma }_{k} \) . However, we can use the polynomia... | No |
Lemma 8.5.3 Let \( \Gamma \) be a finitely generated non-elementary subgroup of \( \mathrm{{SL}}\left( {2,\mathbb{C}}\right) \) such that \( \{ \operatorname{tr}\gamma : \gamma \in \Gamma \} \) consists of algebraic integers. Let \( R \) denote the ring of integers in \( \mathbb{Q}\left( {\operatorname{tr}\Gamma }\righ... | Proof: Recall that \( \{ I, g, h,{gh}\} \) spans \( {A}_{0}\left( \Gamma \right) \) over \( \mathbb{Q}\left( {\operatorname{tr}\Gamma }\right) \) . It is clear that \( R\left\lbrack {I, g, h,{gh}}\right\rbrack \) is a complete \( R \) -lattice containing 1, so that it remains to show that it is a ring. This can easily ... | Yes |
Theorem 9.1.1 The cusp set of \( {\Gamma }_{d} \) is \( \mathcal{C}\left( {\Gamma }_{d}\right) = \mathbb{P}{K}_{d} \subset \mathbb{{PC}} \), where \( \mathbb{{PC}} \) is identified with \( \partial {\mathbf{H}}^{3} = \mathbb{C} \cup \{ \infty \} \) and the number of ends of \( {\mathbf{H}}^{3}/{\Gamma }_{d} = \) \( \le... | Proof: Clearly every cusp of \( {\Gamma }_{d} \) can be identified with an element \( \left\lbrack {x, y}\right\rbrack \in \) \( \mathbb{P}{K}_{d} \) . Conversely, for \( \left\lbrack {x, y}\right\rbrack \in \mathbb{P}{K}_{d} \), with \( x, y \in {O}_{d} \), the parabolic element of \( {\Gamma }_{d} \) given by \( \lef... | No |
Theorem 9.2.1 There are infinitely many links whose complements are arithmetic hyperbolic 3-manifolds. | Proof: Let \( L \) denote the Borromean rings so that none of the components are knotted and linking numbers are zero. Thus if we take cyclic covers of \( {S}^{3} \smallsetminus L \) branched over one component of \( L \), then the cover is still a link complement in \( {S}^{3} \) . The corresponding fundamental groups... | No |
Theorem 9.2.2 Let \( {\mathbf{H}}^{3}/\Gamma \) be an arithmetic link complement. Then \( \Gamma \) is a subgroup of \( P\left( {\mathcal{O}}^{1}\right) \), where \( \mathcal{O} \) is a maximal order in \( {M}_{2}\left( {\mathbb{Q}\left( \sqrt{-d}\right) }\right) \) for some \( d \) . | Proof: By Corollary 4.2.2, \( {k\Gamma } = \mathbb{Q}\left( {\operatorname{tr}\Gamma }\right) \) . Then if \( \Gamma \) is arithmetic, \( \Gamma \) must be derived from a quaternion algebra by Corollary 8.3.6. | No |
Corollary 9.3.3 For all but a finite number of \( d,\alpha {\left( {\Gamma }_{d}\right) }_{\mathbb{Q}} \) at (9.2) is not surjective. | The corollary follows immediately from the theorem and the properties of \( \widetilde{f} \) outlined above. | No |
Theorem 9.3.6 There are only finitely many values of \( d \) such that \( {\Gamma }_{d} \) can contain a subgroup \( \Gamma \) of finite index such that \( {\mathbf{H}}^{3}/\Gamma \) is the complement of a link in \( {S}^{3} \) . | Our argument shows that whenever \( {\Gamma }_{d} \) has non-trivial cuspidal cohomology (suitably interpreted), then \( \Gamma \) has non-trivial cuspidal cohomology. Thus the finite values of \( d \) such that \( {\Gamma }_{d} \) can contain an arithmetic link complement group will be among those values of \( d \) fo... | No |
Theorem 9.5.1 There exist infinitely many commensurability classes of compact hyperbolic 3-manifolds which have no immersed totally geodesic surfaces. These may be chosen to have the same trace field. | Proof: Let \( k \) be any field with one complex place and odd prime degree over \( \mathbb{Q} \) . There are infinitely many isomorphism classes of quaternion algebras over \( k \) which ramify at all real places. Each such quaternion algebra yields a commensurability class of arithmetic Kleinian groups by Theorem 8.4... | Yes |
Theorem 9.5.2 Let \( F \) be a non-elementary Fuchsian subgroup of an arithmetic Kleinian group \( \Gamma \) . Then \( F \) is a subgroup of an arithmetic Fuchsian group \( G \) . | Proof: Let \( \mathcal{C} \) denote the circle or straight line in \( \mathbb{C} \) left invariant by \( F \) so that\n\n\( F \subset G \mathrel{\text{:=}} \{ \gamma \in \Gamma \mid \gamma \left( \mathcal{C}\right) = \mathcal{C} \) and \( \gamma \) preserves the components of \( \widehat{\mathbb{C}} \smallsetminus \mat... | Yes |
Corollary 9.5.3 With \( \Gamma \) and \( G \) as in Theorem 9.5.2,\n\n1. \( \left\lbrack {{k\Gamma } : {kG}}\right\rbrack = 2 \) and \( {kG} = {k\Gamma } \cap \mathbb{R} \) .\n\n2. \( {A\Gamma } \cong {AG}{ \otimes }_{kG}{k\Gamma } \) . | Proof: If \( \left\lbrack {{k\Gamma } : {kG}}\right\rbrack > 2 \), the identity embedding of \( {kG} \) would be the restriction of an embedding \( \sigma \) of \( {k\Gamma } \), different from the identity and complex conjugation. Thus exactly as in the proof of the theorem, we would have \( {F}^{\left( 2\right) } \su... | Yes |
Theorem 9.5.4 Let \( k \) be a field with one complex place and let \( A \) be a quaternion algebra over \( k \) which is ramified at all real places. Assume that \( \left( {k, A}\right) \) satisfy the additional two conditions:\n\n1. \( \left\lbrack {k : \ell }\right\rbrack = 2 \), where \( \ell = k \cap \mathbb{R} \)... | Proof: Implicit in this statement is the fact that \( B \) does yield arithmetic Fuchsian groups. Note, first of all, that \( \ell \) is totally real. Also, if \( v \) is any valuation on \( \ell \) and \( w \) a valuation on \( k \) such that \( w \mid v \), then\n\n\[ \left( {B{ \otimes }_{\ell }k}\right) { \otimes }... | No |
Theorem 9.5.5 Let \( k \) be a number field with one complex place and let \( A \) be a quaternion algebra over \( k \) which is ramified at all real places. Let \( k \) be such that \( \left\lbrack {k : \ell }\right\rbrack = 2 \), where \( \ell = k \cap \mathbb{R} \) . Let \( B \) be a quaternion algebra over \( \ell ... | Proof: Note that \( {\operatorname{Ram}}_{f}\left( A\right) \) is empty when \( r = 0 \), and in that case, \( {\operatorname{Ram}}_{f}\left( B\right) \) may also be empty.\n\nWe make use of the isomorphism at (9.5). Suppose that \( \mathcal{P} \) is an ideal in \( {R}_{k} \) and \( p = \mathcal{P} \cap {R}_{\ell } \) ... | Yes |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.