The Arithmetic of Quaternion Algebra

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80 CHAPTER 4. APPLICATIONS TO ARITHMETIC GROUPS where Hv denotes the quaternion field over Kv. From this we obtain an inclusion ϕ : H → � M(2, Kv) = G v∈S,v /∈Ram(H) which send O ∞ on a subgroup of G 1 . By abuse of notations, we identify H ′ v with Hv in sequel. Note that two inclusion ϕ, ϕ ′ is different by an inner automorphism of G × Theorem 4.1.1. (1) The group ϕ(O 1 ) is isomorphic to O 1 . It is a discrete subgroup, of finite covolume of G 1 . It is cocompact if H is a field. (2) The projection of ϕ(O 1 ) onto a factor G ′ = � v SL(2, Kv) of G 1 , with 1 �= G ′ �= G 1 , is equal to O 1 . It is dense in G ′ . Proof. The nontrivial part of the theorem is an application of the fundamental theorem III.1.4 and III.2.3. The isomorphism with O1 is trivial since the image of O1 in G ′ with 1 �= G ′ is � ϕv(O1 ) which is isomorphic to O1 . The idea is to describe the group H1 A /H1 K . Set U = G 1 � � · C with C = O 1 v. H v∈S and v ∈ Ram(H) 1 v v /∈S The group U is an open subgroup in H 1 A satisfying: H 1 A = H 1 K From this we deduce a bijection between and H 1 K U = O 1 . H 1 A/H 1 KU and U/O 1 . According to III.1.4, and III.2.3, we have (1) H 1 K is discrete in H1 A of the finite covolume being equal to τ(H1 = 1, cocompact if H is a field. According to III.4.3, we have (2) H 1 K G′′ is dense in H 1 A if G′′ = � SL(2, Kv) with i �= G ′′ . It follows (1) O 1 is discrete in U of finite covolume being equal to 1 for the Tamagawa measures, cocompact if H is a field. (2) The image of O 1 in G · C is dense. We thus utilize the following lemma for finishing the proof of theorem 1.1. Lemma 4.1.2. Let X be a locally compact group, Y a compact group, Z the direct product X · Y , and T a subgroup of Z with its projection V in X. We have the following properties: a) If T is discrete in Z, then V is discrete in X. Moreover, T is of a finite covolume (resp. cocompact) in Z if and only if V has the same property in X. b) If T is dense in Z, then V is dense in X. Proof. a) Suppose K to be discrete in Z. For every compact neighborhood D of the unit in X, we show that V ∩ D has only a finite number of elements. In fact, X ∩ (D · C) has a finite number of elements, being great or equal to that of V ∩ D. Hence V is discrete in X. Let FT ⊂ Z, FV ⊂ X be the fundamental sets of T in Z, and of V in X. It is clear that FV · C contains a fundamental
4.1. QUATERNION GROUPS 81 set of T in Z, and the projection of FT in X contains a fundamental set of V in X. (a) has been deduced. b) Suppose that T is dense in Z. every point (x, y) ∈ X · Y is the limit of a sequence of points (v, w) ∈ T . Hence every point x ∈ X is the limit of a sequence of points v ∈ V , and V is dense in X. Therefore, the theorem is proved. Definition 4.1. Two subgroups X, Y of a group Z are commensurable if their intersection X ∩ Y is of a finite index in X and Y . The commensurable degree of X with respect to Y is The commensurator of X in Z is [X : Y ] = [X : (X ∩ Y )][Y : (X ∩ Y )] −1 . CZ(X) = {x ∈ Z|X and xXx −1 is commensurable}. Definition 4.2. We call the group ϕ(O 1 ) a quaternion group of G 1 . A subgroup of G 1 which is conjugate in G 1 to a commensurable group with a quaternion group (hence of the form ϕ(O 1 ) for an appropriate choice of a given K, H, S, ϕ, Ω ) is called an arithmetic group. We leave the verification of the following elementary lemma as an exercise to readers. Lemma 4.1.3. Let Z be a locally compact group, X and Y be two subgroups of Z Which are commensurable. Therefore X is discrete in Z if and only if Y is discrete in Z. Moreover, X is of a finite covolume (resp. cocompact) if and only if Y is of a finite covolume (resp. cocompact). In this case, we have : vol(Z/X)[X : Y ] = vol(Z/Y ). EXAMPLE. A subgroup Y of a finite index of a group X is commensurable to X. The commensurable degree [X : Y ] is the index of Y in X. The commensurator of Y in X is equal to X. For every x ∈ X, we have [X : xY x −1 ] = [x : Y ]. Remark 4.1.4. Takeuchi ([1]-[4]) determined all the arithmetic subgroup of SL(2, R) which is triangular, that is to say, it admit a presentation: Γ =< γ1, γ2, γ3|γ e1 1 = γe2 2 = γe3 3 = γ1γ2γ3 = ∓1 >, where ei are integers, 2 ≤ ei ≤ ∞. He determined the commensurable class of a quaternion group in SL(2, R) too. Proposition 4.1.5. The groups O1 for O ∈ Ω are pairwise commensurable. The commensurator of a pair in G × = � GL(2, Kv) v∈S,v /∈Ram(H) is equal to Zϕ(H × ), where Z is the center of G × .
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The Arithmetic of Quaternion Algebra

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