The Arithmetic of Quaternion Algebra

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82 CHAPTER 4. APPLICATIONS TO ARITHMETIC GROUPS Proof. The first part comes from Proposition 1.4. If x ∈ G × belongs to the commensurator of ϕ(O 1 ), it induces an inner automorphism ˜x fixing ϕ(H). Every automorphism of ϕ(H) fixing ϕ(K) pointwise is inner. Therefore x ∈ Zϕ(H × ). Inversely it is clear that Zϕ(H × ) is contained in the commensurator of ϕ(O 1 ) in G × . Definition 4.3. Let I be a two-sided integer of an order O ∈ Ω. The kernel O 1 (I) in O 1 of the canonical homomorphism : O → O/I is called the principal congruent group of O 1 modulo I. A congruent group of O 1 modulo I is a subgroup of O 1 containing O 1 (I). The congruent group are of the commensurable groups among them. We have [O 1 : O 1 (I)] ≤ [O : I]. If O ′ is an Eichler order of level N contained in a maximal order O, then the group O ′ 1 is a congruent group of O 1 modulo the two-sided ideal NO. The groups so constructed with the Eichler orders and the principal groups are the groups for which we have certain arithmetic information: – the value of covolume, indices (Theorem 1.7), – the value of the number of conjugate classes of a given characteristic polynomial(III. 5.14,and 5.17). Partially for this reason we often encounter them. Another collection of groups we encounter sometimes( for the same reason). They are the normalizers N(ϕ(O 1 )) in G 1 of groups ϕ(O 1 ), where O is an Eichler order. The quotient groups N(ϕ(O 1 ))/ϕ(O 1 are of type (2, 2, ...). We deduce from IV.5.14,5.16,5.17, and exercise 5.12 the next proposition: Proposition 4.1.6. Every group O 1 for O ∈ Ω contains a subgroup of finite index which contains only the the elements different from unit and of finite orders. The relation τ(H 1 ) = 1 , in the form vol(G 1 ·C/O 1 ) = 1, allow us to compute the covolume of ϕ(O 1 ) in G 1 : vol(G 1 /ϕ(O 1 )) = vol(C) −1 for Tamagawa measures. By using the definition of C = � � H v∈S, and v ∈ Ram(H) 1 v O p/∈S 1 p we can then compute the the global commensurable degree from the local commensurable degrees. Theorem 4.1.7. The commensurable degree of two groups O 1 , O ′ 1 for O, O ′ ∈ Ω is equal to the product of the local commensurable degrees: For Tamagawa measure, [O 1 : O ′ � 1 ] = [O 1 p : O ′ 1 p ] = � p/∈S vol(G 1 /ϕ(O 1 )) −1 = � p/∈S v∈Ram(H)and v ∈ S vol(O 1 p)vol(O ′ 1 p ) −1 . vol(H 1 v ) � vol(O 1 p). p/∈S
4.1. QUATERNION GROUPS 83 The explicit formulae (II,exercise 4.2,4.3) of the local volumes for the Tamagawa measures have obtained already for the principal congruent groups obtained with the Eichler orders. By using these we obtain for example Corollary 4.1.8. If O is a maximal order, then vol(G 1 /ϕ(O 1 )) −1 = ζK(2)(4π 2 ) −|Ram∞(H)| D 3/2 K We shall give another examples. EXAMPLES. � p∈Ramf (H) (Np−1) � p∈S∩P,p /∈Ramf (H) 1. H is an indefinite quaternion algebra over Q, i.e. HR = M(2, R), hence the covolume of O 1 if O is a maximal Z-order is π 2 6 � (p − 1), p|D where D is the reduced discriminant of H. 2. H = M(2, Q( √ −1)) and O 1 = SL(2, Z( √ −1)), then the covolume is 8ζ Q( √ −1) (2) times by a number for which we are ignorant of its arithmetic nature: we don’t know if it is transcendental. The group O 1 is called now and then the Picad group. 3. H is a quaternion algebra over Q ramified at the infinity and unramified at p and S = {∞, p}. For a maximal order O, the group O 1 is a cocompact discrete subgroup of SL(2, Qp) and of the covolume 1 � (1 − p−2) q|D(q − 1), 24 where D is the reduced discriminant of H. 4. Congruent groups. Let K be a non-archimedean local field of an integer ring R, and let p be a uniform parameter of R. For every integer m ≥ 1, we defined ( II.exercise 4.3) in the canonical Eichler order of levle p m R of M(2, K) the groups Γ0(p m ) ⊃ Γ1(p m ) ⊃ Γ(p m ), by which we computed the volumes for Tamagawa measure. Consider now a global field K, a set of places S satisfying C¿E¿ for a quaternion algebra H/K, and R the ring of elements of K which are integral for v /∈ S. For every ideal N of R being prime to the reduced discriminant D of H/K, let O be an Eichler R-order in H of level N. For every prime ideal p|N, and such that p m ||N, we choose an inclusion ip : K → Kp, where Kp is a non-archimedean local field. We can extend ip to an inclusion of H in M(2, Kp) ,denoted by the same way, such that ip(O) is the canonical Eichler order of level p m Rp. The preimage by ip of the groups Γ0(p m ) and Γ(p m ) is O 1 and O 1 (p m ) respectively. We define the congruent groups of mixed type by considering the subgroup Γ of O1 defined by Γ = {x ∈ O 1 ⎧ ⎪⎨ Γ0(p |ip(x) ∈ ⎪⎩ m ) p|N0 Γ1(pm ) if p|N1 , or pm Γ(p ||N m ) if p|N2 D −3/2 p (1−Np −2 ).
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The Arithmetic of Quaternion Algebra

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