The Arithmetic of Quaternion Algebra

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62 CHAPTER 3. QUATERNION ALGEBRA OVER A GLOBAL FIELD We apply this construction to S = Ram(H) and hence obtain this norm theorem. We can even obtain a little bit stronger form of it. Corollary 3.4.2. Every element of KH which is integral for every place excludin possibly one w /∈ Ram(H) is the reduced norm of ann element of H which is integral excluding possibly for w. The strong approximation theorem. Let S be a nonempty set of places of K containing at least an infinite place if K is a number field. Let H 1 be the algebraic group induced by the quaternions which belongs to a quaternion field H over K and is of reduced norm 1. For a finite set S ′ ⊂ V put H 1 � S ′ = v∈S ′ H 1 v . Recall that H 1 v is compact if and only if v ∈ Ram(H). Otherwise, H 1 v = SL(2, Kv). Theorem 3.4.3. (Strong approximation). If H S 1 is not compact, then H 1 K H1 S is dense in H 1 A . The theorem was proved by Kneser [1], [2], [3] as an application of Eichler’s norm theorem if K is a number field and S ⊃ ∞. The condition is natural. If is closed, and hence different H1 S is compact, since H1 K is discrete in H1 A H1 SH1 K from H1 A definitely. The condition introduced in the statement of this theorem play a basic role in the quaternion arithmetics. Definition 3.14. A nonempty finite set of the places of K satisfies the Eichler condition for Hdenoted by C.E., if it contains at least a place of K which is unramified in H. Proof. of the theorem 4.3. Let H1 KH1 S be the closure of H1 KH1 S in H1 A . It is stable under multiplication. It then suffices prove the theorem for every place v /∈ S, for every element � av, integral over Rv,if w = v (1) a = (aw), with aw = 1, if w �= v for every neighborhood U of a, we have H1 KH1 S ∩U �= ∅. For that, it is necessary t(H1 KH1 S ) ∩ t(U) �= ∅, where t is the reduced trace which has been extended to adeles (see III, §3, the Example above Lemma 3.2). We have � t(av), if w = v (2) t(a) = tw with tw = 2, if w �= v . Since t is an open mapping, it suffices to prove that for every neighborhood W ⊂ KA of t(a), we have t(H1 KH1 S ) ∩ W �= ∅. It suffices to prove there exists t ∈ K satisfying the following conditions: ∗ the polynomial p(X, t) = X2−tX+1 is irreducible over Kv if v ∈ Ram(H), (3) ∗ t is near to t(a) in KA, that is to say, t is near to t(av) in Kv for a finite number of places w �= v, w /∈ S.
3.5. ORDERS AND IDEALS 63 We can prove these conditions in virtue of 3.6 and 1.4. Two elements of the same reduced trace and the same reduced norm are conjugate (I,2.1), and H × A = H × KH× Recall that if x ∈ H × , we denoted earlier ˜x(y) = xyx−1 , y ∈ H × , and if S D−1 where D is compact in HA by (3.4) hence H 1 K H1 S ∩ ˜ D(U) �= ∅. Z ⊂ H × , we denote ˜ Z = {˜z|z ∈ Z}, see I,§4, Exercise. There exists then d ∈ D such that ˜ d(a) ∈ H1 KH1 S . Let (b) be an sequence of elements of H× K which converges in Hv to the v-adic part of d−1 . Therefore, bd(a) ∈ H1 KH1 S converges to a: it is true for v-adic by constructions, and if w �= v, aw=1. concludes that a ∈ H Finally it 1 KH1 S . It will be found in 5.8 and 5.9 the applications of the theorem. 3.5 Orders and ideals Fix a nonempty set S of the places of K, which contains the infinite places if K is a number field. Thus the ring R = RS = {x ∈ K|x ∈ Rv, ∀v /∈ S} is a Dedekind ring (Weil [1]). Example. Let S = ∞, and K ⊃ Q, then R is the integer ring of K. If S is reduce to one place, and K is a function field, then R � Fp[T ]. Let H/K be a quaternion algebra over K; the lattices, orders, and ideals in H are relative to R (see definition I,4).We study the orders and the ideals in virtue of their local properties. The present section is consists of three parts: A: General properties of orders and ideals. B: class Numbers and order types. C: Trace formulae for the maximal inclusion. We suppose frequently that S satisfying the Eichler condition defined above and denoted by C.E., in order to obtain the results more simply. The case where C.E. not satisfied will be treated in chapter V. A:General properties. Let Y be a lattice of H. We write Yv = Rv ⊗R Y if v ∈ V . When v ∈ S we have Rv = Kv, and Yv = Hv. Definition 3.15. For every complete R-lattice Y of H, and for every place v /∈ S of K, the Rv-lattice Yv = Rv ⊗R Y is called the localization of lattice Y at v. Since S ⊃ ∞, the places which not belong to S are finite. If (e) is a basis of H/K, the lattice X generated over R by (e) is a global lattice in H which can be obtained from the local lattices in Hv, v /∈ S in the way described in the following proposition. Proposition 3.5.1. Let X be a lattice of H. There exists a bijection between the lattices Y of H and the set of lattices {(Yp)|Yp is the lattice of Hp, Yp = Xp, p.p.} given by the inverse mapping of one to another: Y ↦→ (Yp) p/∈S and (Yp) p/∈S ↦→ Y = {x ∈ H|x ∈ Yp, ∀p /∈ S}.
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The Arithmetic of Quaternion Algebra

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