The Arithmetic of Quaternion Algebra

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70 CHAPTER 3. QUATERNION ALGEBRA OVER A GLOBAL FIELD Remark. (Beck)The non-maximal R-orders are never euclidean for the norm, if K is a number field Proof. If O ′ � O is non-maximal R-order, it exists x ∈ O but x /∈ O ′ , and for every c ∈ O ′ , Nn(x − c) ≥ 1. If x = b −1 a, where a, b ∈ O ′ , the division of a by b in O ′ is impossible. In the contra-example, n(b) and n(a) can not be rendered as being prime to each other. C:Trace formula for the maximal inclusions. Let X be a nonempty finite set of places of K containing the infinite place if K is a number field. Let L/K be a quadratic algebra and separable over K, and B be a R-order of L. Let O be an Eichler order over R of level N in H, and DN be the discriminant of O (D is the product of places, identifying to the ideals of R and being ramified in H and not belonging to S). For each p /∈ S, it can be given a group Gp such that O × p ⊂ Gp ⊂ N(Op. For v ∈ S, set Gv = H × v . The group Ga = � v∈V Gv is a subgroup of H × A . Denote G = GA ∩ H × We intend to consider the inclusions of L in H which is maximal with respect to O/B modulo the inner automorphisms induced by G. cf. I.5 and II.3. We obtain by an adele argument a ”trace formula” which can be simplified if S satisfying Eichler’s condition. Theorem 3.5.11. (Trace formula). Let mp = mp(D, N, B, O × ) be the number of the maximal inclusions of Bp in Op modulo O × p for p /∈ S. Let (Ii), 1 ≤ ileqh be a system of representatives of classes of ideals to the left of O, O (i) be the right order of Ii, and m (i) O × be the number of maximal inclusions of B in O (i) modulo O (i)× . we have h� i=1 m (i) O × = h(B) � p∈S where h(B) equals the class number of ideals in B. Proof. If � mp = 0, the formula is trivial, so suppose it is nonzero. We then can embed L in H so that for every finite place p /∈ S of K we have Lp ∩ Op = Bp; we identify L with its image by a given inclusion. Consider then the set of the adeles TA = {x = (xv) ∈ H × A }, such that for every finite place p /∈ S of K, we have xpLpx−1 p ∩ Op = xpBpx−1 p which describes the set of the maximal local inclusions of Lp in Hp with respect to Op/Bp. The trace formula is resulted from the evaluation by two different methods of number Card(GA\TA/L × ). (1) Card(GA\TA/L × ) = Card(B × A \LA/L × )Card(GA\TA/L × A ) , where B × A = BA ∩ GA. we notice firstly that Card(GA\TA/L × A ) is equal to the product of the numbers mp of maximal inclusions of Bp in Op modulo Gp, and since the numbers are finite and almost always equal to 1 (cf ch.II,§3), then it is a finite number. Let X be a system of representatives of these double classes. The equivalent relation: mp gAtAlAl = t ′ Al ′ A,t A ′ t′ A ∈ X, lA ′l′ A ∈ LA, l ∈ L × , gA ∈ GA
3.5. ORDERS AND IDEALS 71 is equivalent to tA = t ′ A, l ′ A = g ′ AlAl, g ′ A ∈ t −1 A GAtA ∩ LA = B ′ A, from (1). The second evaluation uses the disjoint union: and the adele objects O (i) A H × A = t� i=1 N(OA)xiH × K = x−1 i OAxi, G (i) A = x−1 i GAxi correspond globally to a system of representatives of the type of Eichler orders of level N ,O (〉) = H ∩O (i) A and to the groups G (i) = H ∩ G (i) A . We shall prove : (2) Card(GA\TA/L × ) = t� i=1 Card(G (i) A \N(O(i) A /H(i) )Card(G (i) \T (i) /L × ) where H (i) = N(O (i) A ∩H× , T (i) = TA ∩O (i) . Remark that Card(G (i) \T (i) /L × ) is the number of the maximal inclusions of B in O (i) modulo G (i) . we have the disjoint union TA = t� N(O)xiTi/L × disjoint union. i=1 On other hand, Card(GA\N(OA)xiTi/L × ) = Card(G (i) A \N(O(i) A /L× ) = Card(G (i) A \N(O(i) A /H(i) Card(G (i) \Ti/L × ). We denote H ′(i) G = Card(G(i) A \N(O(i) and hG(B) = Card(B ′ A \LA/L × ). When G = O × the numbers are respectively the class number of two-sided ideals of O (i) and the class number of the ideals of B. The expressions (1) and (2) gives the Theorem(bis). Let mp = mp(D, N, B, G) the number of the maximal inclusion of Bp in Op modulo Gp, if p /∈ S. Let O (i) , 1 ≤ i ≤ t be a system of the representatives of the type of Eichler order of level N, and m (i) G be the number of the maximal inclusion of B in O (i) modulo Gi. We have by the precedent definition: The theorem is proved. t� i=1 H ′ (i) m (i) G � = hG(B) mp. Definition 3.21. Let L/K be a separable quadratic extension. If p is a prime ideal of K, we define Artin symbol ( L p by ⎧ ( L ) = p p/∈S ⎪⎨ 1, if p can be decomposed in L (Lp is not a field) −1, ⎪⎩ 0, if p is inertia in L (Lp/Kp is an unramified extension) . if p is ramified in L (Lp/Kp is a ramified extension) Definition 3.22. Let B be a R-order of a separable quadratic extension L/K. We define Eichler symbol ( B p ) to be equal to Artin symbol if p ∈ S or Bp is a maximal order, and equal to 1 otherwise. The conductor f(B) of B is the integral ideal f(B) of R satisfying f(B)p = f(Bp), ∀p /∈ S. A /Hi)
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The Arithmetic of Quaternion Algebr
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The Arithmetic of Quaternion Algebra

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