The Arithmetic of Quaternion Algebra

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74 CHAPTER 3. QUATERNION ALGEBRA OVER A GLOBAL FIELD Proof. The approximation theorem 4.3 of H1 and the fact that S satisfies E.C. (thus GA ⊃ H × v , v /∈ Ram(H)) lead to 1) H ′ (i) (〉) is independent of O , it equal the class number of two-sided ideals of an Eichler order of level N. 2) If TA �= ∅, the type number of Eichler orders of level N which B can be embedded maximally in is equal to 1/[K × A : R× An(TA)] times the total type number. In fact, if B is embedded maximally in one of these orders O, the other orders in which B is embedded maximally are the right order of ideals I with Ip = Opxp if p /∈ S, where (xp) ∈ T � A∩ p /∈S H× p . We utilize then the theorem 5.7,5.8 of ideal class when E.D. is satisfied. 3) The number of maxima inclusions of B in O modulo G, if it is not zero, is independent of the choice of the Eichler order O of level N. Actually, the natural mapping G\T/L × → GA\T ′ A /L× is bijective if T ′ A = {x ∈ Ta|n(x) ∈ K ′ }. It is obviously injective, and it is surjective because of T ′ A ⊂ GA(H ∩ Ta) ⊂ GAT . The properties 1), 2), 3) complete the proof of the theorem. In order that the theorem 5.15 to be applicable it is useful to know when the number d in consideration is equal to 1. In this case all of the Eichler orders of a given level play the same role. Proposition 3.5.16. Suppose S satisfies C.E.. With the notations of the precedent theorem, the number of the maximal inclusions of B in an Eichler order of level N modulo G is independent of the choice of the order, and is equal to m if H �= M(2, K) or if it exists a place such that: 1) v is ramified in L, or 2) v ∈ S, v is not decomposed in L. Proof. Since TA ⊃ L × AN(OA), we raise d to d ′ = [K × A : K× n(L × A )R× An(N(OA))] and use the theorem 3.7. If there exists a place v such that K × v �= n(L × v ), or turning back to the same thing that v is not decomposed in L, and such that K × v is contained in the group K × n(L × A )R× An(N(OA)), then the degree d ′ is 1.Since K × A ⊂ R× A if v ∈ S, the condition 2) is valid immediately for whole H. It is automatically satisfied if there exists an infinite place ramified in H. If p is a finite place ramified in L, then K × v = R × v = R × v n(L × v ) and d ′ = 1. If p is a finite place ramified in H, then K × v = n(N(Ov)) and hence d ′ = 1. When the number of quadratic extension L/K unramified is finite, we can say that in general the number of maximal inclusion of B in O only dependent on O by intervening of its level. Therefore, in general, the number of conjugate classes in O modulo G , with the given characteristic polynomial, only depends on O by its its level, if S satisfies C.E. Corollary 3.5.17. Suppose that S satisfies C.E. With the notations of 5.12 if K(h)/K satisfies the condition of 5.16 and if N is square-free, then � h(B) � (1 − ( h B p)) � (1 + ( B p )) h∈B b|D is equal to the number of conjugate classes in O modulo O × , of the characteristic polynomial being equal to that of h. We shall obtain easily with 5.12 and 5.13 the correspondent formula for the conjugate classes modulo O 1 or N(O). p|N
3.5. ORDERS AND IDEALS 75 Exercise 1. Prove the quaternion field over Q of reduced discriminant 46 is generated by i, j satisfying i 2 = −1, j 2 = 23, ij = −ji and O = Z[1, i, j, (1 + i + j + ij)/2] is a maximal order. 2. Prove the quaternion field over Q is a maximal order if: p = 2, {a, b} = {−1, −1}, O = Z[1, i, j, (1 + i + j + ij)/2]; p ≡ −1mod4, {a, b} = {−1, −p}, O = Z[1, i, (i + j)/2, (1 + ij)/2]; p ≡ 5mod8, {a, b} = {−2, −p}, O = Z[i, (1 + i + j)/2, j, (2 + i + ij)/4]; p ≡ 1mod8, {a, b} = {−p, −q}, O = Z[(1 + j)/2, (1 + aij)/2, ij] where q is a positive integer which is congruent to −1 modulo 4p, and a is an integer which is congruent to ∓1 modulo q. We can find in Pizer [6] a method which allows to obtain the Eichler order of level N explicitly (one is permitted p|N to the condition that the local order at p is isomorphic to the canonical order of Exercise II,4.4. 3. Let p be a prime ideal of R, which is prime to the reduced discriminant D of H/K, where R, K, H are defined as that in §5. Using II. 2.4, II. 2.6, III. 5.1, prove a) ∀n ≥ 2, there exist the orders in H of the reduced discriminant Dp n which are not the Eichler orders. b) every order in H of reduced discriminant Dp is an Eichler order. 4. Prove the normalizer N(O) of an order of H/K (with the notation in §5) satisfies N(O) = {x ∈ H|x ∈ N(Op), ∀p /∈ S}. Suppose that O is an Eichler order. Prove the group N(O)/K × O × is a finite group which is isomorphic to (Z/2Z) m where m is less or equal to the number of prime divisors of the reduced discriminant of O. 5. Let S = ∞, K a number field and h + the class number of ideals of K in the restrict sense induced by all of the classes real infinite places of K. Prove a) if h + is odd, every quaternion algebra over K which is unramified at least at one infinite place, and contains a single type of Eichler order (over the integer ring of K) of a given level. b) if h + = 1, with the same hypothesis as in a) all of the Eichler orders are principal. In particular if K = Q, every quaternion algebra H/Q such that H ⊗ R � M(2, R) contains a unique Eichler order O of a given level up to conjugations. This Eichler order is principal. If DN is its level, then the group N(O)/Q × O × is isomorphic to (Z/2Z) m , where m is the number of prime divisors of DN (Exercise 5.4). 6. Tensor product. With the notations of this section, let Hi/K be the quaternion algebras such that D = H1 ⊗ H2 = H0 ⊗ H3
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The Arithmetic of Quaternion Algebra

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