The Arithmetic of Quaternion Algebra

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66 CHAPTER 3. QUATERNION ALGEBRA OVER A GLOBAL FIELD Therefore, O is an order.It is maximal because the reduced discriminant |det(eiej)| 1 2 of the order Z[e1, ..., e4] = Z[1, i, j, i] is 13 · 8hence the reduced discriminant of O which is deduced from the above order by a base change of determinant 1/4 equals 13 · 8/4 = 26. We shall see other examples in exercise 5.1,5.2,5.6. The properties of normal ideals. These are such ideals whose left and right orders are maximal. The local-global correspondence in Lattices, and the the properties mentioned in chapter II show that these ideals are locally principal. We leave as an exercise the following properties ( utilize the definitions of chapter I,8.5 and the properties of normal ideals of a quaternion algebra over a local field as we saw in chapter II,§1,2): (a) A ideal to the left of a maximal order has a maximal right order. (b) If the right order of ideal I is equal to the left order of ideal J, then the product IJ is an ideal and n(IJ) = n(I)n(J). Its left order equals that of I, and its right order equals that of J. (c) The two-sided ideals ”commute” with the ideals in a sense of CI = IC ′ , where C is a two-sided ideal of the left order of I and C ′ is the unique two-sided ideal of the right order of I such that n(C) = n(C ′ ). (d)If I is an integral ideal of reduced norm AB, where A and B are integral ideals of R, then I can be factorized into a product of two integral ideals of reduced norm A and B. (e) The two-sided ideals of a maximal order O constitute a commutative group generated by the ideals of R and the ideals of reduced norm P , where P runs through the prime ideals of R which are ramified in H. We shall utilize the fact that the single two-sided ideal of a maximal order Op of Hp of norm Rp is Op. These properties are true too for the the locally principal ideals of Eichler orders of a square-free level N. B:The class number of ideals and the type number of orders . Unfortunately, the property for an ideal being principal is not a local property. It is just one of the reasons that we work very often on adeles instead of working globally. It means that we often like to replace a lattice Y by the set (Yp) of its localizations (5.1). We denote YA = � Yv, with Yv = Hv if v ∈ S. v∈V From now on the orders in consideration will always be Eichler orders, and the ideals will be principal locally. Fix an Eichler order O of level N. The adele object associated with it is denoted by OA, the units of OA by O × A , the normalizer of OA in HA by N(OA). The global-adele dictionary. Ideals: The left ideals of O correspond bijectively with the set O × A /H× A ; the ideal I is associated with (xv) ∈ H × A such that Ip = Opxp if p /∈ S. Two-sided ideals: correspond bijectively with O × A \N(OA). Eichler orders of level N: correspond bijectively with N(OA)\H × A ; the order O′ is associated with (xv) ∈ H × A such that O′ p = x−1 p Opxp. Classes of ideals: The classes of left ideals of O correspond bijectively with O × A \H× A /H× × K . The classes of two-sided ideals correspond bijectively with O× A \N(OA)/(H N(OA)), the types of Eichler order of level N correspond bijectively with H × K \H× A /N(OA). Theorem 3.5.4. The class number of the ideals to the left of O is finite. K ∩
3.5. ORDERS AND IDEALS 67 Proof. According to the fundamental theorem 1.4, we have H × A = H× A H× v C for every place v,and infinity if K is a number field, and where C is a compact set (dependent of v). Since O × A is open in H× A by the definition of the topology on it, and O × A ⊃ H× v , where v satisfies the above conditions, then it follows that the class number of ideals is finite by using the global-local dictionary. Corollary 3.5.5. The class number of two-sided ideals is finite. the type number of Eichler orders of level N is finite. In fact, these numbers are less than or equal to the number of the classes of left ideals of O. Two Eichler orders of the same level being always tied by an ideal (of which the left order is one of these order, and the right order is the other one) since two Eichler orders of the same level are locally conjugate (ch.II), the number of classes of two-sided ideals of O not depends on the choice of O, but more precisely on its level N. By contrast, the number of classes of two-sided ideals of O possibly depends on the choice of O, or more precisely on the type of O. NOTATION. we denote by h(D, N) = h(Ram(H), N) the class number of the ideals to the left of O, by t(D, n) = t(Ram(H), N) the type number of Eichler order of level N, and for 1 ≤ i ≤ t, by h ′ i (D, N) the class number of two-sided ideals of an order of the type of Oi, where Oi runs through a system of representative of Eichler orders of level N. Lemma 3.5.6. We have h(D, N) = �t i=1 h′ i (D, N). Proof. The types of orders correspond to the decomposition H × A = �t × i=1 N(OA)xiH K . Let Oi be the right order of ideal Oxi. We have N(Oi,A) = x −1 i N(OA)xi and O × i,A = x−1 i O× i,Axi. It follows N(OA)xiH × K = xiN(Oi,A)H i × K and O× A \N(OA)xiH K /H× K = (D, N). O × i,A \N(Oi,A)/(HK ∩ N(Oi,A)) = h ′ i In particular, if the class number of two-sided ideals does not depend on the chosen type, and is denoted by h ′ (D, N), we then have the relation h(D, N) − t(D, N)h ′ (D, N). This is the case when S satisfies the Eichler’s condition (see the beginning of this section): it is an application of the strong approximation theorem(Thm 4.1 and Thm. 4.3) Definition 3.18. Let KH = n(H) and let PH be the group of the ideals of R generated by the elements of KH. Two ideals I and J in R are equivalent in the restrict sense induced by H if IJ −1 ∈ PH. Since H/K is fixed, we simply say ”in the restrict sense”. We denote by h the class number of ideals of K in the restrict sense. Recall KH = {x ∈ K|x is positive for the real places being ramified in H. Therefore h only depends on K and the real places of Ram(H)∞. Theorem 3.5.7. (Eichler,[3],[4]). If S satisfies C.E., an ideal to the left of an Eichler order is principal if and only if its reduced norm belongs to PH Corollary 5.7 (bis.) If S satisfies C.E., then
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The Arithmetic of Quaternion Algebra

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