The Arithmetic of Quaternion Algebra

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48 CHAPTER 3. QUATERNION ALGEBRA OVER A GLOBAL FIELD so that the volumes of x −1 C ′′ and C ′′ x are strictly greater than the volume of XA/XK. We set then C = C ′′ − C ′′ = {x − y|x, y ∈ C ′′ }. It is a compact set of XA since the mapping (x, y) ↦→ x − y is continuous. There exist a, b ∈ XK such that xa ∈ C, bx −1 ∈ C. Now we suppose that X is a field, then we can choose a, b in X × K . We have ba ∈ C2 which is compact in XA. The number of possible value for ba = c is then finite, and hence we choose C ′ = � c −1 C. 4)In view of Fusijaki’s theorem, it is evident for a field X. In fact, with the choice of v, the group of modulus of X × v is of finite index in that of X × A , and if X × A,1 denotes the elements of XA with modulus 1, we then prove immediately that XA,1/XK is compact. It remains to the case of M(2, K) to prove. It is well known that we can use the existence of the ”Siegel sets”. But in the very simple case which we are interested in, the proof is quite easy. Let P be the group of upper triangular matrices, D the group of diagonal matrices, and N the unipotent group of P .By triangulation (II,lemma 2.2 for v ∈ P ), we have GL(2, A) = PA · C = DANAC where C equals a maximal compact subgroup of GL(2, A). According to the theorem of approximation in the adeles A � NA, and the property 4) having been proved for K, we have PA = DKDvC ′ · NKNvC ′′ . The elementary relation of permutation � a 0 � � 0 1 b 0 � � x 1 = 1 0 � � ax/b a 1 0 � 0 b implies PA = PKPvC ′′ where C ′′ ⊂ PA is compact. 4) is proved. Exercise 1. Let X be a global field K, or a quaternion field H/K. Prove that X × A /X× K is the direct product of the compact group XA,1/X × K and a group being isomorphic to R+ = {x ∈ R|x > 0} or to Z, depending on the characteristic of K is zero or not. It follows that the group of quasi-characters (continuous homomorphisms to C × ) of X × A being trivial on X× K is isomorphic to the direct product of the group of characters (homomorphism with value in {z ∈ C| |z| = 1} ) of XA,1/X × K by the group of quasi-characters of R+ or Z. Prove then that every quasi-character has the form of X × A being trivial on X× K χ(x) = c(x)||x|| s where s ∈ C, and c is a character of X × A being trivial on X× K . 3.2 Zeta function, Tamagawa number Definition 3.6. The Classic zeta function of X, where X is a global field K or a quaternion field H/K, is the product of the zeta functions of Xv, here v ∈ P .
3.2. ZETA FUNCTION, TAMAGAWA NUMBER 49 The product is absolutely convergent when the complex variable s has a real part Res > 1. We therefore have ζA(s) = � ζv(s), Res > 1. v∈P It follows from II,4.2 the following formula, called the multiplicative formula: ζH(s/2) = ζK(s)ζK(s − 1) � v∈Ramf H (1 − Nv 1−s ) where Nv is the number of the prime ideal associated with the finite place v ∈ P . This formula plays a basic role in the classification of the quaternion algebra over a global field. The definition of general zeta function is intuitive: not only restrict to the finite places. Definition 3.7. The zeta function of X is the product ZX(s) = � ZXv v∈V (s) of the local zeta functions of Xv for v ∈ V . By abuse of terms we call by zeta function of X the product of ZX by a non-zero constant too, The functional equation is not modified. Proposition 3.2.1. (Multiplicative formula). The zeta function of global field K equals ZK(s) = ZR(s) r1 Z r2 C ζ K(s), where r1, r2 denote the numbers of real places, complex places of K respectively, and the the archimedean local factors are the gamma functions: ZR(s) = π −s/2 Γ(s/2), ZC(s) = (2π) −s Γ(s). The zeta function of quaternion algebra H/K equals ZH(s) = ZK(2s)ZK(2s − 1)JH(2s), whereJH(2s) depends on the ramification of H/K, and JH(s) = � v∈RamH Jv(s), � 1 − Nv with Jv(s) = 1−s , if v ∈ P s − 1, if s ∈ ∞ . Now we shall use the following adele measures: textoverXA, dx ′ a = � dx ′ vwith dx ′ � dxv, x ∈ ∞ v = D −1/2 v dxv , v ∈ P. over X × A , dx∗A = � v v dx ∗ vwithdx ∗ v = � dx · v, v ∈ ∞ D −1/2 v dx × v , v ∈ P For the local definition see II,4. From this we obtain by the compatibility the adele measures on the groups XA,1, H 1 A , H1 A /K′ A , denoted by dxA,1, dx 1 A , dxA,P respectively. We denote by the same way the adele measure on GA, and that on GA/GK obtained by compatibility with the discrete measure assigning every element of GK with value 1 if GK is a discrete subgroup of GA.
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The Arithmetic of Quaternion Algebra

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