The Arithmetic of Quaternion Algebra

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38 CHAPTER 2. QUATERNION ALGEBRA OVER A LOCAL FIELD Lemma 2.4.4. We have � ZX(s) = N(x) B −s dx · ⎧ ⎪⎨ ζK(s), = ζH(s) ⎪⎩ ζK(2) if X = K · � (1 − q−1 ) −1 , 1, if X = H is a field, if X = M(2, K) . Proof. The number of the elements of B modulo B × with norm q n , n ≥ 0 is the number of the integral ideals of B with norm q n . The integral is then equal to ζX(s)vol(B × ). The function ζX(s) is hence given by Proposition 4.2. Definition 2.17. Let dx be the Lebesque measure on R. Let X ⊂ R, and (ei) be a R-basis of X. For x = � xiei ∈ X, we denote by TX(x) the common trace of the R-endomorphisms of X given by the multiplication by x to the left and to the right. We denote by dxX the additive Haar measure on X such that We denote by dx · X dxX = |det(TX(eiej))| 1 2 Πdxi. the multiplicative Haar measure ||x||−1 X dxX. We can verify that the above definition is given explicitly by (1) dxC = 2dx1dx2, if x = x + ix, xi ∈ R, (2) dxH = 4dx1...dx4, if x = x1 + ix2 + jx3 + ijx4, xi � � ∈ R, x x (3) dxM(2,K) = Π(dxi)K, if x = ∈ M(2, K), K = R or C. x x We denote by tx the transpose of x in a matrix algebra. By an explicit manner the real number TX( tx¯x) equals to (0)’ x2 , if X = R, (1)’ 2x¯x, if X = C, (2)’ 2n(x), if X = H, (3)’ � x2 i , if X = M(2, R), (3)” 2 � xi¯x, if X = M(2, C). We put � ZX(s) = exp(−πTX( t x¯x))Nx −s dx Lemma 2.4.5. We have X × ZR = ∗π −s/2 Γ(s/2), ZC(s) = ∗(2π) −s Γ(s), � ZH + ∗ZK(s)ZK(s − 1) · (s − 1) 1, if H is a field if H = M(2,K) where ∗ represents a constant independent of s. Leave the proof of the lemma as an exercise. If X = M(2, K) we shall utilize the Iwasawa’s decomposition of GL(2, K). Every element x ∈ GL(2, K) can be written by unique way as x = � � y t u, y, z ∈ R 0 z + , t ∈ K, u ∈ U
2.4. ZETA FUNCTION 39 , where U is the group consisting of the matrices satisfying t ¯yy = 1. If n = [K : R], then the integrand function is (yz) 2ns exp(−nπ(y 2 + z 2 + t¯t)) Definition 2.18. The Schwartz-Bruhat space S of X is S = � the infinitely differentiable functions with fast decreasing if X ⊃ R the locally constant functions with compact support, if X � R. A quasi-character of a locally compact group G is a continuous homomorphism of G in C. If the modulus of its value is always 1, we call it a character An example of quasi-character on X is x ↦→ N(x) s . It is a character if and only if s is a pure imaginary. The quasi-characters of H × are trivial on the commutator group of it. According to I,3.5, the commutator group of H × equals the group of the quaternions of reduced norm 1. Every quasi-character of H × has the form χH = χK ◦ n, where χK is a quasi-character of K. Definition 2.19. The zeta function of a function f of the Schwartz-Bruhat and a quasi-character χ is the integral � ZX(f, χ) = f(x)χ(x)dx · . The canonical function Φ of X is Φ = X × � the characteristic function of a maximal order if X � R, exp(−πTX( t ¯xx)), if X ⊃ R Therefore the function ZX(s) of Lemma 4.4 and 4.5 are equal to ZX(Ψ, Nx−s ). We include this section with the definition of Tamagawa measure, a notion more or less equivalent to that of discriminant. We choose on X a character ψX ,called a canonical character, defined by the conditions: – ψR(x) = exp(−2iπx), – ψK ′ is trivial on the integer ring RK ′ = R′ and R ′ is self-dual with respect to ψK ′, if K′ is a non-archimedean prime field. – ψK(x) = ψK ′ ◦ TX(x), if K ′ is the sub prime field of K. We shall see in exercise 4.1 the explicit construction of ψK ′. The isomorphism x ↦→ (y ↦→ ψX(xy)) between X and its topological dual can be written as the Fourier transformation on X too: f ∗ � = f(y)ψX(xy)dy, X where dy = dyX is the the additive measure normalized as above. The dual measure is the measure d∗y such that the following inversion formula is valid: � f(x) = f X ∗ (y)ψX(−yx)dy ∗ .
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The Arithmetic of Quaternion Algebra

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