The Arithmetic of Quaternion Algebra

More documents

Recommendations

Info

54 CHAPTER 3. QUATERNION ALGEBRA OVER A GLOBAL FIELD the results obtained by the two methods: we shall have a chance to obtain some apparently different results but which essentially should be the same. It will deduce from it in §3 a large part of the theorem of classification. 4)Convergence. The Riemann zeta function converges absolutely for Res = σ > 1 since ζ(σ) = n −σ satisfies 1 < ζ(σ) < 1 + � ∞ 1 t −σ dt. If K is a finite extension of Q of degree d, there are in K at most d prime ideals over a ideal of Z, and 1 < ζK(σ) < � P (1 − NP −σ ) −1 ≤ ζ(σ) d , where P runs through the prime ideals of K. Therefore the zeta function converges for Res > 1. If K is a function field Fq(T ) , the zeta function is a rational fraction in q −s and the problem of convergence do not arise. Convergence of the quadratic zeta functions.Let f be a function of Schwartz- Bruhat space, and c be a character of XA,1. There are positive real numbers M.N such that NΦ < r < MΦ, and |c| = 1, hence the integral ZX(f, c, s) converges absolutely because of that the the zeta function of X which we denote by ZX(s) converges absolutely. We have seen that it can be expressed as a product of zeta functions of the center: ZK(2s)ZK(1 − s), by use of this the convergence is no problem. We see that ZX(s) is defined by an absolutely convergent integral for Res > 1. Definition 3.12. The Tamagawa measure on XA, where X = H or K , is the Haar measure dx ′ A . the Tamagawa measure on X× A is the Haar measure m∗K . the measures dx ′ A , dx∗A have been defined already in §2, this chapter, and mK is the residue at the point s = 1 of the classical zeta function ζK of K. We introduce the Tamagawa measures on XA,1, H1 A , H× A /K× A respectively in a standard way as the kernels of the modulus || · ||X on X, of the reduced norm, projective group. Definition 3.13. The Tamagawa number of X = H, or K, X1, H 1 , G = H × /K times are the volumes computed for the canonical measures obtained from the Tamagawa measures τ(X) = vol(XA/XK) τ(X1) = vol(X × A,1 /X× K ) τ(H 1 ) = vol(H 1 A /H1 K ) τ(G) = vol(H× A /K× A H× K ). In these definition it is assumed these volumes are finite. It is true actually in our cases. We have Theorem 3.2.4. The Tamagawa numbers of X, X1, H 1 , G have the following values: τ(X) = τ(X1) = τ(H 1 ) = 1, τ(G) = 2.
3.2. ZETA FUNCTION, TAMAGAWA NUMBER 55 Proof. When X is a field, the computation of Tamagawa number is implicitly contained in the theorem 2.2 of the functional equation. If X = M(2, K) , it can compute directly. The theorem 2.3 can be extended to the central simple algebra X. In this case we have τ(X) = τ(X1) = τ(H 1 ) = 1 and τ(G) = n, if [X : K] = n 2 . Reference: Weil [2]. By the definition of Tamagawa measure, τ(X) = 1. We shall prove τ(G) = 2τ(H 1 ) and τ(H1) = τ(H 1 ), after that then τ(H 1 ) = 1. The proof is analytic, and the Poisson formula is involved. The exact sequence which is compatible with thee Tamagawa measures 1 −−−−→ H 1 A /H× K −−−−→ HA,1/H × K n −−−−→ KA,1/K × −−−−→ 1 proves that τ(H 1 ) = τ(H1)τ(K −1 1 ). The theorem 2.2 shows τ(H1) = τ(K1) = 1 if H is a field because of the definition of Tamagawa measure itself. Therefore τ(H 1 = τ(H1) for every quaternion algebra H/K. It follows from the proof of Theorem 2.2 that � 2 K × A /K× H × A /H× f(||h||H)dhA K ·τ(H1 ) f(||k||K)dk ∗ A = K × A /K× � K × A /K× f(||k|| K 2)dk ∗ A for every function f such that the integrals converge absolutely. Applying ||h||H = ||n(h)|| K2 if h ∈ H times A , we see that � � f(||k||K)dk ∗ � A = τ(G) K × A /K× f(||k||K)dkA∗, it follows τ(G) = 2τ(H 1 ). The Theorem has been proved when X = H or K is a field. It remains to prove τ(SL(2, K)) = 1. The starting point is the formula � � (2) f(x)dx = [ � f(ua)]τ(u), A 2 SL(2,A)/SL(2,K) a∈K 2 − where f is an admissible function on A 2 , cf. II,§2, and τ(u) is a Tamagawa measure on SL(2, A)/SL(2, K), and where A 2 is identified with the column vectors of two elements in A on which SL(2, A) operates by � � � � � � a b x ax + by = . c d y cx + dy � � 0 The orbit of is A 1 2 � � � � 0 1 x − and its isotropic group is NA = { |x ∈ A}. 0 0 1 We apply Poisson formula, � f(ua) = � f ∗ ( t u −1 a) a∈K 2 a∈K 2 because of det(u) = 1. Here we give an other expression for the integral (2) in function of f ∗ . In fact, it prefers to write the integral in f ∗ into the function on f ∗∗ = f(−x). Since τ( tu−1 ) = τ(u), we obtain (3) � A 2 f ∗ � (x)dx = SL(2,A)/SL(2,K) a∈K2 ⎛ ⎝0 0 ⎞ ⎠ [ � f(ux) − f ∗ (0)]τ(u).
Page 1:
The Arithmetic of Quaternion Algebr
Page 4 and 5:
ii CONTENTS
Page 6 and 7:
2 CHAPTER 1. QUATERNION ALGEBRA OVE
Page 8 and 9: 4 CHAPTER 1. QUATERNION ALGEBRA OVE
Page 14 and 15: 10 CHAPTER 1. QUATERNION ALGEBRA OV
Page 84 and 85: 80 CHAPTER 4. APPLICATIONS TO ARITH
Page 104 and 105: 100 CHAPTER 4. APPLICATIONS TO ARIT
Page 106 and 107: 102 CHAPTER 4. APPLICATIONS TO ARIT
Page 108 and 109:
104 CHAPTER 4. APPLICATIONS TO ARIT
Page 110 and 111:
106CHAPTER 5. QUATERNION ARITHMETIC
Page 112 and 113:
Page 114 and 115:
Page 116 and 117:
Page 118 and 119:
Page 120 and 121:
Page 122:
show all

The Arithmetic of Quaternion Algebra

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?