The Arithmetic of Quaternion Algebra

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40 CHAPTER 2. QUATERNION ALGEBRA OVER A LOCAL FIELD Definition 2.20. The Tamagawa measure on is the Haar measure on X, which is self-dual for the Fourier transformation associated with the canonical character ψX. Lemma 2.4.6. The Tamagawa measure of X is the measure dx if K ′ = R. If K ′ �= R, the Tamagawa measure is the measure D −1/2 X dx, where DXis the discriminant of X, that is to say, DX = ||det(TX(eiej)|| −1 K ′ , where (ei) is a R ′ -basis of a maximal order of X. Proof. If K ′ = R, the global definition of dx shows it is self-dual (i.e. equals its dual measure) for ψX. Suppose then K ′ �= R and to choose a R ′ -maximal order which we denote by B. Let Φ denote its characteristic function. The Fourier transform of Φ is the the characteristic function of the dual B∗ of B with respect to trace. By the same way, the bidual of B equals B itself, we see that Φ∗∗ = vol(B∗Φ. The self-dual measure of X is thus vol(B∗−1/2dx. If (ei) is a R ′ -basis of B, we denote by e∗ i ) its dual basis defined by TX(ei, ej) = 0 if i �= j and TX(ei, ei) = 1. The dual basis is a R ′ -basis of B∗ . If e∗ j = � aijei, let A be the matrix (aij). We have vol(B∗ ) = ||det(A)||K ′ · vol(B) = det(A)−1 for the measure dx. On the other hand, it is clear, det(TX(eiej)) = det(A) −1 . We then have vol(B) = ||det(TX(eiej))|| −1 ′ . By the same reason we prove the dual measure of measure dx is D −1 X dx. Lemma 2.4.7. The discriminant of H and of K are connected by the relation K DH = D 4 KNK(d(O)) 2 , where d(O) is the reduced discriminant of a R ′ -maximal order O in H. Proof. With the notations in §1, we have O = {h ∈ H|t(hO) ⊂ R∗ }, it follows easily that O ∗ � R = ∗ , R if H = M(2, K) ∗u − 1, . if H is a field We have then DH = vol(O ∗ ) = NH(O ∗−1 ) = NKn 2 (R ∗−1 )NK(d(O)) 2 = D 4 K N(d(O))2 . Remark 2.4.8. If K ′ �= R, the modulus group ||X × || is a discrete group. We endow it a measure which assigns every element its proper value. In all of the other cases, the discrete group considered in the following chapters will be endowed with the discrete measure which assigns every element with the value 1. compatible measure. Let Y, Z, T be the topological groups equipped with Haar measure dy, dz, dt and there be an exact sequence of continuous mappings 1 −−−−→ Y i −−−−→ Z j −−−−→ T −−−−→ 1 . We say the measure dy, dz, dt are compatible with this exact sequence, or either say that dz = dydt, or dy = dz/dt or dt = dz/dy, if for each function f such that the integral below exists and the equality is valid: � � � f(z)dz = dt f(i(y)z)dy, with t = j(z). Z T Y
2.4. ZETA FUNCTION 41 From this , while knowing two of these measures and the exact sequence we can define a third measure by compatibility. Such a construction will be applied very frequently. but it must be careful: the third measure depends on the exact sequence. Take for example, let X1 be the kernel of modulus, X 1 be the kernel of the reduced norm. We give them the natural measures which deduced the normalized measures above , and the exact sequence suggested by their definitions. We denote their measure by dx1 and dx 1 respectively. These measures are different, though the sets X1 and X 1 may equal. We shall compute explicitly the volume in the exercises of this chapter. If K ′ �= R, we notice that dx1 is the restriction of the measure dx · to X1 because of its naturalness. Excercise 1. Prove the following characters ψK are the canonical characters. If K = Qp, ψK(x) = exp(2iπ < x >), where < x > is the unique number ap −m , m ≥ 0, which is a rational locating between 0 and 1 such that x− < x >∈ Zp, where Zp is the integer ring of Qp. If K = Fp[[T ]], ψ K(x) = exp(2iπ, < x >) where < x >= a−1p −1 if x = � aiT i , 0 ≤ ai ≤ p. If x ∈ Q, we denote ψp(x) = ψQp(x), and ψinfty = ψR(x), where psiR(x) = exp(−2iπx) is the canonical character of R. Prove ψ = ψinftyπpψp defines on Q a character which equals trivial character. 2. Computing volumes. With the measure defined by compatibility coming from the canonical measures (Remark 4.8) prove the formula vol(R1) = 2, vol(C1) = 2π, vol(H1) = 2π 2′ vol(H 1 ) = 4π 2 . we notice that 2vol(H1) = vol(H1 ) for the chosen measures (Remark 4.8) though the sets H1, H1 are the same. Calculate the integral � � R R 3. The volume of groups in Eichler orders. Let Om = pm � be the R R order of The canonical Eichler order of level Rpm with m �= 0 in M(2, K), with K non archimedean and p be a uniform parameter of K. Set Γ0(p) = O 1 m = SL2(R) ∩ Om, Γ1(p m ) = {x ∈ Γ0(p m � 1 )|x ≡ 0 � ∗ mod(O0p 1 m )}, Γ(p m ) = {x ∈ Γ1(p m � 1 )|x ≡ 0 � 0 mod(O0p 1 m )}. H e−n(h) n(h) 2 4dh/n(h) 2 . On X = K, H, M(2, K) we choose the Tamagawa measure D −1/2 X dx, and on X × choose the measure ||x|| −1D −1/2 X 4.8. Verify the following formulae: formulae. dx, cf. Lemma 4.6 and Remark vol(Γ0(p m )) = D −3/2 K (1 − Np−2 )(Np + 1)Np 1−m , vol(Γ1(p m )) = D −3/2 K Np−2m ,
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