The Arithmetic of Quaternion Algebra

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