The Arithmetic of Quaternion Algebra

50 CHAPTER 3. QUATERNION ALGEBRA OVER A GLOBAL FIELD
Definition 3.8. The discriminant of X is the product of the local discriminants
Dv. We denote it by DX = �
v∈P Dv.
The number DX is well-defined, because of Dv = 1, p.p.. We also have
DH = D 4 KN(dH) 2 or N(dH) =
�
Nv
v∈Ramf H
is the norm of reduced discriminant of H/K.
Fourier transformation. It is defined with the canonical character ψA = �
and the self-dual dx ′ A on XA:
f ∗ �
(x) = f(y)ψA(xy)dy ′ A.
XA
The group XK is discrete, cocompact, of covolume
vol(XA/XK) = 1
in XA for the measure dx ′ A , then according to theorem 1.4, we have the
POISSON FORMULA �
a∈XK
f(a) = �
a∈XK
f ∗ (a)
for every admissible function f, i.e. f, f ∗ are continuous and integrable, and for
every x ∈ XA, �
�
f(x + a) and a∈XK a∈XK f ∗ (x + a) converge absolutely and
uniformly with respect to parameter x.
Definition 3.9. The Schwartz-Bruhat functions on XA are the linear combination
of the functions of the form
f = �
v∈V
where fv is a Schwartz-Bruhat function on Xv. We denote by S(Xa) the space
of these functions.
EXAMPLE. The canonical function of XA equals the product of the local
canonical functions : Φ = �
v∈V Φv.
The general definition of zeta functions brings in the quasi-characters χ of XA ×
being trivial on X ×
K If X is a field, Fusijiki’s theorem (theorem 1.4 and exercise
1.1) proves that
χ(x) = c(x)||x|| s , x ∈ C,
where c is a character of X ×
A
f ′ v
being trivial on X×
K .
Definition 3.10. The zeta function of a Schwartz-Bruhat function f ∈ S(XA,
and of a quasi-character χ(x) = c(x)||x|| s of X ×
A being trivial on X×
K is defined
by the integral
�
ZX(f, χ) = f(x)χ(x)dx ∗ A,
denoted also by
�
ZX(f, c, s) =
X ×
A
X ×
A
when the integral converges absolutely.
f(x)c(x)||x|| s dx ∗ A,
v ψv