The Arithmetic of Quaternion Algebra

52 CHAPTER 3. QUATERNION ALGEBRA OVER A GLOBAL FIELD
a change of variables x ↦→ x −1 . The Poisson formula allow us to find again
an entire function , plus a rational fraction with simple poles 0 and 1. Sine it
is already allowed a constant, ZX(f, c, s) is a generalization of Riemann zeta
function. The method for proving the functional equation is the same.
2) Applying to ZX(f, c, s). We shall treat the problem of convergence far behind.
Temporarily we admit that ZX(f, c, s) converges for Res sufficiently large, and
that X is a field. We choose a function ϕ which separates R+ into two parts
[0, 1] and [1, ∞] by putting
⎧
⎪⎨ 0, if 0 ≤ x < 1
ϕ = 1/2,
⎪⎩
1,
if
if
x = 1
x > 1
.
We consider firstly the integral taking for ||x|| −1 ∈ [0, 1],
Z 1 �
X(f, c, s) = f(x)c(x)ϕ(||x||)||x|| s dx ∗ A,
X ×
A
which defines an entire function on C. In fact if Z1 X (f, c, s) converges absolutely
for Res ≥ Res0, it converges also absolutely for Res ≤ Res0 because of ||x|| s ≤
||x|| s0 if ||x|| ≥ 1. The remaining integral is taken for ||x|| −1 ∈ [1, ∞], after the
change of variables x ↦→ x−1 , it can be written as
�
I =
X ×
A
f(x −1 )c(x −1 )ϕ(x||)||x|| −s dx ∗ A.
After seeing that every term in the symbol of the integral except for f(x−1 only depend on the class of x in X ×
A /X× K we can apply Poisson formula to it.
Utilizing that X is a field, writing it as XK = X ×
K ∪ {0}.
�
I = c(x −1 )ϕ(||x||)||x|| −s { �
f(ax −1 − f(0))}dx ∗ A,
X ×
A \X× K
a∈XK
where the terms in the embrace, transformed by Poisson formula, is
||x||[f ∗ (0) + �
f ∗ (xa)] − f(0).
a∈X ×
K
Regrouping the terms, I can be written as the sum of one entire function and
the other two terms:
with
I = Z 1 (f ∗ , c −1 , 1 − s) + J(f ∗ , c, 1 − s) − J(f, c, −s)
�
J(f, c, −s) = f(0)
X ×
A /X×
c(x
K
−1 )||x|| −s ϕ(||x||)dx ∗ A.
applying the exact sequence
we obtain
1 → XA,1/X ×
K → X×
A /X× K
→ ||X× A || → 1
�
J(f, c, −s) = f(0)
||X ×
A ||
t −s ϕ(t)dt ·
�
XA,1/X ×
c
K
−1 (y)dy.