The Arithmetic of Quaternion Algebra

36 CHAPTER 2. QUATERNION ALGEBRA OVER A LOCAL FIELD
where P is the two-sided integral maximal ideal of a maximal order O of H.
The norm of a principal ideal Oh is naturally equal to the norm of the ideal
hO. By the Corollary 1.7 and the Theorem 2.3, we have
Lemma 2.4.1. The number of the integral ideals to the left ( to the right) of a
maximal order of H with norm q n , n ≥ 0, is equal to
�
1, if n is even
0, if n is odd,
if H is a field;
if H � M(2, K).
1 + q + ... + q n ,
Definition 2.13. The zeta function of X = H or K is a complex function of
a complex variable
ζX(s) = �
N(I) −s
I⊂B
where the sum is taken over the the integral ideal to the left (right) of a maximal
order B of H.
The above lemma allow to compute explicitly ζH(s) as the function of ζK(s).
We have
ζK = �
ζH = �
0 �
n≥
0≤d≤n
n≥0
q−ns = (1 − q −s ) −1 ,
ζH = �
q −2ns ζ(2s), if H is a field,
n≥0
if H � M(2, K). We then have
q d−2ns = � �
q d−2(d+d′ )s
= ζK(2s)ζK(2s − 1),
d≥0 d ′ ≥0
Proposition 2.4.2. The zeta function of X = K or H equals
or
ζH =
ζK(s) = (1 − q −s ) −1
�
ζK(2s), if H is a field
ζ(2s)ζK(2s − 1), if H = M(2, K) .
There is a more general definition of zeta function available for X ⊃ R.
The idea of such function comes from Tate [1] in the case of local field. Their
generation to the cental simple algebra is due to Godement [1], and to Jacquet-
Godemant [1]. The crucial point is to observe that the classical zeta function
can also be defined as the integral over the locally compact group X × of the
character function of a maximal order multiplied by χ(x) = N(x) −s for a certain
Haar measure. This definition can be generalized then to the zeta function of a
so called Schwartz-Bruhat function, of a quasi-character , and extends naturally
to the archimedean case. This is what we shall do. We proceed along Weil’s
book [1], for more details one can refer to it.