The Arithmetic of Quaternion Algebra

72 CHAPTER 3. QUATERNION ALGEBRA OVER A GLOBAL FIELD
With these definitions the theorem II.3.1 and II,3.2 show that if the level N
of the Eichler order O is square-free,
�
mp(D, N, B, O × ) �
(1 − ( B �
)) (1 + (
p B
p ))
p/∈S
p|D
and according to the number is zero or not, we have
�
mp(D, N, B, N(O × )) = 0 or 1.
It follows
p/∈S
Corollary 3.5.12. If O is an Eichler order of a square-free level N,
and
h�
i=1
m (i)
O×
p|N
�
= h(B) (1 − ( B �
)) (1 + (
p B
p ))
h�
i=1
p|D
p|N
m (i)
N(O) = 0 or h′ (B)
according to the precedent number is zero or not, where h ′ (B) is the quotient of
h(B) by the class number of group of ideals of B generated by:
– the ideals of R,
– the prime ideal of B which is over an ideal of R and ramified in H and in B.
We compute h(B) in practice by the Dedkind’s formula [1], if K is a number
field and S = {∞}:
h(B) = h(L)N[f(B)] �
p|f(B)
(1 − ( L
p )Np−1 ) · [B ×
L : B× ] −1
where h(L) is the class number of a maximal R-order BL of L, and N is the
norm of K over Q. By definition, if I is an integral ideal of R,
N(I) = Card(R/I).
It is useful to extend the trace formulae (theorem 5.11 and 5.11 bis.) to all of
the groups G which are contained in the normalizer of O, and containing the
kernel O 1 of the reduced norm in O.
Corollary 3.5.13. With the notations of theorem 5.11 and 5.11 bis, if G is a
group such that O 1 ⊂ G ⊂ N(O, the number of the maximal inclusion of B in
O modulo G satisfies
mG = mO[n(O × ) : n(G)n(B × )].
The above index is finite by the Dirichlet’s theorem on the units, cf. chapter
V below. In fact, it suffices to write mG = Card(G\T/L × ) and to notice
that the inclusion f of L in H is maximal with respect to O/B, Then we have