The Arithmetic of Quaternion Algebra

74 CHAPTER 3. QUATERNION ALGEBRA OVER A GLOBAL FIELD
Proof. The approximation theorem 4.3 of H1 and the fact that S satisfies E.C.
(thus GA ⊃ H × v , v /∈ Ram(H)) lead to
1) H ′ (i) (〉) is independent of O , it equal the class number of two-sided ideals of
an Eichler order of level N.
2) If TA �= ∅, the type number of Eichler orders of level N which B can be
embedded maximally in is equal to 1/[K ×
A : R×
An(TA)] times the total type
number. In fact, if B is embedded maximally in one of these orders O, the
other orders in which B is embedded maximally are the right order of ideals
I with Ip = Opxp if p /∈ S, where (xp) ∈ T �
A∩ p /∈S H× p . We utilize then the
theorem 5.7,5.8 of ideal class when E.D. is satisfied.
3) The number of maxima inclusions of B in O modulo G, if it is not zero, is
independent of the choice of the Eichler order O of level N. Actually, the natural
mapping G\T/L × → GA\T ′ A /L× is bijective if T ′ A = {x ∈ Ta|n(x) ∈ K ′ }. It is
obviously injective, and it is surjective because of T ′ A ⊂ GA(H ∩ Ta) ⊂ GAT .
The properties 1), 2), 3) complete the proof of the theorem.
In order that the theorem 5.15 to be applicable it is useful to know when the
number d in consideration is equal to 1. In this case all of the Eichler orders of
a given level play the same role.
Proposition 3.5.16. Suppose S satisfies C.E.. With the notations of the precedent
theorem, the number of the maximal inclusions of B in an Eichler order of
level N modulo G is independent of the choice of the order, and is equal to m
if H �= M(2, K) or if it exists a place such that:
1) v is ramified in L,
or
2) v ∈ S, v is not decomposed in L.
Proof. Since TA ⊃ L ×
AN(OA), we raise d to d ′ = [K ×
A : K× n(L ×
A )R× An(N(OA))] and use the theorem 3.7. If there exists a place v such that K × v �= n(L × v ), or
turning back to the same thing that v is not decomposed in L, and such that K × v
is contained in the group K × n(L ×
A )R× An(N(OA)), then the degree d ′ is 1.Since
K ×
A ⊂ R×
A if v ∈ S, the condition 2) is valid immediately for whole H. It is
automatically satisfied if there exists an infinite place ramified in H. If p is a
finite place ramified in L, then K × v = R × v = R × v n(L × v ) and d ′ = 1. If p is a
finite place ramified in H, then K × v = n(N(Ov)) and hence d ′ = 1.
When the number of quadratic extension L/K unramified is finite, we can
say that in general the number of maximal inclusion of B in O only dependent
on O by intervening of its level. Therefore, in general, the number of conjugate
classes in O modulo G , with the given characteristic polynomial, only depends
on O by its its level, if S satisfies C.E.
Corollary 3.5.17. Suppose that S satisfies C.E. With the notations of 5.12 if
K(h)/K satisfies the condition of 5.16 and if N is square-free, then
� h(B) �
(1 − (
h
B p)) �
(1 + ( B
p ))
h∈B
b|D
is equal to the number of conjugate classes in O modulo O × , of the characteristic
polynomial being equal to that of h.
We shall obtain easily with 5.12 and 5.13 the correspondent formula for the
conjugate classes modulo O 1 or N(O).
p|N