The Arithmetic of Quaternion Algebra

70 CHAPTER 3. QUATERNION ALGEBRA OVER A GLOBAL FIELD
Remark. (Beck)The non-maximal R-orders are never euclidean for the
norm, if K is a number field
Proof. If O ′ � O is non-maximal R-order, it exists x ∈ O but x /∈ O ′ , and for
every c ∈ O ′ , Nn(x − c) ≥ 1. If x = b −1 a, where a, b ∈ O ′ , the division of a by
b in O ′ is impossible. In the contra-example, n(b) and n(a) can not be rendered
as being prime to each other.
C:Trace formula for the maximal inclusions.
Let X be a nonempty finite set of places of K containing the infinite place if K
is a number field. Let L/K be a quadratic algebra and separable over K, and B
be a R-order of L. Let O be an Eichler order over R of level N in H, and DN
be the discriminant of O (D is the product of places, identifying to the ideals
of R and being ramified in H and not belonging to S).
For each p /∈ S, it can be given a group Gp such that O × p ⊂ Gp ⊂ N(Op. For
v ∈ S, set Gv = H × v . The group Ga = �
v∈V Gv is a subgroup of H ×
A . Denote
G = GA ∩ H ×
We intend to consider the inclusions of L in H which is maximal with respect
to O/B modulo the inner automorphisms induced by G. cf. I.5 and II.3. We
obtain by an adele argument a ”trace formula” which can be simplified if S
satisfying Eichler’s condition.
Theorem 3.5.11. (Trace formula). Let mp = mp(D, N, B, O × ) be the number
of the maximal inclusions of Bp in Op modulo O × p for p /∈ S. Let (Ii), 1 ≤ ileqh
be a system of representatives of classes of ideals to the left of O, O (i) be the
right order of Ii, and m (i)
O × be the number of maximal inclusions of B in O (i)
modulo O (i)× . we have
h�
i=1
m (i)
O × = h(B) �
p∈S
where h(B) equals the class number of ideals in B.
Proof. If � mp = 0, the formula is trivial, so suppose it is nonzero. We then can
embed L in H so that for every finite place p /∈ S of K we have Lp ∩ Op = Bp;
we identify L with its image by a given inclusion. Consider then the set of the
adeles TA = {x = (xv) ∈ H ×
A }, such that for every finite place p /∈ S of K, we
have xpLpx−1 p ∩ Op = xpBpx−1 p which describes the set of the maximal local
inclusions of Lp in Hp with respect to Op/Bp. The trace formula is resulted
from the evaluation by two different methods of number Card(GA\TA/L × ).
(1) Card(GA\TA/L × ) = Card(B ×
A \LA/L × )Card(GA\TA/L ×
A )
, where B ×
A = BA ∩ GA. we notice firstly that Card(GA\TA/L ×
A ) is equal to the
product of the numbers mp of maximal inclusions of Bp in Op modulo Gp, and
since the numbers are finite and almost always equal to 1 (cf ch.II,§3), then it
is a finite number. Let X be a system of representatives of these double classes.
The equivalent relation:
mp
gAtAlAl = t ′ Al ′ A,t A ′ t′ A ∈ X, lA ′l′ A ∈ LA, l ∈ L × , gA ∈ GA