The Arithmetic of Quaternion Algebra

3.5. ORDERS AND IDEALS 63
We can prove these conditions in virtue of 3.6 and 1.4. Two elements of the
same reduced trace and the same reduced norm are conjugate (I,2.1), and H ×
A =
H ×
KH× Recall that if x ∈ H × , we denoted earlier ˜x(y) = xyx−1 , y ∈ H × , and if
S D−1 where D is compact in HA by (3.4) hence H 1 K H1 S ∩ ˜ D(U) �= ∅.
Z ⊂ H × , we denote ˜ Z = {˜z|z ∈ Z}, see I,§4, Exercise. There exists then d ∈ D
such that ˜ d(a) ∈ H1 KH1 S
. Let (b) be an sequence of elements of H×
K which
converges in Hv to the v-adic part of d−1 . Therefore, bd(a) ∈ H1 KH1 S converges
to a: it is true for v-adic by constructions, and if w �= v, aw=1.
concludes that a ∈ H
Finally it
1 KH1 S .
It will be found in 5.8 and 5.9 the applications of the theorem.
3.5 Orders and ideals
Fix a nonempty set S of the places of K, which contains the infinite places if K
is a number field. Thus the ring
R = RS = {x ∈ K|x ∈ Rv, ∀v /∈ S}
is a Dedekind ring (Weil [1]).
Example. Let S = ∞, and K ⊃ Q, then R is the integer ring of K. If S is
reduce to one place, and K is a function field, then R � Fp[T ].
Let H/K be a quaternion algebra over K; the lattices, orders, and ideals in H
are relative to R (see definition I,4).We study the orders and the ideals in virtue
of their local properties. The present section is consists of three parts:
A: General properties of orders and ideals.
B: class Numbers and order types.
C: Trace formulae for the maximal inclusion.
We suppose frequently that S satisfying the Eichler condition defined above and
denoted by C.E., in order to obtain the results more simply. The case where
C.E. not satisfied will be treated in chapter V.
A:General properties.
Let Y be a lattice of H. We write Yv = Rv ⊗R Y if v ∈ V . When v ∈ S we have
Rv = Kv, and Yv = Hv.
Definition 3.15. For every complete R-lattice Y of H, and for every place
v /∈ S of K, the Rv-lattice Yv = Rv ⊗R Y is called the localization of lattice Y
at v.
Since S ⊃ ∞, the places which not belong to S are finite. If (e) is a basis
of H/K, the lattice X generated over R by (e) is a global lattice in H which
can be obtained from the local lattices in Hv, v /∈ S in the way described in the
following proposition.
Proposition 3.5.1. Let X be a lattice of H. There exists a bijection between the
lattices Y of H and the set of lattices {(Yp)|Yp is the lattice of Hp, Yp = Xp, p.p.}
given by the inverse mapping of one to another:
Y ↦→ (Yp) p/∈S and (Yp) p/∈S ↦→ Y = {x ∈ H|x ∈ Yp, ∀p /∈ S}.