The Arithmetic of Quaternion Algebra

1.4. ORDERS AND IDEALS 19
(3)O and O ′ are tied, and if I, J are the ideals having the left order O and the
right order O ′ respectively, we have I = J(A)h, where h ∈ H and (A) is a model
of two-sided ideal of O.
Proof. If O ′ = h −1 Oh, the principal ideal Oh ties O to O ′ and reciprocally. If
O ′ = h −1 Oh, then J −1 Ih is a two-sided ideal of O ′ . Inversely, if O and O ′ are
tied, and if I and J satisfy the conditions of (3), then O ′ = J −1 J = hOh −1 .
Corollary 1.4.11. The number of the orders of type t which are related to a
given order is less than or equal to the class number h of this order, if h is finite.
The number of order type of H is the number of the maximal order type.
Definition 1.17. Let L/K be a separable algebra of dimension 2 over K. let
B be a R-order of L and Obe a R-order of H. An inclusion f : L → H is a
maximal inclusion respect to O/B if f(L) ∩ O = B.Since the restriction of f to
B determines f, we also say that f is a maximal inclusion of B in O.
Suppose L = K(h) to be contained in H. By theorem 2.1 the conjugate class
of h in H ×
C(h) = {xhx −1 |x ∈ H × }
corresponds bijectively to the set of the inclusions of L to H. We also have
C(h) = {x ∈ H|t(x) = t(h), n(x) = n(h)}
. The set of maximal inclusions of B to O corresponds bijectively to a subset
of the conjugate class of h ∈ H × which equals
C(h, B) = {xhx −1 |x ∈ H × , K(xhx −1 ) ∩ O = xBx −1 }
and we have the disjoint union
C(h) = �
C(h, B).
B
where B runs through the orders of L. Consider a subgroup G of the normalization
of O in H × :
N(O) = {x ∈ H × |xOx −1 = O.
For x ∈ H × , we denote ˜x : y → xyx −1 the inner automorphism of H associated
with x, and ˜ G = {˜x|x ∈ G}. The set C(h, B) is stable under the right operation
of ˜ G.
Definition 1.18. . A maximal inclusion class of B in O mod G is the class
of maximal inclusion of B in O under the equivalent relation f = ˜xf ′′ ˜x ∈ ˜ G.
The conjugate class mod G of h ∈ H ⊗ is CG(h) = {xhx −1 |x ∈ G}.
We see also that the set of conjugate class modG of the elements x ∈ H such
that t(h) = t(x), n(x) = n(h) is equal to
˜G\C(h) = �
˜G\C(h, B)
B