The Arithmetic of Quaternion Algebra

28 CHAPTER 2. QUATERNION ALGEBRA OVER A LOCAL FIELD
2.2 Study of M(2, K)
Let V be a vector space of dimension 2 over K. Suppose a basis (e1, e2) of V/K
to be fixed such that V = e1K +e2K. This basis allowed�us to identify � M(2, K)
a b
with the ring of endomorphisms End(V ) of V . If h = ∈ M(2, K), it
c d
associates an endomorphism :v ↦→ v.h, which is defined by the product of the
row matrix (x, y) by h, if v = e1x + e2y. Recall that a complete lattice in V is a
R-module containing a basis of V/K. If L, M are two complete lattices in V , we
denote the ring of R-endomorphisms from L to M by End(L, M), or End(L) if
L = M.
Lemma 2.2.1. (1) The maximal orders of End(V ) are the rings End(L), where
L runs through the complete lattices of V .
(2) The normal ideals of End(v) are the ideals End(L, M), where L, M runs
through the complete lattices of V .
Proof. (1) Let O be an order of End(V ) and M a complete lattice of V . Set L =
{m ∈ M|m ∈ O}. It is a R-module contained in V . There exists a ∈ R such that
aEnd(M) ⊂ M}. It follows aM ⊂ L ⊂ M, hence L is a complete lattice. It is
clear that O ⊂ End(L). (2) Let I be an ideal to the left of End(L). We identify
I to a R-module f(I) of V 2 by the mapping h ↦→ f(h) = (e1h, e2h). Let xi,jbe
the endomorphism permuting e1 and e2 if i �= j, but fixing ei if i = j, and taking
the other element of the basis to zero. Choosing all the possibility for (i, j), and
computing f(xi,j, we see that f(I) contains (e1.h, 0), (0, e2.h) (e2h, e1.h).
Therefore f(I) = M + M by putting M = L.I. We then see easily that M is a
complete lattice. It follows finally I = End(L, M).
We recall here some classical results about the elementary theory of divisors.
Lemma 2.2.2. Let L ⊂ M be two complete lattices of V .
(1) There exists a R-basis (f1, f2) of M and a R-basis (f1π a , f2π b ) of L whee
a, b are integers uniquely determined.
(2) If (f1, f2) is a R-basis of L, there exists a unique basis of M/R of the form
(f1π n , f1r + f2π m ), where n, m are integers, and r belongs to a given set Um of
the representation of R/(π m R) in R.
Proof. (1) is classical. We� prove �(2).
The basisf1a + f2b, f1c + f2d) of M are
a b
such that the matrix A = satisfies L.A = M. We can replace A by XA
c d
if X ∈ M(2, R) × � . We verify without difficulty that it can be modified again to
n π r
A =
0 πm �
where n, m are integers and r ∈ Um
We are going to express these results in terms of matrix.
Theorem 2.2.3. (1) The maximal orders of M(2, K) are conjugate to M(2, R).
(2) The two-sided ideal of M(2, R) forms a cyclic group generated by the prime
ideal P = M(2, R)π.
(3)The integral ideals to the left of M(2, R) are the distinct ideals
�
n π r
M(2, R)
0 πm �