The Arithmetic of Quaternion Algebra

30 CHAPTER 2. QUATERNION ALGEBRA OVER A LOCAL FIELD
Use N(O) to denote the normalizer of an Eichler order O of M(2, K) in
GL(2, K). By definition N(O) = {x ∈ GL(2, K)|xOx−1 = O}. Let O1, O2
be the maximal orders containing O. The inner automorphism associating to
an element of N(O) fixes the couple (O1, O2). The study of two-sided ideal
of maximal order has showed that the two-sided ideal of a maximal order is
generated by the nonzero elements of K. Hence it follows O = On �
with
�
n ≥ 1.
0 1
. We can
therefore we see that N(On) is generated by K × On × and
π n 0
verify without difficulty that the reduced discriminant of an Eichler order equals
its level.
The tree of maximal order
Definition 2.8. (Serre [3], Kurihara [1]). A graph is given by
– a set S(X) whose element is called a vertex of X,
– a set Ar(X) whose element is called an edge of X,
– a mapping: Ar(X) → S(X) × S(X) defined by y ↦→ (s, s ′ ) where s is called
the origin of y and s ′ the extremity of y,
– an involution of Ar(X) denoted by y ↦→ ¯y such that the origin of y to be the
extremity of ¯y and such that
(1) y �= ¯y
.
A chain of a graph X is a sequence of edges (y1, ..., yi+1...) such that the extremity
of yi to be the origin of yi+1 for all i. To give a chain is equivalent to
give a sequence of vertices such that two consecutive vertices to be always the
origin and the extremity of an edge. A finite chain (y1, ..., yn) is called to have
the length n, and we say it joins the origin of y1 and the extremity of yn. A pair
(yi, ¯yi) in a chain is called a loop. A finite chain without any loop such that the
origin of y1 to be the extremity of yn is called a circuit. A graph is connect if
it always exists a chain joining two distinct vertices. A tree is a connect graph
and without circuit.
We see the set X of maximal orders of M(2, K) is provided with a structure
of graph denoted still by X , such that these maximal orders are the vertices of
X and the pair (O, O ′ ) of maximal orders of distance 1 are the edges of X.
Lemma 2.2.5. Let O be a maximal order. The maximal orders at distance n
to O are the extremities of some chains which have no loops, their origins are
O and the length are n.
Proof. Let O ′ be a maximal order such that d(O, O ′ ) = n. Therefore, O =
End(e1R + e2R) and O ′ = End(e1R + e2π n R), for an appropriate basis (e1, e2)
of V . The sequence of vertices (O, O1, ..., Oi, ..., O ′ ), where Oi = End(e1R +
e2π i R), 1 ≤ i ≤ n − 1, is a chain without loops, joining O and O ′ , of length n.
Inversely, provided there is a chain of length n > 2 given by a sequence of
vertices (O0, ..., On). There exist the R-lattices Li ⊃ Li+1 ⊃ Liπ such that
Oi = End(Li) for 0 ≤ i ≤ n. the chain has no loops if Liπ �= Li+2 for
0 ≤ i ≤ n − 2. WE have
Li+1 ⊃ Liπ ⊃ Li+1π
Li+1 ⊃ Li+2 ⊃ Li+1π