The Arithmetic of Quaternion Algebra

5.3. EXAMPLES 115
that the difference of K is totally principal in the restrict sense and denote by
H the unique quaternion field( up to isomorphism )which is totally defined over
K and unramified at the finite places. Let R, Rd be the integer ring of K and
its difference respectively.
Proposition 5.3.6. The maximal R-orders of H equipped with the scalar product:
x · y = T K/Q(d −1 t(x¯y)
are the unimodular, even lattice of dimension 8n.
Proof. Recall that the inverse of the difference is the dual of the integer ring R
of K with respect to the bilinear form T K/Q(x¯y) defined by the trace T K/Q of
K over Q. Let O be a maximal R-order of H. It is clear that O is isomorphic
to a Z-lattice of dimension 8n. It should prove that the bilinear form defined in
the proposition is equivalent to the usual scalar product, or in another words,
the quadratic form defined by q(x) = 2T K/Q(d −1 n(x)) is positively definite. In
fact, x ∈ H × implies That d −1 n(x) is totally positive and the trace is strictly
positive.
We verify that
a) x · y ∈ Z and x · x ∈ 2Z, because the inverse of the difference Rd −1 were sent
to each other by the trace in Z.
b) O is equal to its dual O ′ = {x ∈ H|T K/Q(d −1 t(xO)) ⊂ Z = {x ∈ H|t(xO) ∈
R} because H is not ramified at the finite places.
The construction of Leech lattice.
For the reasons prior to the curiosity of a non-specialist in the theory od finite
groups which is justified by the presence of the binary icasahedral group in the
automorphisms of Leech’s lattice, The construction of Tits for the lattice utilizes
the quternion field H which is totally defined and unramified over K = Q( √ 5).
We have seen that a maximal R-order O equipped with the scalar product in
the precedent proposition is a unimodular even latticeof order 8, and recall that,
if τ = (1 + √ 5)/2, Then
R = Z[1, τ] and x · y = T K/Q(2x¯y/(5 + √ 5)).
We shall observe later that the only integers x which are totally positive in R
of trace T K/Q(x) ≤ 4 are
(1) 0, 1, 2, τ 2 = (3 + √ 5)/2, τ −2 = (3 − √ 5)/2.
Although it is not used here, it bears in mind that we have given an explicit
R-basis of an order O in exercise. The unit group of reduced norm 1 of O
denoted by O 1 is isomorphic to the icosahedral binary group of order 120, and
contains the cubic roots of unit. Let x be one of them, put e = x + τ. We can
prove immediately that n(e) = 2, and e 2 = emod(2).
We denote by h the standard hermitian form of the H-vector space H 3 :
h(x, y) = � xi¯yi, if x = (xi) and y = (yi) belong to H 3 ,
from it we deduce on that on R 24 there is a scalar product induced by the
Q-bilinear form of Q-vector space H 3 of dimension 24:
x · y = T K/Q(2h(x, y)/(5 + √ 5))