Complex Multiplication - Elliptic Curves: Number Theory and Cryptography - page 322

the case. This was in contradiction to the Gelfond-Schneider theorem, which
implies that such a number must be transcendental.
We now start the proof of the theorem. If L = Z ω 1 + Z ω 2 is a lattice, we
may divide by ω 2 and thus assume that

L = Z τ + Z ,

with τ ∈H .If β ∈ R ,then βL ⊆ L implies that there exist integers j, k, m, n
with

β τ
1

= jk
mn

τ
1

.

Let N = jn−km be the determinant of the matrix. Rather than concentrating
only on β , it is convenient to consider all 2 × 2 matrices with determinant N
simultaneously.

LEMMA 10.10
Let N be a positive integer and let S N be the set of m atrices of the form
ab
0 d

with a, b, d

2 matrixwithinteger
entries and determ inant N ,then there isaunique m atrix S

∈ Z , ad = N ,and 0 ≤ b<d .If M isa 2 ×

∈ S N su ch that

MS − 1 ∈ SL 2 ( Z ) .

In o ther w ords,ifwesay thattwomatrices M 1 ,M 2 are left SL 2 ( Z ) -eq u ivalent
when there existsamatrix X

SL 2 ( Z ) with XM 1 = M 2 ,then S N con tains
exactlyonee em ent ineachequivalence class of the set of integer m atrices of
determ inant N .

∈

Let pq

be an integer matrix with determinant N .Write

PROOF

rs

= x
y

p
r

−

with gcd( x, y ) = 1. There exist w, z ∈ Z such that xz − wy =1. Then
zw
yx

∈ SL 2 ( Z )

and

zw
yx

pq
rs

= ∗ 0 ∗ .