Complex Multiplication - Elliptic Curves: Number Theory and Cryptography - page 323

Therefore, we may assume at the start that r = 0, and hence ps = N .By

multiplying by −

if necessary, we may also assume that s> 0. Choose

10
0 − 1

t ∈ Z such that

0 ≤ q + ts < s.

Then

1 t
01

pq
0 s

= pq + ts
0

∈ S N .

s

Therefore, the elements of S N represent all SL 2 ( Z )-equivalence classes for
matrices of determinant N .

For the uniqueness, suppose that M i = a i b i ∈ S N for i =1 , 2areleft

0 d i

SL 2 ( Z )-equivalent. Then,
a 1 /a 2 ( b 1 a 2 − a 1 b 2 ) /N
0

= a 1 b 1 a 2 b 2

0 d 2

− 1

∈ SL 2 ( Z ) .

d 1 /d 2 0 d 1

Therefore, a 1 /a 2 and d 1 /d 2 are positive integers with product equal to 1, so
they are both equal to 1. Consequently, a 1 = a 2 and d 1 = d 2 . This implies
that

b 1 a 2 − a 1 b 2

= b 1 a 1 −

a 1 b 2
a 1 d 1

= b 1 −

b 2

.

N d 1

Since this must be an integer (because the matrix is in SL 2 ( Z )), we have

b 1 ≡ b 2 (mod d 1 ) .

Since 0 ≤ b 1 ,b 2 <d 1 = d 2 ,wehave b 1 = b 2 . Therefore, M 1 = M 2 .This
proves the uniqueness.
For S = ab

∈ S N , the function

0 d

S )( τ )= j aτ + b
d

( j ◦

is analytic in H . Define
F N ( X, τ )=
S

S )( τ )) =
k

a k ( τ ) X k , ( X − ( j ◦

∈ S N

so F N is a polynomial in the variable X with coe cients a k ( τ ) that are ana-
lytic functions for τ ∈H .

LEMMA 10.11
a k ( Mτ )= a k ( τ ) for all M ∈ SL 2 ( Z ) .