Elliptic Curves over C - Elliptic Curves: Number Theory and Cryptography - page 276

Since ab
cd

has determinant 1, we have

ab
cd

− 1

= d −b

.

− ca

Let

( m ,n )=( m, n ) ab
cd

=( ma + nc, mb + nd ) .

Then

( m, n )=( m ,n ) d −b

,

− ca

so there is a one-to-one correspondence between pairs of integers ( m, n )and
pairs of integers ( m ,n ). Therefore,
G k aτ + b

=( cτ + d ) k

1
( m τ + n ) k

cτ + d

( m ,n ) =(0 , 0)

=( cτ + d ) k G k ( τ ) .

Since g 2 and g 3 are multiples of G 4 and G 6 ,wehave
g 2 aτ + b

=( cτ + d ) 4 g 2 ( τ ) ,

3 aτ + b

=( cτ + d ) 6 g 3 ( τ ) .

cτ + d cτ + d

Therefore, when we substitute these expressions into the definition of j ,all
the factors ( cτ + d ) cancel.

Let F be the subset of z ∈H such that

1
2 ≤

( z ) < 1

π
3

<θ< π = e iθ

| z |≥ 1 , −

2 ,

z for

2 .

Figure 9.3 is a picture of F . Since we will need to refer to it several times, we

let

ρ = e 2 πi/ 3 .

PROPOSITION 9.14
Given τ

∈H ,there exists

ab
cd

∈ SL 2 ( Z )

su ch that

aτ + b
cτ + d

= z ∈F .

M oreover, z ∈F isuniquelydeterm ined by τ .