Elliptic Curves over C - Elliptic Curves: Number Theory and Cryptography - page 282

SL 2 ( Z ), as in Proposition 9.14. Then, by Proposition 9.13,

j ( τ 1 )= j ( τ 1 )= j ( τ 2 )= j ( τ 2 ) .

By Corollary 9.18, τ 1 = τ 2 . Since an element of SL 2 ( Z )maps τ 1 to τ 1 ,and
an element of SL 2 ( Z )maps τ 1 = τ 2 to τ 2 , the product of these two matrices
(see Exercise 9.2) maps τ 1 to τ 2 , as desired.

There is also a version of Corollary 9.19 for lattices (the j -invariant of a
lattice is defined on page 276).

COROLLARY 9.20
Let L 1 ,L 2 ⊂ C be lattices. T hen j ( L 1 )= j ( L 2 ) ifand onlyifthere exists
0 = λ ∈ C su ch that λL 1 = L 2 .

PROOF One direction was proved on page 276. Conversely, suppose
j ( L 1 )= j ( L 2 ). Write L i =( λ i )( Z τ i + Z )with τ i ∈F , as in Corollary 9.15.
Then j ( τ 1 )= j ( L 1 )= j ( L 2 )= j ( τ 2 ), so Corollary 9.18 implies that τ 1 = τ 2 .
Let λ = λ 2 /λ 1 .Then λL 1 = L 2 .

We can now show that every elliptic curve over C corresponds to a torus.

THEOREM 9.21
Let y 2 =4 x 3

− Ax − B define an ellipticcurve E over C .Thenthere isa
lattice L su ch that

g 2 ( L )= A

and g 3 ( L )= B.

Thereisanisom orphism of groups

C /L E ( C ) .

PROOF Let

A 3

27 B 2 .
By Corollary 9.18, there exists a lattice L = Z τ + Z such that j ( τ )= j ( L )= j .
Assume first that g 2 ( L )

j = 1728

A 3 −

C × =0. Then j = j ( L ) =0,so A = 0. Choose λ ∈

such that

g 2 ( λL )= λ − 4 g 2 ( L )= A.
The equality j = j ( L ) implies that

g 3 ( λL ) 2 = B 2 ,

so g 3 ( λL )= ±B .If g 3 ( λL )= B , we're done. If g 3 ( λL )= −B ,then

g 3 ( iλL )= i − 6 g 3 ( λL )= B g 2 ( iλL )= i 4 g 2 ( λL )= A. and