Elliptic Curves over C - Elliptic Curves: Number Theory and Cryptography - page 256

Chapter 9

Elliptic Curves over C

The goal of this chapter is to show that an elliptic curve over the complex
numbers is the same thing as a torus. First, we show that a torus is isomor-
phic to an elliptic curve. To do this, we need to study functions on a torus,
which amounts to studying doubly periodic functions on C , especially the
Weierstrass ℘ -function. We then introduce the j -function and use its proper-
ties to show that every elliptic curve over C comes from a torus. Since most
of the fields of characteristic 0 that we meet can be embedded in C ,many
properties of elliptic curves over fields of characteristic 0 can be deduced from
properties of a torus. For example, the n -torsion on a torus is easily seen to
be isomorphic to Z n ⊕ Z n , so we can deduce that this holds for all elliptic
curves over algebraically closed fields of characteristic 0 (see Corollary 9.22).

9.1 Doubly Periodic Functions

Let ω 1 ,ω 2 be complex numbers that are linearly independent over R .Then

L = Z ω 1 + Z ω 2 = { n 1 ω 1 + n 2 ω 2 | n 1 ,n 2 ∈ Z }

is called a lattice . The main reason we are interested in lattices is that C /L
is a torus , and we want to show that a torus gives us an elliptic curve.
The set

F = { a 1 ω 1 + a 2 ω 2 | 0 ≤ a i < 1 ,i =1 , 2 }

(see Figure 9.1) is called a fundamental parallelogram for L .Adiffer-
ent choice of basis ω 1 ,ω 2 for L will of course give a different fundamental
parallelogram. Since it will occur several times, we denote

ω 3 = ω 1 + ω 2 .

A function on C /L can be regarded as a function f on C such that f ( z +
ω )= f ( z ) for all z ∈ C and all ω ∈ L . We are only interested in meromorphic