ABSTRACT
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 isomorphic 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 properties 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 Zn ⊕ Zn, so we can deduce that this holds for all elliptic curves over algebraically closed fields of characteristic 0 (see Corollary 9.22).
