 
Summary: Basic Theorems of Elliptic Curves
Jonathan Cass
10/14/09
Abstract
An elliptic curve E is the locus of solutions to a degree 3 equation. Many
interesting results are obtained by looking at the structure of the solutions
whose coordinates are in various fields. We denote the solutions to E with
coordinates in a ring F by E(F). We can introduce a group structure on E(F),
and then examine what kinds of groups we get when F is one of a number of
different fields. In this talk we will discuss the structure of E(C), E(R), E(Q),
and E(Zp). The main theorems to be proven are that E(C) is isomorphic to a
torus and that E(Q) is a finitely generated group.
Content
The Basic Definition
An elliptic curve is the locus of solutions to a polynomial equation in two vari
ables of the form
y2
= x3
+ ax2
+ bx + c
