Galois Representations and Elliptic Curves

By Wade Hindes

October 19, 2011

Abstract

The study of Galois representations has been a rich source of exciting mathematics in recent years. From The Langlands Program to Fermat's Last Theorem, the representation theory of Galois groups continues to generate surprising and deep connections to algebraic geometry, complex analysis, discrete subgroups of Lie Groups and so much more. In this talk I will discuss one of the most important Galois modules, the Tate Module, associated to the prime powered torsion points on an elliptic curve. I will begin by giving a "gentle" introduction to elliptic curves, as far as is necessary to state results, define the Tate module, and sketch the proofs of Serre's celebrated Finite Index and Surjectivity Theorems.