This talk on Basic Elliptic Curves was held on Friday November 18, 2016 in MC 4020. The talk was given by Akshay Tiwary.

## Abstract

Elliptic Curves lie in the intersection of Algebra, Geometry, Number Theory and Complex Analysis. While my talk won't require any experience with complex analysis or algebraic geometry, I hope to expose you to this active area of research. Although I *could* tell you that the meat of Wiles' proof of Fermat's Last Theorem involved proving a special case conjecture previously known as the Taniyama Shimura conjecture involving elliptic curves, or that the Birch and Swinnerton-Dyer conjecture is an open millenium prize problem that talks about the relation between the arithmetic behaviour of elliptic curves and the analytic behaviour of an L-function, or that elliptic curves over finite fields is so useful for cryptography that there was a memo that recommended elliptic curves for federal government use in 1999, I don't have to because you're a math student who will attend this talk without needing to be wowed with cool applications, right?

## Outline

Introduction (25 mins)

Why rational points on curves?

The Projective Plane

Introduction to Elliptic Curves

Why Elliptic Curves?

Weirstrauss Normal Form

The Group Law

Points of Finite Order (10 mins)

Nagell Lutz

The Torsion subgroup of $E(\mathbb{Q})$ and Mazur's Theorem

Mordell's Theorem (15 mins)

Neron Tate Height

Mordell's Theorem

Rank of an elliptic curve

Integer Points on Cubic Curves (10 mins)

Siegel's Theorem

Taxicab Numbers and Sums of two cubes