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