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


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?


  • 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