### MA 353: Elliptic Curves

#### Pre-requisites :

1. Algebra I and II
2. a working knowledge of basic algebraic number theory

Elliptic curves are smooth projective curves of genus 1 with a marked point. Over a field of characteristic zero they are given by an equation of the form $y^2 = x^3+ax+b$. They are at the boundary of our (conjectural) understanding of rational points on varieties and are subject of many famous conjectures as well as celebrated results. They play an important role in number theory.

The course will begin with an introduction to algebraic curves. We will then study elliptic curves over complex number, over finite fields, over local fields of characteristic zero and finally over number fields. Our goal will be to prove the Mordell-Weil theorem.

#### Suggested books and references:

1. Joseph Silverman, The arithmetic of elliptic curves, Springer GTM 106, 2009.
2. Joseph Silverman and John Tate, Rational points on elliptic curves, Springer UTM, 1992.
3. J.W.S. Cassels, Lectures on elliptic curves, Cambridge University Press, 2012.

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Contact: +91 (80) 2293 2711, +91 (80) 2293 2265 ;     E-mail: chair.math[at]iisc[dot]ac[dot]in
Last updated: 20 Mar 2023