Manjul Bhargava: Rational points on elliptic and hyperelliptic curves

Abstract:

Understanding whether (and how often) a mathematical expression takes a square value is a problem that has fascinated mathematicians since antiquity. In this talk I will give a survey of this problem, and will then concentrate on the case where the mathematical expression in question is simply a polynomial in one variable. The main result in this case—proved just recently—is that if the degree of the polynomial is at least 6, then it is not very likely to take even a single square value!

I’ll explain how this was proved, and how the question relates to the very active and exciting area of mathematics today known as “arithmetic geometry”.