Lecture: Rational points on elliptic and hyperelliptic curves

Manjul Bhargava

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".

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