Title: Quantifying congruences between Eisenstein series and cusp forms

Date: 01/10/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Consider the following two problems in algebraic number theory:
1. For which prime numbers p can we easily show that the Fermat equation x^p + y^p =z^p has no non-trivial integer solutions?
2. Given an elliptic curve E over the rational numbers, what can be said about the group of rational points of finite order on E?
These seem like very different problems, but, surprisingly, they share a common theme: they are both related to the existence of congruences between two types of modular forms, Eisenstein series and cusp forms. We will explain these examples, and discuss a new technique for giving quantitative information about these congruences (for example, counting the number of cusp forms congruent to an Eisenstein series). We will explain how this can give finer arithmetic information than simply knowing the existence of a congruence. This is joint work with Carl Wang-Erickson.