Department of Mathematics

Colloquium

  •  Quantifying congruences between Eisenstein series and cusp forms
  •  01/10/2018
  •  4:10 PM - 5:00 PM
  •  C304 Wells Hall
  •  Preston Wake, UCLA

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.

 

Contact

Department of Mathematics
Michigan State University
619 Red Cedar Road
C212 Wells Hall
East Lansing, MI 48824

Phone: (517) 353-0844
Fax: (517) 432-1562

College of Natural Science