Department Colloquium: Prof. Chandrashekhar Khare:  University of California Los Angeles

Title: Modularity of elliptic curves over number fields

Abstract: I will give an account of developments arising from Wiles’s  work on modularity of elliptic curves over the rational numbers  and Fermat’s Last Theorem.  I will focus on recently announced results of Ana Cariani and James Newton which prove modularity of all elliptic curves over Gaussian numbers. Their result uses as a key step a result of Patrick Allen, Jack Thorne and myself which proves the modularity of mod 3 representations arising from such elliptic curves. This provides a starting point for Cariani and Newton in the same way as a result of Langlands-Tunnel was a starting point for Wiles.

This lecture will be a guided tour of Wiles’s breakthrough in 1994 and the numerous developments since in this very active area of number theory.