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.