Number Theory Seminar
Wiles's proof of the modularity of semistable elliptic curves over the rationals uses the Langlands-Tunnell theorem as a starting point. In order to feed this into a modularity lifting theorem, one needs to use congruences between modular forms of weight one and modular forms of higher weight. Similar congruences are not known over imaginary quadratic fields and Wiles's strategy runs into problems right from the start. We circumvent this congruence problem and show that mod 3 Galois representations over imaginary quadratic fields arise from automorphic forms that are the analog of higher weight modular forms. Our argument relies on a 2-adic automorphy lifting theorem over CM fields together with a "2-3 switch." As an application, we deduce that a positive proportion of elliptic curves over imaginary quadratic fields are modular. This is joint work with Chandrashekhar Khare and Jack Thorne.