Joint IAS/Princeton University 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, implying that the mod 3 Galois representation attached to the elliptic curve arises from a modular form of weight one. 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 CM 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 CM field arise from cohomological automorphic forms. Our argument relies on a 2-adic automorphy lifting theorem over CM fields together with a "2-3 switch" that gives a criterion for when a given mod 6 representation arises from an elliptic curve. This is joint work in progress with Chandrashekhar Khare and Jack Thorne.