Arithmetic Geometry and Number Theory RTG Seminar: Automorphy of mod 3 representations over CM fields

Seminar | February 5 | 3:10-5 p.m. | 748 Evans Hall

 Patrick Allen, UIUC

 Department of Mathematics

Wiles's proof of the modularity of semistable elliptic curves over the rationals uses, as a starting point, the Langlands-Tunnell theorem, which implies that the mod 3 Galois representation attached to an elliptic curve over the rationals arises from a modular form of weight one. In order to feed this into modularity lifting theorems, 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 representations over CM field arise from the "correct" 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.

Seminar Format: The seminar consists of two 50-minute talks, a pre-talk (3:10-4:00) and an advanced talk (4:10-5:00), with a 10-minute break (4:00-4:10) between them. The advanced talk is a regular formal presentation about recent research results to general audiences in arithmetic geometry and number theory; the pre-talk (3:10-4:00) is to introduce some prerequisites or background for the advanced talk to audiences consisting of graduate students.