Automorphy of mod 3 representations over CM fields

Workshop on Motives, Galois Representations and Cohomology Around the Langlands Program
Topic:Automorphy of mod 3 representations over CM fields
Speaker:Chandrashekhar Khare
Affiliation:University of California, Los Angeles
Date:Tuesday, November 7
Time/Room:4:00pm - 5:00pm/S-101
Video Link:https://video.ias.edu/MotivesGaloisRepsandCohomology/2017/1107-ChandrashekharKhare

Abstract:  Wiles' work on modularity of elliptic curves over the rationals, used as a starting point that odd, irreducible represenations $G_Q \rightarrow GL_2 (F_3)$ arise from cohomological cusp forms (i.e. new forms of weight $K \geq 2$). In ongoing work with Patrick Allen and Jack Thorne we address the question of showing that representations $G_K \rightarrow GL_2 (F_3)$ arise from cohomological cusp forms on $GL_2(A_K)$ for $K$ a CM field like $Q(i)$.  I will describe some of the main ideas of this work, in which instead of invoking the Langlands-Tunnell theorem like Wiles (which does not seem directly useful in this setting), we isntead rely on (restricted) 2-adic automorphy lifting theorems (extending results in the 10-author paper Patrick Allen will talk about at the conference) in the residually dihedral case. A starting point is a Diophantine argument ("2-3" switch) that gives a criterion for mod 6 represenations to arise from elliptic curves over $K$.