# Potential Automorphy for Certain Galois Representations to GL(n)

 JOINT IAS/PU NUMBER THEORY SEMINAR Topic: Potential Automorphy for Certain Galois Representations to GL(n) Speaker: Thomas Barnet-Lamb Affiliation: Harvard University Date: Thursday, February 19 Time/Room: 4:30pm - 5:30pm/Fine Hall -- 214

I will describe recent generalizations of mine to a theorem of Harris, Shepherd-Barron, and Taylor, showing that have certain Galois representations become automorphic after one makes a suitably large totally-real extension to the base field. The main innovation is that the result applies to Galois representations to $\GL_n$, where the previous work dealt with representations to $\Sp_n$; I can also dispense with certain congruence conditions which existed in the earlier work, and work over a CM, rather than a totally-real, field. The main technique is the consideration of the cohomology the Dwork hypersurface, and in particular, of pieces of this cohomology other than the invariants under the natural group action.