Potential automorphy for certain Galois representations to GL(n)

Potential automorphy for certain Galois representations to GL(n)

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Thomas Barnet-Lamb, Harvard University
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.