Potential automorphy for certain Galois representations to GL_2n

TL;DR

This paper proves that certain Galois representations to GL_{2n} become automorphic after a large totally-real extension, extending previous results from GSp_n to a broader class using cohomology of Dwork hypersurfaces.

Contribution

It extends potential automorphy results from GSp_n to GL_{2n} representations, introducing new cohomological techniques involving Dwork hypersurfaces.

Findings

Galois representations to GL_{2n} become automorphic after extension.

Utilizes cohomology of Dwork hypersurfaces for the proof.

Generalizes previous automorphy results to a larger class of representations.

Abstract

Building upon work of Clozel, Harris, Shepherd-Barron, and Taylor, this paper shows that certain Galois representations become automorphic after one makes a suitably large totally-real extension to the base field. The main innovation here is that the result applies to Galois representations to GL_{2n}, where previous work dealt with representations to GSp_n. 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.