| Building upon work of Clozel, Harris, Shepherd-Barron, and Taylor, this thesis 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 GLn, where previous work dealt with representations to GSpn. We can prove results unconditionally for n = 3, n = 5 and for all even n; our remaining results are conditional on certain conjectures of Michael Harris and coworkers which are expected to be theorems by 2010. The main technique for the case of even n is the consideration of the cohomology of Dwork families of hypersurfaces, and in particular, of pieces of their cohomology other than the invariants under the natural group action. For the odd n cases, we must additionally use results which give the automorphy or potential automorphy of symmetric powers of elliptic curves, most notably those of Gelbert-Jacquet, Kim-Shahidi and Harris.;As an application of these results, we can show the Galois representations coming from the cohomology of the one particular Dwork family are potentially automorphic. (In particular, we work with the family of hypersurfaces in P5 over Q with projective equation: &parl0;X51+X52 +X53+X54+X5 5&parr0;=5lX1X2&ldots;X 5 where gamma is the parameter.) Hence we show that the zeta function of the family has meromorphic continuation throughout the complex plane. |
