Potential automorphy and change of weight

Abstract

We prove an automorphy lifting theorem for l-adic representations where we impose a new condition at l, which we call "potentential diagonalizabil- ity." This result allows for "change of weight" and seems to be substantially more exible than previous theorems along the same lines. We derive sev- eral applications. For instance we show that any irreducible, totally odd, essentially self-dual, regular, weakly compatible system of l-adic represen- tations of the absolute Galois group of a totally real field is potentially automorphic and hence is pure and its L-function has meromorphic con- tinuation to the whole complex plane and satisfies the expected functional equation. © 2014 Department of Mathematics, Princeton University.