Automorphy lifting for residually reducible -adic Galois representations, II

Patrick B Allen,Jack A Thorne,James Newton

doi:10.1112/s0010437x20007484

Abstract

We revisit the paper [Automorphy lifting for residually reducible$l$-adic Galois representations, J. Amer. Math. Soc. 28 (2015), 785–870] by the third author. We prove new automorphy lifting theorems for residually reducible Galois representations of unitary type in which the residual representation is permitted to have an arbitrary number of irreducible constituents.

Highlights

In this paper, we prove new automorphy lifting theorems for Galois representations of unitary type
The first theorems of this type were proved in the paper [Tho15], under the assumption that ρhas only two irreducible constituents
Our results are applied to the problem of symmetric power functoriality in [NT19], where they are combined with level-raising theorems to establish automorphy of symmetric powers for certain level 1 Hecke eigenforms congruent to a theta series

Summary

Introduction

We prove new automorphy lifting theorems for Galois representations of unitary type. The first main innovation in this paper that allows us to bypass this is the observation that by fully exploiting the ‘connectedness dimension’ argument to prove that R = T (which goes back to [SW99] and appears in this paper in the proof of Theorem 5.1), one only needs to control the size of the reducible locus in quotients of the universal deformation ring that are known a priori to be finite over the Iwasawa algebra Λ.

Notation

Determinants

Deformations

Automorphic forms and Hecke algebras on unitary groups

Propagation of potential pro-automorphy

The end