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Automorphy lifting for residually reducible l-adic Galois ...

Automorphy lifting for residually reducible l-adic Galois ...

Automorphy lifting for residually reducible l-adic Galois

Automorphy lifting for residually reducible l-adic Galois representations Jack A. Thorne ∗ February 4, 2013 Abstract We prove automorphy lifting theorems for residually reducible Galois representations in the setting of unitary groups. Our methods are inspired by those of Skinner-Wiles in the setting of GL2. Contents 1 Introduction 2 2 Automorphic forms on GLn(AF ) 4 3 Deformation theory 5 3.1 The group Gn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.2 Deformation of Galois representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.3 Local deformation problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.3.1 Unrestricted deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.3.2 Ordinary deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.3.3 Level raising deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.3.4 Steinberg deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.3.5 Taylor-Wiles deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.4 Pseudodeformations of Galois representations . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.5 Reducible deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.6 Twisting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.7 Localizing at a dimension one prime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.8 Connectedness dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4 Automorphic forms 19 4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.2 Galois representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.3 Deformation rings and Hecke algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.4 Auxiliary levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.5 Soluble base change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.6 A patching argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 5 Taylor-Wiles systems 37 5.1 Galois theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 6 The main argument 45 ∗ This research was partially conducted during the period the author served as a Clay Research Fellow. 1

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