Modularity lifting theorems for Galois representations of unitary type

TL;DR

This paper proves modularity lifting theorems for l-adic Galois representations of any dimension with unitary type and Fontaine-Laffaille conditions, extending prior results using advanced Taylor-Wiles methods and base change techniques.

Contribution

It generalizes modularity lifting theorems to higher dimensions and broader conditions, building on and extending previous foundational work.

Findings

Proved modularity lifting theorems for Galois representations of any dimension.

Extended results of Clozel, Harris, Taylor, and others to new settings.

Applied advanced Taylor-Wiles methods to unitary groups and base change techniques.

Abstract

We prove modularity lifting theorems for l-adic Galois representations of any dimension satisfying a unitary type condition and a Fontaine-Laffaille type condition at l. This extends the results of Clozel, Harris and Taylor, and the subsequent work by Taylor. The proof uses the Taylor-Wiles method, as improved by Diamond, Fujiwara, Kisin and Taylor, applied to Hecke algebras of unitary groups, and results of Labesse on stable base change and descent from unitary groups to GL_n.