G-valued local deformation rings and global lifts

TL;DR

This paper investigates G-valued local deformation rings for Galois representations, establishing their smoothness, dimension, and functorial properties, and applies these findings to global deformation rings and Serre's conjecture.

Contribution

It provides new results on the structure and properties of G-valued local deformation rings for arbitrary reductive groups, extending previous work and improving bounds on global deformation rings.

Findings

Local deformation rings are generically smooth and have computable dimensions.

Functorial operations induce maps between irreducible components of deformation rings.

Results lead to improved bounds on global deformation rings and advances in Serre's conjecture.

Abstract

We study G-valued Galois deformation rings with prescribed properties, where G is an arbitrary (not necessarily connected) reductive group over an extension of Z_l for some prime l. In particular, for the Galois groups of p-adic local fields (with p possibly equal to l) we prove that these rings are generically smooth, compute their dimensions, and show that functorial operations on Galois representations give rise to well-defined maps between the sets of irreducible components of the corresponding deformation rings. We use these local results to prove lower bounds on the dimension of global deformation rings with prescribed local properties. Applying our results to unitary groups, we improve results in the literature on the existence of lifts of mod l Galois representations, and on the weight part of Serre's conjecture.