$G$-valued local deformation rings and global lifts

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 ${\mathbb{Z}}_l$ 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 regular, 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.