G-valued local deformation rings and global lifts

Bellovin, Rebecca; Gee, Toby

doi:10.2140/ant.2019.13.333

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1944-7833 (online)

ISSN 1937-0652 (print)

Author index

To appear

Other MSP journals

$G$-valued local deformation rings and global lifts

Rebecca Bellovin and Toby Gee

Vol. 13 (2019), No. 2, 333–378

DOI: 10.2140/ant.2019.13.333

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 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.

Keywords

Galois deformations

Mathematical Subject Classification 2010

Primary: 11F80

Secondary: 11F85

Milestones

Received: 6 October 2017

Revised: 8 November 2018

Accepted: 24 December 2018

Published: 2 March 2019

Authors

Rebecca Bellovin
Department of Mathematics Imperial College London London United Kingdom

Toby Gee
Department of Mathematics Imperial College London London United Kingdom

Vol. 13, No. 2, 2019