On deformation rings of residually reducible Galois representations and R=T theorems

TL;DR

This paper investigates the structure of crystalline deformation rings of certain Galois representations, establishing conditions under which R equals T and applying these results to modularity problems over quadratic fields and the rationals.

Contribution

It introduces new criteria for R=T theorems for residually reducible Galois representations, especially in self-dual cases, and applies these to modularity lifting.

Findings

R/I is cyclic and often finite under certain Selmer group conditions

I is principal for essentially self-dual representations

New algebraic criterion links Hecke data to R=T theorems

Abstract

We study the crystalline universal deformation ring R (and its ideal of reducibility I) of a mod p Galois representation rho_0 of dimension n whose semisimplification is the direct sum of two absolutely irreducible mutually non-isomorphic constituents rho_1 and rho_2. Under some assumptions on Selmer groups associated with rho_1 and rho_2 we show that R/I is cyclic and often finite. Using ideas and results of (but somewhat different assumptions from) Bellaiche and Chenevier we prove that I is principal for essentially self-dual representations and deduce statements about the structure of R. Using a new commutative algebra criterion we show that given enough information on the Hecke side one gets an R=T-theorem. We then apply the technique to modularity problems for 2-dimensional representations over an imaginary quadratic field and a 4-dimensional representation over the rationals.