On the pro-modularity in the residually reducible case for some totally real fields

TL;DR

This paper investigates the connection between deformation rings and Hecke algebras in the residually reducible case for totally real fields, proving a pro-modularity result and a conditional big R=T theorem.

Contribution

It extends the pro-modularity and big R=T results to residually reducible cases over totally real fields, generalizing previous work.

Findings

Proves a pro-modularity result in the residually reducible case.

Establishes a conditional big R=T theorem over some totally real fields.

Generalizes Deo's big R=T result to new settings.

Abstract

In this article, we study the relation between the universal deformation rings and big Hecke algebras in the residually reducible case. Following the strategy of Skinner-Wiles and Pan's proof of the Fontaine-Mazur conjecture, we prove a pro-modularity result. Based on this result, we also give a conditional big $R=\mathbb{T}$ theorem over some totally real fields, which is a generalization of Deo's result.