Non-minimal modularity lifting in weight one

TL;DR

This paper extends the integral R = T theorem to non-minimal cases for certain p-adic Galois representations and demonstrates the existence of weight one Katz modular forms mod p that do not lift to characteristic zero.

Contribution

It generalizes the R = T theorem to non-minimal cases and constructs explicit examples of weight one modular forms mod p that lack characteristic zero lifts.

Findings

Proved an integral R = T theorem for non-minimal cases.

Established existence of weight one Katz modular forms mod p without lifts.

Extended previous results to broader class of Galois representations.

Abstract

We prove an integral R = T theorem for odd two dimensional p-adic representations of the absolute Galois group which are unramified at p, extending results of [CG] to the non-minimal case. We prove, for any p, the existence of Katz modular forms modulo p of weight one which do not lift to characteristic zero.