Explicit non-Gorenstein R=T via rank bounds I: Deformation theory

TL;DR

This paper establishes a Galois-theoretic criterion ensuring the minimality of deformation rings at non-optimal levels, leading to an $R=\mathbb{T}$ theorem in the context of reducible residual Galois representations.

Contribution

It introduces a new criterion for the deformation ring to be minimal at certain non-Gorenstein levels, advancing the understanding of deformation theory and modular forms.

Findings

Proves a criterion for the deformation ring to be as small as possible.

Establishes an $R=\mathbb{T}$ theorem under this criterion.

Analyzes non-optimal levels with product of two primes.

Abstract

Ribet has proven remarkable results about non-optimal levels of residually reducible Galois representations. We focus on a non-optimal level $N$ that is the product of two distinct primes and where the Galois deformation ring is not expected to be Gorenstein. We prove a Galois-theoretic criterion for the deformation ring to be as small as possible -- that is, for there to be a unique newform of level $N$ with reducible residual representation. When this criterion is satisfied, we deduce an $R=\mathbb{T}$ theorem.