Explicit non-Gorenstein R=T via rank bounds II: Computational aspects

TL;DR

This paper develops computational techniques to verify criteria for minimal Galois deformation rings in the context of residually reducible Galois representations, extending previous theoretical results.

Contribution

It adapts Sharifi's method to compute specific number fields and analyze prime splitting, enabling practical verification of Galois deformation ring properties.

Findings

Successfully computed number fields with twisted-Heisenberg Galois groups

Verified splitting behavior of primes in these extensions

Provided computational tools for Galois deformation ring analysis

Abstract

This is the second in a pair of papers about residually reducible Galois deformation rings with non-optimal level. In the first paper, we proved a Galois-theoretic criterion for the deformation ring to be as small as possible. This paper focuses on the computations needed to verify this criterion. We adapt a technique developed by Sharifi to compute number fields with twisted-Heisenberg Galois group and prescribed ramification, and compute the splitting behavior of primes in these extensions.