On flat even deformation rings

TL;DR

This paper proves that certain global even deformation rings are finite and flat over Z_p in cases with a nontrivial dual Selmer group, extending known results from odd parity cases.

Contribution

It establishes flatness of global even deformation rings in new cases, especially when a dual Selmer group is nontrivial, using techniques from global class field theory.

Findings

Global even deformation rings are finite and flat over Z_p under certain conditions.

Explicit examples of even representations are computed where the results apply.

If Leopoldt's conjecture holds for a related number field, the deformation ring is flat at the minimal level.

Abstract

In the presence of a nontrivial dual Selmer group, certain global even deformation rings are shown to be finite and flat over $\mathbb{Z}_p$. Previously, flatness was only known in established cases of Langlands reciprocity in the odd parity. By techniques from global class field theory, explicit examples of even representations are computed to which the results apply. For even representations $\overline{\rho}$ in an explicit family, it is observed that if Leopoldt's conjecture is true for a certain number field attached to $\overline{\rho}$, then the global even deformation ring is flat at the minimal level.