On flat even deformation rings

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 $\overlineρ$ in an explicit family, it is observed that if Leopoldt's conjecture is true for a certain number field attached to $\overlineρ$, then the global even deformation ring is flat at the minimal level.