Deformations of pseudocharacters and Mazur's finiteness condition

TL;DR

This paper proves that deformation rings of G-pseudocharacters are noetherian for groups satisfying Mazur's finiteness condition, extending previous results and constructing related moduli spaces with applications to Galois representations.

Contribution

It establishes the noetherian property of deformation rings for pseudocharacters under Mazur's finiteness condition and constructs associated moduli spaces as rigid analytic spaces.

Findings

Deformation rings $R^{ps}$ are noetherian for groups satisfying Mazur's finiteness condition.

The functor from rigid analytic spaces to G-pseudocharacters is representable by a quasi-Stein space.

Results are applicable to the study of global Galois representations.

Abstract

We show that deformation rings $R^{\mathrm{ps}}$ of $G$-pseudocharacters of a profinite group $\Gamma$ are noetherian, when $\Gamma$ satisfies Mazur's finiteness condition. The proof proceeds by reduction to the case when $\Gamma$ is finitely generated, where the result was previously established by the second author. This enables us to extend our work on moduli spaces of $R^{\mathrm{ps}}$-condensed representations of a finitely generated profinite group $\Gamma$, to the groups satisfying Mazur's finiteness condition. We also show that the functor from rigid analytic spaces over $\mathbb{Q}_p$ to sets, which associates to a rigid space $Y$ the set of continuous $\mathcal{O}(Y)$-valued $G$-pseudocharacters of $\Gamma$ is representable by a quasi-Stein rigid analytic space, and we study its general properties. We expect these results to be useful, when studying global Galois representations.