Deformations of pseudocharacters and Mazur's finiteness condition
Vytautas Paškūnas, Julian Quast
TL;DR
This work develops a deformation theory for $G$-pseudocharacters of profinite groups $\Gamma$, proving that the universal deformation ring $R^{\mathrm{ps},\Gamma}_{\overline{\Theta}}$ is noetherian under Mazur's finiteness condition $\Phi_p$ by reducing to the finitely generated case and transferring results through a quotient $Q$. The key technical device is the isomorphism $R^{\mathrm{ps},\Gamma}_{\overline{\Theta}}=R^{\mathrm{ps}, Q}_{\overline{\Theta}}$, enabling finite-type and condensed-representation moduli to be controlled via geometric invariant theory and the moduli spaces $X^{\mathrm{gen},\Gamma}_{\overline{\Theta}}$, which become representable by finite-type algebras. The authors also construct a rigid analytic moduli space $X_G$ representing continuous $G$-pseudocharacters, with $\overline{L}$-points corresponding to $G^0(\overline{L})$-conjugacy classes of continuous $G$-semisimple representations, thus linking algebraic and analytic deformation theories. These results extend prior GL$_n$ determinant-law methods to general reductive groups and provide a framework for studying global Galois representations via moduli of pseudocharacters and condensed representations in both algebraic and rigid-analytic settings.
Abstract
We show that deformation rings $R^{\mathrm{ps}}$ of $G$-pseudocharacters of a profinite group $Γ$ are noetherian, when $Γ$ satisfies Mazur's finiteness condition. The proof proceeds by reduction to the case when $Γ$ 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 $Γ$, 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 $Γ$ 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.