The p-adic analytic space of pseudocharacters of a profinite group and pseudorepresentations over arbitrary rings

TL;DR

This paper constructs a rigid analytic space parametrizing pseudocharacters of a profinite group over arbitrary rings, extending determinant theory and providing a framework for studying pseudodeformation rings in number theory.

Contribution

It introduces a new theory of determinants over arbitrary rings and constructs a moduli space of pseudocharacters as a quasi-Stein rigid analytic space.

Findings

The functor of continuous pseudocharacters is representable by a quasi-Stein space.

The paper extends determinant theory to arbitrary base rings.

Application to number theory via pseudodeformation rings, especially in residually reducible cases.

Abstract

Let G be a profinite group which is topologically finitely generated, p a prime number and d an integer. We show that the functor from rigid analytic spaces over Q_p to sets, which associates to a rigid space Y the set of continuous d-dimensional pseudocharacters G -> O(Y), is representable by a quasi-Stein rigid analytic space X, and we study its general properties. Our main tool is a theory of "determinants" extending the one of pseudocharacters but which works over an arbitrary base ring; an independent aim of this paper is to expose the main facts of this theory. The moduli space X is constructed as the generic fiber of the moduli formal scheme of continuous formal determinants on G of dimension d. As an application to number theory, this provides a framework to study the generic fibers of pseudodeformation rings (e.g. of Galois representations), especially in the "residually reducible" case, and including when p <= d.