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

Gaëtan Chenevier (1)

1 CMLS - Centre de Mathématiques Laurent Schwartz

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.

Domaines: Théorie des nombres [math.NT]; Théorie des représentations [math.RT]

Gaëtan Chenevier. The p-adic analytic space of pseudocharacters of a profinite group and pseudorepresentations over arbitrary rings. Automorphic forms and Galois representations, 1, London Math. Soc., pp.221-285, 2014, Proceedings of the LMS Durham Symposium 2011. ⟨hal-00824274⟩