# zbMATH — the first resource for mathematics

The $$p$$-adic analytic space of pseudocharacters of a profinite group and pseudorepresentations over arbitrary rings. (English) Zbl 1350.11063
Diamond, Fred (ed.) et al., Automorphic forms and Galois representations. Proceedings of the 94th London Mathematical Society (LMS) – EPSRC Durham symposium, Durham, UK, July 18–28, 2011. Volume 1. Cambridge: Cambridge University Press (ISBN 978-1-107-69192-6/pbk; 978-1-107-44633-5/ebook). London Mathematical Society Lecture Note Series 414, 221-285 (2014).
Author’s abstract: Let $$G$$ be a profinite group which is topologically finitely generated, $$p$$ a prime number and $$d\geq 1$$ an integer. We 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 $$d$$-dimensional pseudocharacters $$G\to{\mathcal O}(Y)$$, 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 chapter 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 rigid analytic families of Galois representations (e.g., eigenvarieties) and generic fibers of pseudodeformation spaces (especially in the “residually reducible” case, including when $$p\leq d$$).
For the entire collection see [Zbl 1310.11002].

##### MSC:
 11F80 Galois representations 22E50 Representations of Lie and linear algebraic groups over local fields 14G22 Rigid analytic geometry 13A99 General commutative ring theory 11F85 $$p$$-adic theory, local fields