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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].

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