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

Abstract

Let 1 G be a profinite group which is topologically finitely generated 2 , p a prime number and d ≥ 1 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 rigid analytic families of Galois representations (e.g. eigenvarieties) and generic fibers of pseudodeformation spaces (especially in the "residually reducible" case, including when p ≤ d).σ∈S n+1 ε(σ)T σ (g 1 , g 2 , . . . , g n+1 ) = 0, 1 The author is supported by the C.N.R.S., as well as by the A.N.R. project ANR-10-BLAN 0114. 2 Actually, we only assume that for any normal open subgroup H ⊂ G, there are only finitely many continuous group homomorphisms H −→ Z/pZ.3 This expression has the following important interpretation, due to Kostant. Assume that T : GL m (A) → A is the trace map, if g 1 , . . . , g n ∈ GL m (A) and if σ ∈ S n , then T σ (g 1 , . . . , g n ) coincides with the trace of the element (g 1 , . . . , g n )σ acting on V ⊗An , where V := A m . 1 2 GAËTAN CHENEVIERwhere ε(σ) denotes the signature of the permutation σ. We say that T is a d-dimensionalThe main interest of pseudocharacters lies in the close relations they share with traces of representations : by an old result of Frobenius [Fr, p. 50], the trace of a representation G −→ GL d (A) is a d-dimensional pseudocharacter 4 , and it is known that the converse holds when A is an algebraically closed field with d! ∈ A * (Procesi [P3], Taylor [T] for Q-algebras, [Rou] in general 5 ) as well as in various other situations (see below). In particular, we obtain this way an interesting parametrization of the isomorphism classes of semisimple representations of G over such algebraically closed fields. As the covariant functor from the category of Z[1/d!]−commutative algebras with unit to the category Ens of sets, which associates to A the set of d-dimensional pseudocharacters G → A, is obviously representable 6 , it turned out to be an interesting substitute for the quotient functor Hom(G, GL d (−))/PGL d (−) of isomorphism classes of d-dimensional representations of G. Indeed, since they have been introduced in number theory by Wiles [W] (when d = 2), and by Taylor [T] under the form above (sometimes under the name of pseudorepresentations), they have proved to be a successful tool, first to actually construct some (Galois) representations, and then to study Galois representations and Hecke-algebras.Over Q-algebras, most of the basic properties of pseudocharacters follow ac...

Irreducibility of versal deformation rings in the (p,p)-case for 2-dimensional representations

“…In Corollary 4.3, we prove analogs of Theorems 1.9 and 1.12 for pseudo-representations, in the sense of Chenevier [9].…”

“…For the following result, we assume that the reader is familiar with the theory of determinants as introduced in [9]. Following [33] we shall call them pseudorepresentations.…”

Pseudodeformations

“…It is defined as the sum, over all simple closed paths γ ∈ SC P(n − 1) and all simple cycles If an element γ of [γ ] may be obtained from γ by inserting a new vertices between γ (0) and γ (1) (explicitly: γ (0) = γ (0), γ (l) = γ (l − 1) for 2 ≥ l ≥ n, and γ (1) is any element of I different from γ (0) and γ (1)), then this element γ is clearly unique, and we define h n,[γ ],γ as the tensor product of the Yoneda product map Ext 1 (ρ γ (1),γ (0) ) ⊗ Ext 1 (ρ γ (2) ,ρ γ (1) ) → Ext 2 (ρ γ (1) ,ρ γ (0) ) and the identity maps between the other Ext 1 . Otherwise, we set h n,[γ ],γ = 0.…”

“…. , r. (2) Then condition (1) is not satisfied if r > 1, and Mazur's deformation functor Dρ is not prorepresentable in this case. There are two ways to circumvent this problem, both introduced in [8]: the first way is to try to replaceρ by a representationρ whose semi-simplification is isomorphic toρ but that satisfies (1), and to study the deformations ofρ .…”

“…Letρ be as in (2). Moreover we assume throughout the paper that theρ i are pair-wise non isomorphic, in order to be able to apply the results on residually multiplicity-free pseudocharacters of [1].…”