Arithmetic families of (φ, Γ)-modules and locally analytic representations of GL_2(Q_p)

Abstract

Let A be a Q p -affinoid algebra, in the sense of Tate. We develop a theory of locally convex A-modules parallel to the treatment in the case of a field by Schneider and Teitelbaum. We prove that there is an integration map linking a category of locally analytic representations in A-modules and separately continuous relative distribution modules. There is a suitable theory of locally analytic cohomology for these objects and a version of Shapiro's Lemma. In the case of a field this has been substantially developed by Kohlhaase. As an application we propose a p-adic Langlands correspondence in families: For a regular trianguline (ϕ, Γ)-module of dimension 2 over the relative Robba ring RA we construct a locally analytic GL2(Qp)-representation in A-modules.

Representations of $p$-adic groups over coefficient rings

“…For instance, if two integral irreducible Q ac ℓ -representations of GL(n, F ) become isomorphic modulo ℓ, then Z acts on these representations by scalars congruent modulo ℓ. 59 It is a slight modification of the Breuil and Schneider correspondence transferred to ℓ-adic representations; the socle of V is the maximal semi-simple subrepresentation of V 60 ρ identifies with a representation Gal(F ac /F ) → GL(n, E) 61 For an example of a local p-adic Langlands correspondence in families for GL (2,Qp), see Ildar Gaisin and Joaquin Rodrigues Jacinto [74] For any R and G quasi-split, Dat, Helm, Kurinczuk and Moss [51] studied the scheme of Langlands parameters of G with coefficient the smallest possible ring R = Z[1/p] for a local Langlands correspondence in families. In particular, this allows to study chain of congruences of Langlands parameters modulo several different primes.…”