An infinite dimensional Hodge-Tate theory

Abstract

L etG be the absolute Galois group of a p-adic field K and R a Banach algebra over K. Given a continuous homomorphism ρ : G→R ∗ (R ∗ = units of R) we construct a canonical operator ϕ ∈R ⊗K C which determines the C-extension of ρ upto local isomorphism ( ⊗ = complete tensor product, and C = completion of an algebraic closure of K). If R is a matrix ring over a suitable power series ring one obtains information about the variation of the Hodge-Tate structure in families of finite-dimensional representations of G.