Locally analytic vectors and decompletion in mixed characteristic

Abstract In p -adic Hodge theory and the p -adic Langlands program, Banach spaces with Qₚ -coefficients and p -adic Lie group actions are central. Studying the subrepresentation of G -locally analytic vectors, W^ {la}, is useful because W^ {la} can be studied via the Lie algebra Lie (G), which simplifies the action of G. Additionally, W^ {la} often behaves as a decompletion of W, making it closer to an algebraic or geometric object. This article introduces a notion of locally analytic vectors for W in a mixed characteristic setting, specifically for Zₚ -Tate algebras. This generalization encompasses the classical definition and also specializes to super-Hölder vectors in characteristic p. Using binomial expansions instead of Taylor series, this new definition bridges locally analytic vectors in characteristic 0 and characteristic p. Our main theorem shows that under certain conditions, the map W W^ {la} acts as a descent, and the derived locally analytic vectors R ₋₀ⁱ (W) vanish for i 1. This result extends Theorem C of Por24, providing new tools for propagating information about locally analytic vectors from characteristic 0 to characteristic p. We provide three applications: a new proof of Berger-Rozensztajn’s main result using characteristic 0 methods, the introduction of an integral multivariable ring {A} ₋ₓ^, {la} in the Lubin-Tate setting, and a novel interpretation of the classical Cohen ring {A} ₐₚ from the theory of (, ) -modules in terms of locally analytic vectors.