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In $p$-adic Hodge theory and the $p$-adic Langlands program, Banach spaces with $\mathbb{Q}_p$-coefficients and $p$-adic Lie group actions are central. Studying the subrepresentation of $\Gamma$-locally analytic vectors, $W^{\mathrm{la}}$, is useful because $W^{\mathrm{la}}$ can be analyzed via the Lie algebra $\mathrm{Lie}(\Gamma)$, which simplifies the action of $\Gamma$. Additionally, $W^{\mathrm{la}}$ 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 $\mathbb{Z}_p$-Tate algebras. This generalization encompasses the classical definition and also specializes to super-H\"older vectors in characteristic $p$. Using binomial expansions instead of Taylor series, this new definition bridges locally analytic vectors in characteristic $0$ and $p$. Our main theorem shows that under certain conditions, the map $W \mapsto W^{\mathrm{la}}$ acts as a descent, and the derived locally analytic vectors $\mathrm{R}_{\mathrm{la}}^i(W)$ vanish for $i \geq 1$. This result extends Theorem C of \cite{Po24}, providing new tools for propagat

Author

Porat, Gal

Published

2024

Language

EN