Locally analytic vector bundles on the Fargues–Fontaine curve

Porat, Gal

doi:10.2140/ant.2024.18.899

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Locally analytic vector bundles on the Fargues–Fontaine curve

Gal Porat

Vol. 18 (2024), No. 5, 899–946

DOI: 10.2140/ant.2024.18.899

Abstract

We develop a version of Sen theory for equivariant vector bundles on the Fargues–Fontaine curve. We show that every equivariant vector bundle canonically descends to a locally analytic vector bundle. A comparison with the theory of $(φ, Γ)$ -modules in the cyclotomic case then recovers the Cherbonnier–Colmez decompletion theorem. Next, we focus on the subcategory of de Rham locally analytic vector bundles. Using the $p$ -adic monodromy theorem, we show that each locally analytic vector bundle $ℰ$ has a canonical differential equation for which the space of solutions has full rank. As a consequence, $ℰ$ and its sheaf of solutions $Sol (ℰ)$ are in a natural correspondence, which gives a geometric interpretation of a result of Berger on $(φ, Γ)$ -modules. In particular, if $V$ is a de Rham Galois representation, its associated filtered $(φ, N, G_{K})$ -module is realized as the space of global solutions to the differential equation. A key to our approach is a vanishing result for the higher locally analytic vectors of representations satisfying the Tate–Sen formalism, which is also of independent interest.

Keywords

$p$-adic Hodge theory, $p$-adic Galois representations

Mathematical Subject Classification

Primary: 11F80

Secondary: 11S25

Milestones

Received: 17 May 2022

Revised: 12 April 2023

Accepted: 3 July 2023

Published: 16 April 2024

Authors

Gal Porat
Department of Mathematics University of Chicago Eckhart Hall, 5734 S University Ave Chicago, IL 60637 United States

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Vol. 18, No. 5, 2024