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Locally analytic vector bundles on the Fargues–Fontaine curve

Gal Porat

Vol. 18 (2024), No. 5, 899–946

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 has a canonical differential equation for which the space of solutions has full rank. As a consequence, and its sheaf of solutions Sol () 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,GK)-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.

$p$-adic Hodge theory, $p$-adic Galois representations
Mathematical Subject Classification
Primary: 11F80
Secondary: 11S25
Received: 17 May 2022
Revised: 12 April 2023
Accepted: 3 July 2023
Published: 16 April 2024
Gal Porat
Department of Mathematics
University of Chicago
Eckhart Hall, 5734 S University Ave
Chicago, IL 60637
United States

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