Solid locally analytic representations of $p$-adic Lie groups

Jacinto, Joaquín Rodrigues; Camargo, Juan Esteban Rodríguez

doi:10.48550/arxiv.2110.11916

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“…The functor W → W G-an is left exact. Following §2.2 of [Pa21] and [RJRC21], define right derived functors for i ≥ 0:…”

Section: Rings Of Analytic Functionsmentioning

confidence: 99%

Locally analytic vector bundles on the Fargues-Fontaine curve

Porat¹

2022

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In this article, 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) satisfy a Riemann-Hilbert type relation, 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. Locally analytic vectorbundles 14 5. Acyclicity of locally analytic vectors for semilinear representations 18 5.1. Statement of the results 18 5.2. Vanishing of H-cohomology 21 5.3. Descent of semilinear representations 23 5.4. Descent of C an (G 0 , M) 24 5.5. Computation of higher locally analytic vectors I 26 5.6. Computation of higher locally analytic vectors II 27 5.7. Computation of higher locally analytic vectors III 29 6. Descent to locally analytic vectors 31 6.1. Computations at the stalk 31 6.2. Descent to locally analytic vectors 32 7. The comparison with (ϕ, Γ)-modules 34 7.1. Galois representations and (ϕ, Γ)-modules 34 7.2. The comparison with locally analytic vector bundles 36 8. Locally analytic vector bundles and p-adic differential equations 37 8.1. Modifications of locally analytic vector bundles 37 8.2. de Rham and C p -admissible locally analytic vector bundles 38 8.3. The surfaces Y log,L and X log,L 38 8.4. Sheaves of smooth functions 40 8.5. The solution functor 41 References 44

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“…The functor W → W G-an is left exact. Following §2.2 of [Pa21] and [RJRC21], define right derived functors for i ≥ 0:…”

Section: Rings Of Analytic Functionsmentioning

confidence: 99%