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| Format: | Preprint |
| Published: |
2022
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| Online Access: | https://arxiv.org/abs/2204.11354 |
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| _version_ | 1866909231973662720 |
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| author | Porat, Gal |
| author_facet | Porat, Gal |
| contents | 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 $\mathcal{E}$ has a canonical differential equation for which the space of solutions has full rank. As a consequence, $\mathcal{E}$ and its sheaf of solutions $\mathrm{Sol}(\mathcal{E})$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_11354 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Locally analytic vector bundles on the Fargues-Fontaine curve Porat, Gal Number Theory 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 $\mathcal{E}$ has a canonical differential equation for which the space of solutions has full rank. As a consequence, $\mathcal{E}$ and its sheaf of solutions $\mathrm{Sol}(\mathcal{E})$ 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. |
| title | Locally analytic vector bundles on the Fargues-Fontaine curve |
| topic | Number Theory |
| url | https://arxiv.org/abs/2204.11354 |