Analytic de Rham stacks of Fargues-Fontaine curves
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915829502705664 |
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| author | Anschütz, Johannes Bosco, Guido Bras, Arthur-César Le Camargo, Juan Esteban Rodríguez Scholze, Peter |
| author_facet | Anschütz, Johannes Bosco, Guido Bras, Arthur-César Le Camargo, Juan Esteban Rodríguez Scholze, Peter |
| contents | We define and initiate the study of analytic de Rham stacks of relative Fargues-Fontaine curves. To this end, we develop a theory of analytic de Rham stacks with sufficiently strong descent and approximation properties. Specializing to the de Rham stack of the Fargues-Fontaine curve attached to $\mathbb{C}_p$, we apply the general theory to obtain a new geometric proof of the $p$-adic monodromy theorem, avoiding any reliance on earlier results on $p$-adic differential equations. Building on the foundations established here, we plan in a sequel to investigate the cohomology of de Rham stacks of relative Fargues-Fontaine curves in geometric situations and, in particular, provide a stack-theoretic definition of Hyodo-Kato cohomology. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_15196 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Analytic de Rham stacks of Fargues-Fontaine curves Anschütz, Johannes Bosco, Guido Bras, Arthur-César Le Camargo, Juan Esteban Rodríguez Scholze, Peter Algebraic Geometry Number Theory 14F40, 14F30, 14G22, 14G45, 14A20, 11G25 We define and initiate the study of analytic de Rham stacks of relative Fargues-Fontaine curves. To this end, we develop a theory of analytic de Rham stacks with sufficiently strong descent and approximation properties. Specializing to the de Rham stack of the Fargues-Fontaine curve attached to $\mathbb{C}_p$, we apply the general theory to obtain a new geometric proof of the $p$-adic monodromy theorem, avoiding any reliance on earlier results on $p$-adic differential equations. Building on the foundations established here, we plan in a sequel to investigate the cohomology of de Rham stacks of relative Fargues-Fontaine curves in geometric situations and, in particular, provide a stack-theoretic definition of Hyodo-Kato cohomology. |
| title | Analytic de Rham stacks of Fargues-Fontaine curves |
| topic | Algebraic Geometry Number Theory 14F40, 14F30, 14G22, 14G45, 14A20, 11G25 |
| url | https://arxiv.org/abs/2510.15196 |