Geometric Eisenstein series I: finiteness theorems
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914949202182144 |
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| author | Hamann, Linus Hansen, David Scholze, Peter |
| author_facet | Hamann, Linus Hansen, David Scholze, Peter |
| contents | We develop the theory of geometric Eisenstein series and constant term functors for $\ell$-adic sheaves on stacks of bundles on the Fargues-Fontaine curve. In particular, we prove essentially optimal finiteness theorems for these functors, analogous to the usual finiteness properties of parabolic inductions and Jacquet modules. We also prove a geometric form of Bernstein's second adjointness theorem, generalizing the classical result and its recent extension to more general coefficient rings proved in [Dat-Helm-Kurinczuk-Moss]. As applications, we decompose the category of sheaves on $\mathrm{Bun}_G$ into cuspidal and Eisenstein parts, and show that the gluing functors between strata of $\mathrm{Bun}_G$ are continuous in a very strong sense. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_07363 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Geometric Eisenstein series I: finiteness theorems Hamann, Linus Hansen, David Scholze, Peter Number Theory Algebraic Geometry Representation Theory We develop the theory of geometric Eisenstein series and constant term functors for $\ell$-adic sheaves on stacks of bundles on the Fargues-Fontaine curve. In particular, we prove essentially optimal finiteness theorems for these functors, analogous to the usual finiteness properties of parabolic inductions and Jacquet modules. We also prove a geometric form of Bernstein's second adjointness theorem, generalizing the classical result and its recent extension to more general coefficient rings proved in [Dat-Helm-Kurinczuk-Moss]. As applications, we decompose the category of sheaves on $\mathrm{Bun}_G$ into cuspidal and Eisenstein parts, and show that the gluing functors between strata of $\mathrm{Bun}_G$ are continuous in a very strong sense. |
| title | Geometric Eisenstein series I: finiteness theorems |
| topic | Number Theory Algebraic Geometry Representation Theory |
| url | https://arxiv.org/abs/2409.07363 |