Geometrization of the local Langlands correspondence
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866929606542491648 |
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| author | Fargues, Laurent Scholze, Peter |
| author_facet | Fargues, Laurent Scholze, Peter |
| contents | Following the idea of [Far16], we develop the foundations of the geometric Langlands program on the Fargues--Fontaine curve. In particular, we define a category of $\ell$-adic sheaves on the stack $\mathrm{Bun}_G$ of $G$-bundles on the Fargues--Fontaine curve, prove a geometric Satake equivalence over the Fargues--Fontaine curve, and study the stack of $L$-parameters. As applications, we prove finiteness results for the cohomology of local Shimura varieties and general moduli spaces of local shtukas, and define $L$-parameters associated with irreducible smooth representations of $G(E)$, a map from the spectral Bernstein center to the Bernstein center, and the spectral action of the category of perfect complexes on the stack of $L$-parameters on the category of $\ell$-adic sheaves on $\mathrm{Bun}_G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2102_13459 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Geometrization of the local Langlands correspondence Fargues, Laurent Scholze, Peter Representation Theory Algebraic Geometry Number Theory 11S37, 14D24, 22E57, 11F77, 11F85, 14G45 Following the idea of [Far16], we develop the foundations of the geometric Langlands program on the Fargues--Fontaine curve. In particular, we define a category of $\ell$-adic sheaves on the stack $\mathrm{Bun}_G$ of $G$-bundles on the Fargues--Fontaine curve, prove a geometric Satake equivalence over the Fargues--Fontaine curve, and study the stack of $L$-parameters. As applications, we prove finiteness results for the cohomology of local Shimura varieties and general moduli spaces of local shtukas, and define $L$-parameters associated with irreducible smooth representations of $G(E)$, a map from the spectral Bernstein center to the Bernstein center, and the spectral action of the category of perfect complexes on the stack of $L$-parameters on the category of $\ell$-adic sheaves on $\mathrm{Bun}_G$. |
| title | Geometrization of the local Langlands correspondence |
| topic | Representation Theory Algebraic Geometry Number Theory 11S37, 14D24, 22E57, 11F77, 11F85, 14G45 |
| url | https://arxiv.org/abs/2102.13459 |