The categorical local Langlands conjecture
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2026
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866910276959338496 |
|---|---|
| author | Hansen, David Mann, Lucas |
| author_facet | Hansen, David Mann, Lucas |
| contents | We formulate a program to prove the categorical local Langlands conjecture (CLLC) of Fargues-Scholze, for all quasisplit $p$-adic groups where the Fargues-Scholze $L$-parameters agree with the semisimplification of a known "automorphic" local Langlands parametrization. A key working hypothesis - which we expect to prove elsewhere jointly with Hamann - is the compatibility of the enhanced Whittaker coefficient functor $c_ψ$ with Eisenstein series. For $\mathrm{GL}_n$, we show that this hypothesis alone implies the full CLLC. For more general groups $G$, we prove an induction principle which reduces CLLC for $G$ to CLLC for all proper Levi subgroups together with a very small amount of information about $G$. This principle applies unconditionally to many classical groups with current technology.
Along the way, we establish many foundational results. In particular:
- We prove a very strong finiteness theorem for spectral constant term functors.
- We prove a spectral analogue of Bernstein's finite global dimension theorem for $p$-adic Hecke algebras.
- We introduce and develop the theory of admissible ind-coherent sheaves and admissible duality on derived stacks.
- We prove a duality theorem for the spectral action.
Using all of these results, we unconditionally define a new and explicit functor $t_ψ$ from the spectral side to the automorphic side, which is defined on enough ind-coherent sheaves to control the entire conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_00983 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The categorical local Langlands conjecture Hansen, David Mann, Lucas Number Theory Algebraic Geometry Representation Theory We formulate a program to prove the categorical local Langlands conjecture (CLLC) of Fargues-Scholze, for all quasisplit $p$-adic groups where the Fargues-Scholze $L$-parameters agree with the semisimplification of a known "automorphic" local Langlands parametrization. A key working hypothesis - which we expect to prove elsewhere jointly with Hamann - is the compatibility of the enhanced Whittaker coefficient functor $c_ψ$ with Eisenstein series. For $\mathrm{GL}_n$, we show that this hypothesis alone implies the full CLLC. For more general groups $G$, we prove an induction principle which reduces CLLC for $G$ to CLLC for all proper Levi subgroups together with a very small amount of information about $G$. This principle applies unconditionally to many classical groups with current technology. Along the way, we establish many foundational results. In particular: - We prove a very strong finiteness theorem for spectral constant term functors. - We prove a spectral analogue of Bernstein's finite global dimension theorem for $p$-adic Hecke algebras. - We introduce and develop the theory of admissible ind-coherent sheaves and admissible duality on derived stacks. - We prove a duality theorem for the spectral action. Using all of these results, we unconditionally define a new and explicit functor $t_ψ$ from the spectral side to the automorphic side, which is defined on enough ind-coherent sheaves to control the entire conjecture. |
| title | The categorical local Langlands conjecture |
| topic | Number Theory Algebraic Geometry Representation Theory |
| url | https://arxiv.org/abs/2606.00983 |