Geometrization of the local Langlands correspondence

Abstract

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 ${\mathrm{Bun}}_G$.

arXiv comment356 pages, v4: accepted version. v3: improved discussion of spectral action with integral coefficients (now also applying in the usual Betti setting), including new results (Theorem VIII.0.3) on preservation of good filtrations ("Donkin subgroups"). Comments welcome!

References in corpus (8)

• Higher Topos Theory
• Cycles on Shimura varieties via geometric Satake
• G-valued local deformation rings and global lifts
• The Bernstein center of the category of smooth $W(k)[G{L}_n(F)]$-modules
• Chtoucas restreints pour les groupes réductifs et paramétrisation de Langlands locale
• A toy model for the Drinfeld-Lafforgue shtuka construction
• Smoothness of cohomology sheaves of stacks of shtukas
• A $(-q)$-analogue of weight multiplicities

Cited by in corpus (12)

• Coherent Springer theory and the categorical Deligne-Langlands correspondence
• Categorical traces and a relative Lefschetz-Verdier formula
• The Jacobson--Morozov morphism for Langlands parameters in the relative setting
• Diamantine Picard functors of rigid spaces
• Local parameters of supercuspidal representations
• The de Rham-Fargues-Fontaine cohomology
• Geometrization of the Satake transform for mod $p$ Hecke algebras
• Averaging functors in Fargues' program for GL_n
• Progrès récents sur les représentations supercuspidales
• The plectic conjecture over function fields
• The Arinkin-Gaitsgory temperedness conjecture
• Mod $p$ points on Shimura varieties of parahoric level (with an appendix by Rong Zhou)