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Geometrization of the local Langlands correspondence

Laurent Fargues, Peter Scholze

TL;DR

This work provides a conceptual framework and foundational toolkit for geometrizing the local Langlands correspondence via the Fargues–Fontaine curve. It constructs an ℓ-adic sheaf theory on Bun_G, proves a geometric Satake equivalence in this setting, and develops the stack of L-parameters with a spectral action, tying automorphic data to Galois parameters through excursion operators. Finiteness results for local shtuka and Rapoport–Zink-type spaces follow from the geometric setup, while L-parameters arise from a categorical action of the spectral side on D(Bun_G). The paper also builds a robust geometric and categorical infrastructure—diamonds, v-stacks, and universally locally acyclic sheaves—providing a platform for further comparisons between automorphic and Galois perspectives in p-adic contexts. The overarching theme is that the local Langlands program can be viewed as a geometric Langlands program on the Fargues–Fontaine curve, with a rich interplay between Hecke operators, the dual group, and spectral data.

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 $\mathrm{Bun}_G$.

Geometrization of the local Langlands correspondence

TL;DR

This work provides a conceptual framework and foundational toolkit for geometrizing the local Langlands correspondence via the Fargues–Fontaine curve. It constructs an ℓ-adic sheaf theory on Bun_G, proves a geometric Satake equivalence in this setting, and develops the stack of L-parameters with a spectral action, tying automorphic data to Galois parameters through excursion operators. Finiteness results for local shtuka and Rapoport–Zink-type spaces follow from the geometric setup, while L-parameters arise from a categorical action of the spectral side on D(Bun_G). The paper also builds a robust geometric and categorical infrastructure—diamonds, v-stacks, and universally locally acyclic sheaves—providing a platform for further comparisons between automorphic and Galois perspectives in p-adic contexts. The overarching theme is that the local Langlands program can be viewed as a geometric Langlands program on the Fargues–Fontaine curve, with a rich interplay between Hecke operators, the dual group, and spectral data.

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 -adic sheaves on the stack of -bundles on the Fargues--Fontaine curve, prove a geometric Satake equivalence over the Fargues--Fontaine curve, and study the stack of -parameters. As applications, we prove finiteness results for the cohomology of local Shimura varieties and general moduli spaces of local shtukas, and define -parameters associated with irreducible smooth representations of , 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 -parameters on the category of -adic sheaves on .

Paper Structure

This paper contains 145 sections, 363 theorems, 1405 equations.

Table of Contents

  1. Introduction
  2. The local Langlands correspondence
  3. The big picture
  4. The Fargues--Fontaine curve
  5. The geometry of $\mathrm{Bun}_G$
  6. $\ell$-adic sheaves on $\mathrm{Bun}_G$
  7. The geometric Satake equivalence
  8. Cohomology of moduli spaces of shtuka
  9. The stack of $L$-parameters
  10. Construction of $L$-parameters
  11. The spectral action
  12. The origin of the ideas
  13. Acknowledgments
  14. Notation
  15. The Fargues--Fontaine curve and vector bundles
  16. ...and 130 more sections

Key Result

Theorem I.2.1

The prestack $G\text{-}\mathcal{I}\mathrm{soc}$ is a stack for the v-topology on perfect $\overline{\mathbb F}_q$-algebras. It admits a stratification into locally closed substacks for $b\in B(E,G)$, consisting of the locus where at each geometric point, the $G$-isocrystal is isomorphic to $\mathcal{E}_b$. Moreover, each stratum is a classifying stack for the locally profinite group $G_b(E)$.

Theorems & Definitions (509)

  • Theorem I.2.1
  • Definition I.2.2
  • Proposition I.2.3
  • Definition I.2.4
  • Remark I.2.5
  • Theorem I.2.6
  • Theorem I.2.7
  • Remark I.2.8
  • Remark I.2.9
  • Remark I.2.10
  • ...and 499 more