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Categorical local Langlands and torsion classes of some Shimura varieties

Kieu Hieu Nguyen

Abstract

We study the cohomology of various local Shimura varieties for $GL_n$. This provides an explicit description of the spectral action constructed by Fargues-Scholze in certain cases and allows us to prove some strongly generic part of the categorical local Langlands conjecture for $GL_n$ with torsion coefficients. As applications, we are able to prove an analogue of the Harris-Viehmann conjecture and deduce new vanishing results for the cohomology of Shimura varieties of type A in the torsion coefficient setting.

Categorical local Langlands and torsion classes of some Shimura varieties

Abstract

We study the cohomology of various local Shimura varieties for . This provides an explicit description of the spectral action constructed by Fargues-Scholze in certain cases and allows us to prove some strongly generic part of the categorical local Langlands conjecture for with torsion coefficients. As applications, we are able to prove an analogue of the Harris-Viehmann conjecture and deduce new vanishing results for the cohomology of Shimura varieties of type A in the torsion coefficient setting.
Paper Structure (31 sections, 31 theorems, 113 equations)

This paper contains 31 sections, 31 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Some ideas of the proof
  4. Some applications
  5. Generalities on $\mathrm{Bun}_{\mathrm{G}}$
  6. Combinatoric description of $\mathrm{Bun}_{\mathrm{G}}$
  7. Sheaves on $\mathrm{Bun}_{\mathrm{G}}$
  8. Stacks of $L$-parameters
  9. Some connected components of the stack of $L$-parameters
  10. Perfect complexes on the stack of $L$-parameters
  11. Modular representations of inner forms of $\mathrm{GL}_n$
  12. Generalities on modular representations of inner forms of $\mathrm{GL}_n$
  13. Fargues-Scholze correspondence
  14. The supercuspidal case
  15. The categorical equivalence up to some shifts
  16. ...and 16 more sections

Key Result

Theorem 1.1

Theorems & Definitions (66)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • Definition 3.1
  • Proposition 3.2
  • proof
  • ...and 56 more