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Geometrization of the local Langlands correspondence, motivically

Peter Scholze

Abstract

Based on the formalism of rigid-analytic motives of Ayoub--Gallauer--Vezzani, we extend our previous work with Fargues from $\ell$-adic sheaves to motivic sheaves. In particular, we prove independence of $\ell$ of the $L$-parameters constructed there.

Geometrization of the local Langlands correspondence, motivically

Abstract

Based on the formalism of rigid-analytic motives of Ayoub--Gallauer--Vezzani, we extend our previous work with Fargues from -adic sheaves to motivic sheaves. In particular, we prove independence of of the -parameters constructed there.
Paper Structure (7 sections, 23 theorems, 103 equations)

This paper contains 7 sections, 23 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Basic results
  3. $\mathcal{D}_{\mathrm{mot}}(\mathrm{Bun}_G)$
  4. $\mathcal{D}_{\mathrm{mot}}(\mathrm{Div}^1)$
  5. Stacks of $L$-parameters
  6. Geometric Satake
  7. Synopsis

Key Result

Theorem 1.1

The association $\pi\mapsto \varphi_{\pi,\iota}$ is independent of $\ell$ and $\iota$.

Theorems & Definitions (47)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 37 more