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Inductive construction of supercuspidal $L$-packets

Raphaël Beuzart-Plessis, Michael Harris, Jack Thorne

Abstract

Genestier--Lafforgue and Fargues--Scholze have constructed a semisimple local Langlands paramterization for reductive groups over equicharacteristic local fields. Assuming a version of the stable twisted trace formula for function fields, we prove the surjectivity of this parameterization for split groups in sufficiently large characteristic.

Inductive construction of supercuspidal $L$-packets

Abstract

Genestier--Lafforgue and Fargues--Scholze have constructed a semisimple local Langlands paramterization for reductive groups over equicharacteristic local fields. Assuming a version of the stable twisted trace formula for function fields, we prove the surjectivity of this parameterization for split groups in sufficiently large characteristic.

Paper Structure

This paper contains 25 sections, 34 theorems, 106 equations.

Table of Contents

  1. Notation
  2. Review of Galois parameters
  3. Hypotheses about the trace formula
  4. Comparison of $\rho$-traces
  5. Discrete parts of the trace formulas
  6. Elliptic endoscopic data
  7. Assumptions on the stabilization of trace formulas
  8. Proof of Proposition \ref{['hyp rho traces']}
  9. Construction of types
  10. Statement of the main result
  11. Review of Moy-Prasad filtrations
  12. Proof of property (ii)
  13. Approximation of Whittaker coinvariants and properties (iii)-(iv)
  14. Relation to principal congruence subgroups
  15. Globalization
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $K$ be any global function field and $G$ a connected semisimple algebraic group over $K$. (i) Lafa There is a map with the following property: if $v$ is a place of $K$ and $\Pi \in {\mathcal{A}}_0(G)$ is a cuspidal automorphic representation such that $\Pi_v$ is unramified, then ${\mathcal{L}}(\Pi)$ is unramified at $v$, and ${\mathcal{L}}^{ss}(\Pi) ~|_{W_{K_v}}$ is the Satake parameter of $\

Theorems & Definitions (62)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 1.3
  • proof
  • Lemma 1.4
  • proof
  • Lemma 1.5
  • proof
  • Proposition 2.1
  • Proposition 2.3
  • ...and 52 more