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Endoscopic transfer and the wavefront upper bound conjecture

Hiraku Atobe, Dan Ciubotaru

Abstract

We verify the upper bound conjecture of Kim and the second author, and Hazeltine--Liu--Lo--Shahidi, for the geometric wavefront sets of co-tempered representations of split classical $p$-adic groups with $p\gg 0$, under certain technical conditions. The proof uses Waldspurger's work on the endoscopic transfer supplemented by results of Konno and Varma, as well as the wavefront set computations in the unipotent case by Mason-Brown--Okada and the second author.

Endoscopic transfer and the wavefront upper bound conjecture

Abstract

We verify the upper bound conjecture of Kim and the second author, and Hazeltine--Liu--Lo--Shahidi, for the geometric wavefront sets of co-tempered representations of split classical -adic groups with , under certain technical conditions. The proof uses Waldspurger's work on the endoscopic transfer supplemented by results of Konno and Varma, as well as the wavefront set computations in the unipotent case by Mason-Brown--Okada and the second author.
Paper Structure (16 sections, 12 theorems, 90 equations)

This paper contains 16 sections, 12 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Wavefront set
  3. $L$-parameters
  4. $L$-packets and $A$-packets
  5. Conjecture and the main result
  6. Outline of the proof
  7. Twisted endoscopy
  8. Twisted local character expansion
  9. Transfer of nilpotent orbital integrals
  10. Wavefront set for co-tempered parameters
  11. Standard endoscopy
  12. Waldspurger map
  13. Proof of Theorem \ref{['main']}
  14. Preliminaries for proving Proposition \ref{['WS']}
  15. Proof of Proposition \ref{['WS']}
  16. ...and 1 more sections

Key Result

Theorem 1.3

Let $H$ be a split classical group over $F$, and let $\phi \in \Phi_\mathrm{temp}(H(F))$. Suppose that one of the following holds: Set $\psi = \widehat{\phi}$.

Theorems & Definitions (23)

  • Conjecture 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Corollary 2.4
  • proof
  • ...and 13 more