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Sur le comportement, par torsion, des facteurs epsilon de paires
Published online by Cambridge University Press: 20 November 2018
Résumé
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Soient 
$F$ un corps commutatif localement compact non archimédien et 
$\psi$ un caractère additif non trivial de $F$. Soient 
$n$ et 
${n}'$ deux entiers distincts, supérieurs à 1. Soient 
$\pi$ et 
${\pi }'$ des représentations irréductibles supercuspidales de 
$\text{G}{{\text{L}}_{n}}\left( F \right)$, 
$\text{G}{{\text{L}}_{{{n}'}}}\left( F \right)$ respectivement. Nous prouvons qu’il existe un élément 
$c=c\left( \pi ,{\pi }',\psi \right)$ de 
${{F}^{\times }}$ tel que pour tout quasicaractère modéré 
$\mathcal{X}$ de ${{F}^{\times }}$ on ait 
$\mathcal{E}\left( \chi \pi \times {\pi }',s,\psi \right)=\chi {{\left( c \right)}^{-1}}\mathcal{E}\left( \pi \times {\pi }',s,\psi \right)$. Nous examinons aussi certains cas où 
$n={n}',{\pi }'={{\pi }^{\text{v}}}$. Les résultats obtenus forment une étape vers une démonstration de la conjecture de Langlands pour $F$, qui ne fasse pas appel à la géométrie des variétés modulaires, de Shimura ou de Drinfeld.
Abstract
Let $F$ be a non-Archimedean local field, and $\psi $ a non-trivial additive character of $F$. Let $n$ and ${n}'$ be distinct positive integers. Let $\pi $, ${\pi }'$ be irreducible supercuspidal representations of $\text{G}{{\text{L}}_{n}}\left( F \right)$, $\text{G}{{\text{L}}_{{{n}'}}}\left( F \right)$ respectively. We prove that there is $c=c\left( \pi ,{\pi }',\psi \right)$
$\in $${{F}^{\times }}$ such that for every tame quasicharacter $\mathcal{X}$ of ${{F}^{\times }}$ we have $\mathcal{E}\left( \chi \pi \times {\pi }',s,\psi \right)=\chi {{\left( c \right)}^{-1}}\mathcal{E}\left( \pi \times {\pi }',s,\psi \right)$. We also treat some cases where 
$n={n}'$ and 
${\pi }'={{\pi }^{\text{V}}}$. These results are steps towards a proof of the Langlands conjecture for $F$, which would not use the geometry of modular—Shimura or Drinfeld—varieties.
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- Research Article
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- Copyright © Canadian Mathematical Society 2001
References
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