Skip to main content Accessibility help
×
Home

Sur le comportement, par torsion, des facteurs epsilon de paires

  • Colin J. Bushnell (a1) and Guy Henniart (a2)

Résumé

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.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Sur le comportement, par torsion, des facteurs epsilon de paires
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Sur le comportement, par torsion, des facteurs epsilon de paires
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Sur le comportement, par torsion, des facteurs epsilon de paires
      Available formats
      ×

Copyright

References

Hide All
[1] Arthur, J. and Clozel, L., Simple algebras, base change, and the advanced theory of the trace formula. Ann. of Math. Studies 120, Princeton University Press, 1989.
[2] Bushnell, C. J. Hereditary orders, Gauss sums and supercuspidal representations of GL(n) . J. Reine Angew. Math. 375/376(1987), 184210.
[3] Bushnell, C. J., Gauss sums and local constants for GL(N) . In: L-functions and Arithmetic, (eds., Coates, J., Taylor, M. J.), London Math. Soc. Lecture Notes 153, Cambridge University Press, 1991, 6173.
[4] Bushnell, C. J. and Henniart, G., Local tame lifting for GL(n) I: simple characters. Inst. Hautes Études Sci. Publ. 83(1996), 105233.
[5] Bushnell, C. J. and Henniart, G., Local tame lifting for GL(n) II: wildly ramified supercuspidals. Astérisque 254(1999),
[6] Bushnell, C. J. and Henniart, G., Supercuspidal representations of GL n: explicit Whittaker functions. J. Algebra 209(1998), 270287.
[7] Bushnell, C. J. and Henniart, G., Calculs de facteurs epsilon de paires pour GL n sur un corps local I. Bull. London Math. Soc. 31(1999), 534542.
[8] Bushnell, C. J. and Henniart, G., Davenport-Hasse relations and an explicit Langlands correspondence. J. Reine Angew. Math. 519(2000), 171199.
[9] Bushnell, C. J. and Henniart, G., Davenport-Hasse relations and an explicit Langlands correspondence II: twisting conjectures. J. Th. Nombres Bordeaux 12(2000), 309347.
[10] Bushnell, C. J., Henniart, G. and Kutzko, P. C., Local Rankin-Selberg convolutions for GL n: explicit conductor formula. J. Amer. Math. Soc. 11(1998), 703730.
[11] Bushnell, C. J., Henniart, G. and Kutzko, P. C., Correspondance de Langlands locale pour GL n et conducteurs de paires. Ann. Sci. École Norm. Sup. (4) 31(1998), 537560.
[12] Bushnell, C. J. and Kutzko, P. C., The admissible dual of GL(N) via compact open subgroups. Ann. of Math. Studies 129, Princeton University Press, 1993.
[13] Deligne, P., Les constantes des équations fonctionnelles des fonctions L. In: Modular forms of one variable II, Lecture Notes in Math. 349, Springer, Berlin, 501597, 1974.
[14] Deligne, P. and Henniart, G., Sur la variation, par torsion, des constantes locales d’équations fonctionnelles des fonctions L. Invent. Math. 64(1981), 89118.
[15] Godement, R. and Jacquet, H., Zeta functions of simple algebras. Lecture Notes in Math. 260, Springer, Berlin, 1972.
[16] Harris, M. and Taylor, R., On the geometry and cohomology of some simple Shimura varieties. Prépublication, 1999.
[17] Henniart, G., Représentations du groupe de Weil d’un corps local. Enseign. Math. 26(1980), 155172.
[18] Henniart, G., Galois ε-factors modulo roots of unity. Invent. Math. 78(1984), 117126.
[19] Henniart, G., Une preuve simple des conjectures de Langlands pour GL n sur un corps p-adique. Invent. Math. 139(2000), 439455.
[20] Henniart, G. and Herb, R., Automorphic induction for GL(n) (over local non-Archimedean fields). Duke Math. J. 78(1995), 131192.
[21] Jacquet, H., Principal L-functions of the linear group. In: Automorphic forms, representations and L-functions, (eds., Borel, A. and Casselman, W.), Proc. Symposia Pure Math. (2) 33(1979), Amer. Math. Soc., 6387.
[22] Jacquet, H., Piatetskii-Shapiro, I. and Shalika, J., Rankin-Selberg convolutions. Amer. J. Math. 105(1983), 367483.
[23] Laumon, G., Rapoport, M. and Stuhler, U., D-elliptic sheaves and the Langlands correspondence. Invent. Math. 113(1993), 217338.
[24] Sauvageot, F., Principe de densité pour les groupes réductifs. Compositio Math. 108(1997), 151184.
[25] Shahidi, F., Fourier transforms of intertwining operators and Plancherel measures for GL(n) . Amer. J. Math. 106(1984), 67111.
[26] Tate, J., Number theoretic background. In: Automorphic forms, representations and L-functions, (eds., Borel, A. and Casselman, W.), Proc. Symposia Pure Math. (2) 33(1979), Amer. Math. Soc. 326.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

Sur le comportement, par torsion, des facteurs epsilon de paires

  • Colin J. Bushnell (a1) and Guy Henniart (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed