Hostname: page-component-8448b6f56d-t5pn6 Total loading time: 0 Render date: 2024-04-18T18:30:13.881Z Has data issue: false hasContentIssue false
  • English
  • Français

SUR LE COMPTAGE DES FIBRÉS DE HITCHIN NILPOTENTS

Part of: Lie groups Families, fibrations Algebraic number theory: global fields Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  07 August 2014

Pierre-Henri Chaudouard  and
Gérard Laumon
Pierre-Henri Chaudouard
Affiliation:
Université Paris Diderot (Paris 7), Institut de mathématiques de Jussieu-Paris Rive gauche, UMR 7586, Bâtiment Sophie Germain, Case 7012, F-75205 Paris Cedex 13, France (Pierre-Henri.Chaudouard@imj-prg.fr)
Gérard Laumon
Affiliation:
CNRS et université Paris-Sud, UMR 8628, Mathématique, Bâtiment 425, F-91405 Orsay Cedex, France (Gerard.Laumon@math.u-psud.fr)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Cet article est une contribution à la fois au calcul du nombre de fibrés de Hitchin sur une courbe projective et à l’explicitation de la partie nilpotente de la formule des traces d’Arthur-Selberg pour une fonction test très simple. Le lien entre les deux questions a été établi dans [Chaudouard, Sur le comptage des fibrés de Hitchin. À paraître aux actes de la conférence en l’honneur de Gérard Laumon]. On décompose cette partie nilpotente en une somme d’intégrales adéliques indexées par les orbites nilpotentes. Pour les orbites de type «régulières par blocs», on explicite complètement ces intégrales en termes de la fonction zêta de la courbe.

This paper is concerned with two problems. One is to count Hitchin bundles on a projective curve and the other is to get an explicit formula for the nilpotent part of the Arthur–Selberg trace formula for a simple test function. The fact that the two problems are indeed related has been noticed in a previous paper cf. [Chaudouard, Sur le comptage des fibrés de Hitchin. À paraître aux actes de la conférence en l’honneur de Gérard Laumon]. We expand the nilpotent part of the Arthur–Selberg trace formula in a sum of adelic integrals indexed by nilpotent orbits. For «regular by blocks» orbits, we get an explicit formula for these integrals in terms of the zeta function of the curve.

Keywords

Hitchin fibrationArthur–Selberg trace formulanilpotent cone

MSC classification

Primary: 14D20: Algebraic moduli problems, moduli of vector bundles
Secondary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields 11F72: Spectral theory; Selberg trace formula 11R39: Langlands-Weil conjectures, nonabelian class field theory 22E55: Representations of Lie and linear algebraic groups over global fields and adèle rings
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 15 , Issue 1 , January 2016 , pp. 91 - 164
Copyright
© Cambridge University Press 2014 

References

Bibliographie

Arthur, J., A trace formula for reductive groups I. Terms associated to classes in G (ℚ), Duke Math. J. 45 (1978), 911–952.CrossRefGoogle Scholar
Arthur, J., The trace formula in invariant form, Ann. of Math. (2) 114 (1981), 1–74.Google Scholar
Arthur, J., A measure on the unipotent variety, Canad. J. Math. 37 (1985), 1237–1274.Google Scholar
Arthur, J., A local trace formula, Publ. Math. Inst. Hautes Études 73 (1991), 5–96.Google Scholar
Arthur, J., An introduction to the trace formula, in Harmonic analysis, the trace formula, and Shimura varieties, (Clay Math. Proc.), Volume 4, pp. 1–263 (American Mathematical Society, Providence, RI, 2005).Google Scholar
Chaudouard, P.-H., Sur le comptage des fibrés de Hitchin. À paraître aux actes de la conférence en l’honneur de Gérard Laumon.Google Scholar
García-Prada, O., Heinloth, J. et Schmitt, A., On the motives of moduli of chains and Higgs bundles. ArXiv e-prints, April 2011.Google Scholar
Harder, G., Minkowskische Reduktionstheorie über Funktionenkörpern, Invent. Math. 7 (1969), 33–54.Google Scholar
Hausel, T. et Rodriguez-Villegas, F., Mixed Hodge polynomials of character varieties, Invent. Math. 174(3) (2008), 555–624; with an appendix by Nicholas M. Katz.CrossRefGoogle Scholar
Laumon, G., Cohomology of Drinfeld modular varieties. Part II, Cambridge Studies in Advanced Mathematics, Volume 56 (Cambridge University Press, Cambridge, 1997); Automorphic forms, trace formulas and Langlands correspondence, with an appendix by Jean-Loup Waldspurger.Google Scholar
Lusztig, G. et Spaltenstein, N., Induced unipotent classes, J. Lond. Math. Soc. (2) 19 (1979), 41–52.Google Scholar
Mozgovoy, S., Solutions of the motivic ADHM recursion formula. ArXiv e-prints, April 2011.Google Scholar
Nitsure, N., Moduli space of semistable pairs on a curve, Proc. Lond. Math. Soc. (3) 62(2) (1991), 275–300.Google Scholar