Compositio Mathematica

Table of Contents - November 2011 - Volume 147, Issue 06  

Research Article

Cohomologie d’intersection des variétés modulaires de Siegel, suite

Sophie Morela1

a1 Department of Mathematics, Harvard University, One Oxford Street, Cambridge, MA 02138, USA (email: morel@math.harvard.edu)

Abstract

In this work, we study the intersection cohomology of Siegel modular varieties. The goal is to express the trace of a Hecke operator composed with a power of the Frobenius endomorphism (at a good place) on this cohomology in terms of the geometric side of Arthur’s invariant trace formula for well-chosen test functions. Our main tools are the results of Kottwitz about the contribution of the cohomology with compact support and about the stabilization of the trace formula, Arthur’s L2 trace formula and the fixed point formula of Morel [Complexes pondérés sur les compactifications de Baily–Borel. Le cas des variétés de Siegel, J. Amer. Math. Soc. 21 (2008), 23–61]. We ‘stabilize’ this last formula, i.e. express it as a sum of stable distributions on the general symplectic groups and its endoscopic groups, and obtain the formula conjectured by Kottwitz in [Shimura varieties and λ-adic representations, in Automorphic forms, Shimura varieties and L-functions, Part I, Perspectives in Mathematics, vol. 10 (Academic Press, San Diego, CA, 1990), 161–209]. Applications of the results of this article have already been given by Kottwitz, assuming Arthur’s conjectures. Here, we give weaker unconditional applications in the cases of the groups GSp4 and GSp6.

(Received December 16 2008)

(Accepted December 10 2010)

(Online publication September 28 2011)

2010 Mathematics Subject Classification

  • 11G18 (primary);
  • 14F20;
  • 20G35 (secondary)

Keywords

  • Siegel modular varieties;
  • intersection cohomology;
  • discrete automorphic representations of symplectic groups

Footnotes

Ce texte a été écrit pendant que j’étais employée par le Clay Mathematics Institute en tant que Clay Research Fellow, et accueillie en tant que membre à l’Institute for Advanced Study à Princeton. Il a été révisé pendant mon séjour à l’université Harvard en tant que visiteur, puis en tant que professeur. De plus, j’ai bénéficié du soutien financier de la NSF à travers les contrats DMS-0111298 et DMS-0635607.