In this paper, we compute the cohomology with compact supports of a Picardmodular surface as a virtual module over the product of the appropriate Galoisgroup and the appropriate Hecke algebra. We use the method developed by Ihara,Langlands, and Kottwitz: comparison of the Grothendieck-Lefschetz formula andthe Arthur-Selberg trace formula. Our implementation of this method takes asits starting point the work of Laumon and Morel.
(1) In 1997, Laumon computed the cohomology with compact supports for the group @math ; see @cite_2 . Our work is adapted from that of Laumon's.
@cite_2
2008185589
In this paper we compute the cohomology with compact supports of a Siegelthreefold as a virtual module over the product of the Galois group of ( Q ^ ) over ( Q ) and the Hecke algebra. We use a method which has been developed by Ihara, Langlands and Kottwitz: comparison of the Grothendieck--Lefschetz formula and the Arthur--Selberg trace formula.