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.
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.