CONTROL THEOREMS OF p-NEARLY ORDINARY COHOMOLOGY GROUPS FOR SL(n)

— In this paper, we prove control theorems for the p-adic nearly ordinary cohomology groups for SL(n) over an aribitrary number field, generalizing the result already obtained for SL(2). The result should have various implications in the study of p-adic cohomological modulat forms on GL(n). In particular, in a subsequent paper, we will study p-adic analytic families of cuch Hecke eigenforms. RÉSUMÉ. — Dans cet article, on démontre le théorème de contrôle pour les groupes de cohomologie quasi-ordinaire p-adique de SL(n) sur un corps de nombre arbitraire en généralisant le résultat déjà connu pour SL(2). Le résultat doit avoir des implications variées dans la théorie des formes modulaires p-adiques cohomologiques sur GL(n). En particulier, on étudiera des familles p-adiques analytiques des formes propres de Hecke dans un prochain article. Introduction Let p be a prime. In [H2], we have studied the control theorem for p-ordinary cohomology groups for the algebraic group SL(2) defined over an arbitrary number field F . Here we generalize the result to reductive algebraic groups G over Q whose group of Qp-points is isomorphic to GLn(Fp) for Fp = F ⊗Q Qp. We fix such an isomorphism (GL) i :G(Qp) ∼= GLn(Fp). (*) Texte reçu le 13 juin 1994, révisé le 16 août 1994. The author is partially supported by an NSF grant. Writing the paper was basically finished while the author was visiting the Galilée Institute at the University of Paris XIII (France), and some of the results in this paper was presented in a short series of lectures there. The author is grateful to J. Tilouine for the invitation to the Institute. Haruzo HIDA, Department of Mathematics, UCLA, Los Angeles, CA 90024 (USA) and Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo 060 (Japan). Email: [email protected]. AMS classification: 11F33, 11F55, 11F60, 11F67, 11F75, 11F85. BULLETIN DE LA SOCIÉTÉ MATHÉMATIQUE DE FRANCE 0037–9484/1995/425/$ 5 c © Société mathématique de France