The cohomology of \(p\)-adic symmetric spaces. (English) Zbl 0751.14016

For a local field \(K\) the “symmetric space” or “upper half space” \(\Omega^{(d+1)}\) is the \(d\)-dimensional projective space over \(K\) with all \(K\)-rational hyperplanes removed. By defining \(\varepsilon\)- neighbourhoods of rational hyperplanes and thereby exhausting \(\Omega^{(d+1)}\) by affinoid subdomains \(\Omega_ n^{(d+1)}\), V. G. Drinfel’d [Math. USSR, Sb. 23 (1974), 561-592 (1976); translation from Mat. Sb., Nov. Ser. 94(136), 594-627 (1974; Zbl 0321.14014)] showed that \(\Omega^{(d+1)}\) is a rigid-analytic subvariety of \(\mathbb{P}^ d_ K\), and the present authors show that it is a Stein space.
In the first part of the paper (§§1-4), the cohomology of \(\Omega^{(d+1)}\) is computed. The calculations work in any “good” cohomology theory, axiomatically characterized by four standard properties; the basic examples are de Rham and étale cohomology. The authors also define in this abstract setting relative cohomology groups (for an open subvariety) and construct a spectral sequence converging to this relative cohomology. With this technique they can express the cohomology of \(\Omega^{(d+1)}\) in terms of the simplicial homology of a certain generalized Tits building for \(G=GL_{d+1}(K)\). This cohomology in turn can be written as a certain space of locally constant functions on quotients of \(G\). The first explicit formula is obtained from this via Bruhat decomposition and expresses \(H^ s(\Omega^{(d+1)})\) as the \(A\)- valued distributions on the \({1\over 2}s(2d+1-s)\)-dimensional affine space over \(K\) (where the ring \(A\) depends on the chosen cohomology theory).
The second explicit formula expresses \(H^ s(\Omega^{d+1})\) as a subspace of the space of \(s\)-cochains on the Bruhat-Tits building for \(SL_{d+1}(K)\). If \(s=d\) this space is the space of harmonic cochains.
The second part of the paper (§5) is devoted to the cohomology of the quotient \(X_ \Gamma\) of \(\Omega^{(d+1)}\) by a discrete cocompact subgroup \(\Gamma\) of \(\text{PGL}_{d+1}(K)\) that acts without fixed points on \(\Omega^{(d+1)}\). It is known that \(X_ \Gamma\) is a proper smooth rigid-analytic variety (Drinfeld) and even a projective variety (Mustafin). The cohomology of \(X_ \Gamma\) is calculated by the spectral sequence \(H^ r(\Gamma,H^ s(\Omega^{(d+1)}))\Rightarrow H^{r+s}(X_ \Gamma)\). The \(E_ 2\)-terms of this spectral sequence are transformed into certain Ext-groups for smooth \(\text{PGL}_{d+1}(K)\)- representations, and these are calculated using representation theoretic arguments (generalizing results of Casselman). The final result only depends on a single representation theoretic invariant of \(\Gamma\), namely the multiplicity of the Steinberg representation in the representation induced from the trivial character. The result is then applied to give a proof of the \(p\)-adic Shimura isomorphisms which relates modular forms and group cohomology — slightly correcting the corresponding statement in the first named author’s paper in Number Theory, Proc. Journée arithmétique, Noordwijkerhout 1983, Lect. Notes Math. 1068, 216-230 (1984; Zbl 0572.14014); an independent and more elementary proof of this was given by E. de Shalit [J. Reine Angew. Math. 400, 3-31 (1989; Zbl 0674.14031)].
Finally the last part (§6) develops a general setting for the construction of certain resolutions of \(GL_{d+1}(K)\)-modules, which provides in particular a result needed in §5.


14G20 Local ground fields in algebraic geometry
14F20 Étale and other Grothendieck topologies and (co)homologies
14F30 \(p\)-adic cohomology, crystalline cohomology
32P05 Non-Archimedean analysis
Full Text: DOI EuDML


[1] Báyer, P., Neukirch, J.: On Automorphic Forms and Hodge Theory. Math. Ann.257, 137-155 (1981) · Zbl 0476.32039
[2] Bernstein, I. N., Zelevinskii, A. V.: Representations of the group GL(n, F) whereF is a non-archimedean local field. Russ. Math. Surv.31(3), 1-68 (1976) · Zbl 0348.43007
[3] Borel, A., Serre, J.-P.: Cohomologie d’immeubles et de groupes S-arithmétiques. Topology15, 211-232 (1976) · Zbl 0338.20055
[4] Borel, A., Tits, J.: Compléments à l’article: ?Groupes réductifs?, Publ. Math. IHES41, 253-276 (1972) · Zbl 0254.14018
[5] Borel, A., Wallach, N.: Continuous cohomology, discrete subgroups, and representations of reductive groups. Ann. Math. Stud. 94. Princeton University Press 1980 · Zbl 0443.22010
[6] Bosch, S., Güntzer, U., Remmert, R.: Non-Archimedean analysis. Berlin Heidelberg New York: Springer 1984 · Zbl 0539.14017
[7] Bourbaki, N.: Groupes et algèbres de Lie, Chap. 4-6. Paris: Masson 1981 · Zbl 0483.22001
[8] Brown, K.S.: Buildings. Berlin Heidelberg New York: Springer 1989
[9] Bruhat, F., Tits, J.: Groupes réductifs sur un corps local. I. Données radicielles valuées. Publ. Math. IHES41, 5-252 (1972)
[10] Cartier, P.: Les arrangements d’hyperplane: Un chapitre de géométrie combinatoire. Sém. Bourbaki 1980/81, exp. 561. Lect. Notes Math. vol. 901. Berlin Heidelberg New York: Springer 1981
[11] Casselman, W.: Introduction to the theory of admissible representations of ?-adic reductive groups (Preprint)
[12] Casselman, W.: On ap-adic vanishing theorem of Garland. Bull. Soc. AMS80, 1001-1004 (1974) · Zbl 0354.20033
[13] Casselman, W.: A new non-unitarity argument forp-adic representations. J. Fac. Sci. Univ. Tokyo28, 907-928 (1981) · Zbl 0519.22011
[14] Curtis, C.W., Lehrer, G.I., Tits, J.: Spherical Buildings and the Character of the Steinberg Representation. Invent. Math.58, 201-210 (1980) · Zbl 0435.20024
[15] Deligne, P.: La conjecture de Weil H. Publ. Math. IHES52, 137-252 (1980)
[16] Drinfeld, V.G.: Elliptic modules. Math. USSR Sbornik23, 561-592 (1974) · Zbl 0321.14014
[17] Drinfel’d, V.G.: Coverings ofp-adic symmetric regions. Funct. Anal Appl.10, 107-115 (1976) · Zbl 0346.14010
[18] Fresnel, J., van der Put, M.: Géométrie analytique rigide et applications. Boston: Birkhäuser 1981 · Zbl 0479.14015
[19] Garland, H.:p-adic curvature and the cohomology of discrete subgroups ofp-adic groups. Ann. Math.97, 375-423 (1973) · Zbl 0262.22010
[20] Godement, R.: Topologie algébrique et théorie des faisceaux. Paris: Hermann 1964
[21] Goldmann, O., Iwahori, N.: The space of ?-adic norms. Acta Math.109, 137-177 (1963) · Zbl 0133.29402
[22] Hartshorne, R.: Residues and duality. Lect. Notes Math., Vol. 20. Berlin Heidelberg New York: Springer 1966 · Zbl 0212.26101
[23] Hartshorne, R.: On the de Rham cohomology of algebraic varieties. Publ. Math. IHES45, 5-99 (1975) · Zbl 0326.14004
[24] Hiller, H.: Geometry of Coxeter groups. Boston: Pitman 1982 · Zbl 0483.57002
[25] Hyodo, O.: On the de Rham-Witt complex attached to a semistable family. Preprint 1988
[26] Iwahori, N.: Generalized Tits system (Bruhat decomposition) on 122-1 semisimple groups. In: Algebraic groups and discontinuous subgroups. Proc. Symp. Pure Math.9, 71-83 (1966)
[27] Jensen, C.U.: Les foncteurs dérivés de \(\underleftarrow {\lim }\) et leurs applications en théorie des modules. Lect. Notes Math., vol. 254. Berlin Heidelberg New York: Springer 1972
[28] Kiehl, R.: Der Endlichkeitssatz für eigentliche Abbildungen in der nichtarchimedischen Funktionentheorie. Invent. Math.2, 191-214 (1967) · Zbl 0202.20101
[29] Kiehl, R.: Theorem A und Theorem B in der nichtarchimedischen Funktionentheorie. Invent. Math.2, 256-273 (1967) · Zbl 0202.20201
[30] Kurihara, A.: Construction ofp-adic unit balls and the Hirzebruch proportionality. Am. J. Math.102, 565-648 (1980) · Zbl 0498.14011
[31] MacLane, S.: Homology. Berlin Heidelberg New York: Springer 1975 · Zbl 0149.26203
[32] Mehlmann, F.: Ein Beweis für einen Satz von Raynaud über flache Homomorphismen affinoider Algebren. Schriftenr. Math. Inst. Univ. Münster, 2. Serie, Heft 19. Münster: 1981 · Zbl 0455.14014
[33] Milne, J.S.: Etale cohomology. Princeton University Press 1980 · Zbl 0433.14012
[34] Morita, Y.: Analytic representations of SL2 over a ?-adic number field, II. Automorphic forms of several variables, Taniguchi Symp. 1983. Progr. Math.46, 282-297 (1984)
[35] Morita, Y.: Analytic representations of SL2 over a ?-adic number field, III. Automorphic forms and number theory. Adv. Stud. Pure Math.7, 185-222 (1985)
[36] Morita, Y., Schikhof, W.: Duality of projective limit spaces and inductive limit spaces over a nonspherically complete nonarchimedean field. Tôhoku Math. J.38, 387-397 (1986) · Zbl 0649.46069
[37] Mumford, D.: An analytic construction of degenerating curves over complete local rings Compos. Math.24, 129-174 (1972) · Zbl 0228.14011
[38] Mumford, D.: Abelian varieties. Oxford University Press 1974 · Zbl 0326.14012
[39] Mustafin, G.A.: Nonarchimedean uniformization. Math. USSR Sbornik34, 187-214 (1978) · Zbl 0411.14006
[40] van der Put, M.: A note onp-adic uniformization. Proc. Kon. Ned. Akad. Wet. A90, 313-318 (1987). · Zbl 0624.32018
[41] Rapoport, M.: On the bad reduction of Shimura varieties. In: Clozel, L., Milne, J.S. (Eds.) Automorphic forms, Shimura varieties, andL-functions II. Perspect. Math. 11, pp. 253-321. Boston: Academic Press 1990
[42] Raynaud, M.: Geometrie analytique rigide d’apres Tate, Kiehl. Bull. Soc. Math. France, Mémoire39-40, 319-327 (1974)
[43] Robert, A.: Représentationsp-adiques irréductibles de sous-groupes ouverts de SL2(? p ). C.R. Acad. Sci. Paris298, 237-240 (1984)
[44] Schneider, P.: Rigid-analyticL-transforms. In: Number Theory, Noordwijkerhout 1983. Lect. Notes Math., vol. 1068, pp. 216-230, Berlin Heidelberg New York: Springer 1984
[45] Segal, G.: Classifying spaces and spectral sequences. Publ. Math. IHES34, 105-112 (1968) · Zbl 0199.26404
[46] de Shalit, E.: Eichler cohomology and periods of modular forms onp-adic Schottky groups. J. Reine angew. Math.400, 3-31 (1989) · Zbl 0674.14031
[47] Spanier, E.H.: Algebraic topology. New York: McGraw-Hill 1966 · Zbl 0145.43303
[48] Stuhler, U.: Über die Kohomologie einiger arithmetischer Varietäten I. Math. Ann.273, 685-699 (1986) · Zbl 0578.14016
[49] Teitelbaum, J.: Values ofp-adicL-functions and ap-adic Poisson kernel. Invent. Math.101, 395-410 (1990) · Zbl 0731.11065
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.