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
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