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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.

MSC:

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
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