## 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

### Citations:

Zbl 0321.14014; Zbl 0572.14004; Zbl 0674.14031; Zbl 0572.14014
Full Text:

### References:

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