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On arithmetic quotients of the Siegel upper half space of degree two. (English) Zbl 0596.10029

This paper studies the theory of Eisenstein cohomology for the rank 2 symplectic group \(Sp_ 4\). Let \(\Gamma\) be a torsionfree subgroup of finite index in \(Sp_ 4({\mathbb{Z}})\). First the cohomology of the boundary mod \(\Gamma\) of the Borel-Serre compactification X of the corresponding symmetric space is analyzed. Then Eisenstein series E are constructed from cuspidal cohomology classes \(\phi\) of faces of X/\(\Gamma\) corresponding to maximal parabolic \({\mathbb{Q}}\)-subgroups of \(Sp_ 4\). For such an E, which is holomorphic at a certain critical value \(\Lambda\), E(\(\Lambda)\) is a closed and harmonic differential form representing a nontrivial class in \(H^ 4(X/\Gamma)\) which restricts to \(\phi\) on the face we began with and to 0 on the other faces. Complete proofs are given of this part.
A statement of detailed results but only a brief sketch of the proofs is given in the more difficult cases where E may have a pole at \(\Lambda\). The adelic set-up is used here. The main result is a splitting from its image of the restriction map on cohomology in all degrees from X/\(\Gamma\) to its boundary.
Reviewer: A.Ash

MSC:

11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F27 Theta series; Weil representation; theta correspondences
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
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