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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The residual Eisenstein cohomology of $ Sp_{4}$ over a totally real number field


Authors: Neven Grbac and Harald Grobner
Journal: Trans. Amer. Math. Soc. 365 (2013), 5199-5235
MSC (2010): Primary 11F75; Secondary 11F70, 11F55, 22E55
Published electronically: March 12, 2013
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Abstract: Let $ G=Sp_4/k$ be the $ k$-split symplectic group of $ k$-rank 2, where $ k$ is a totally real number field. In this paper we compute the Eisenstein cohomology of $ G$ with respect to any finite-dimensional, irreducible, $ k$-rational representation $ E$ of $ G_\infty =R_{k/\mathbb{Q}}G(\mathbb{R})$, where $ R_{k/\mathbb{Q}}$ denotes the restriction of scalars from $ k$ to $ \mathbb{Q}$. This approach is based on the work of Schwermer regarding the Eisenstein cohomology for $ Sp_4/\mathbb{Q}$, Kim's description of the residual spectrum of $ Sp_4$, and the Franke filtration of the space of automorphic forms. In fact, taking the representation theoretic point of view, we write, for the group $ G$, the Franke filtration with respect to the cuspidal support, and give a precise description of the filtration quotients in terms of induced representations. This is then used as a prerequisite for the explicit computation of the Eisenstein cohomology. The special focus is on the residual Eisenstein cohomology. Under a certain compatibility condition for the coefficient system $ E$ and the cuspidal support, we prove the existence of non-trivial residual Eisenstein cohomology classes, which are not square-integrable, that is, represented by a non-square-integrable residue of an Eisenstein series.


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Additional Information

Neven Grbac
Affiliation: Department of Mathematics, University of Rijeka, Radmile Matejčić 2, HR-51000 Rijeka, Croatia
Email: neven.grbac@math.uniri.hr

Harald Grobner
Affiliation: Fakultät für Mathematik, University of Vienna, Nordbergstrasse 15, A-1090 Wien, Austria
Email: harald.grobner@univie.ac.at

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05796-0
PII: S 0002-9947(2013)05796-0
Keywords: Cohomology of arithmetic groups, Eisenstein cohomology, Eisenstein series, residue of Eisenstein series, automorphic forms, cuspidal automorphic representation, Franke filtration
Received by editor(s): September 15, 2010
Received by editor(s) in revised form: September 26, 2011, January 20, 2012, and January 24, 2012
Published electronically: March 12, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.