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Regular and residual Eisenstein series and the automorphic cohomology of Sp(2,2)

Part of: Lie groups Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  23 November 2009

Harald Grobner
Harald Grobner*
Affiliation:
Faculty of Mathematics, University of Vienna, Nordbergstraße 15, A-1090 Vienna, Austria (email: [email protected])
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Abstract

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Let G be the simple algebraic group Sp(2,2), to be defined over ℚ. It is a non-quasi-split, ℚ-rank-two inner form of the split symplectic group Sp8 of rank four. The cohomology of the space of automorphic forms on G has a natural subspace, which is spanned by classes represented by residues and derivatives of cuspidal Eisenstein series. It is called Eisenstein cohomology. In this paper we give a detailed description of the Eisenstein cohomology HqEis(G,E) of G in the case of regular coefficients E. It is spanned only by holomorphic Eisenstein series. For non-regular coefficients E we really have to detect the poles of our Eisenstein series. Since G is not quasi-split, we are out of the scope of the so-called ‘Langlands–Shahidi method’ (cf. F. Shahidi, On certain L-functions, Amer. J. Math. 103 (1981), 297–355; F. Shahidi, On the Ramanujan conjecture and finiteness of poles for certain L-functions, Ann. of Math. (2) 127 (1988), 547–584). We apply recent results of Grbac in order to find the double poles of Eisenstein series attached to the minimal parabolic P0 of G. Having collected this information, we determine the square-integrable Eisenstein cohomology supported by P0 with respect to arbitrary coefficients and prove a vanishing result. This will exemplify a general theorem we prove in this paper on the distribution of maximally residual Eisenstein cohomology classes.

Keywords

cohomology of arithmetic groupsEisenstein cohomologycuspidal automorphic representationEisenstein seriesresidual spectrum

MSC classification

Secondary: 11F75: Cohomology of arithmetic groups 11F70: Representation-theoretic methods; automorphic representations over local and global fields 11F55: Other groups and their modular and automorphic forms (several variables) 22E55: Representations of Lie and linear algebraic groups over global fields and adèle rings
Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 1 , January 2010 , pp. 21 - 57
Copyright
Copyright © Foundation Compositio Mathematica 2009

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