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Forum Mathematicum

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Volume 31, Issue 5

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A construction of residues of Eisenstein series and related square-integrable classes in the cohomology of arithmetic groups of low k-rank

Neven Grbac
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  • Faculty of Engineering, Juraj Dobrila University of Pula, Zagrebačka 30, HR-52100 Pula, Croatia
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/ Joachim Schwermer
  • Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria; and Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany
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Published Online: 2019-06-13 | DOI: https://doi.org/10.1515/forum-2019-0029

Abstract

The cohomology of an arithmetic congruence subgroup of a connected reductive algebraic group defined over a number field is captured in the automorphic cohomology of that group. The residual Eisenstein cohomology is by definition the part of the automorphic cohomology represented by square-integrable residues of Eisenstein series. The existence of residual Eisenstein cohomology classes depends on a subtle combination of geometric conditions (coming from cohomological reasons) and arithmetic conditions in terms of analytic properties of automorphic L-functions (coming from the study of poles of Eisenstein series). Hence, there are almost no unconditional results in the literature regarding the very existence of non-trivial residual Eisenstein cohomology classes. In this paper, we show the existence of certain non-trivial residual cohomology classes in the case of the split symplectic, and odd and even special orthogonal groups of rank two, as well as the exceptional group of type G2, defined over a totally real number field. The construction of cuspidal automorphic representations of GL2 with prescribed local and global properties is decisive in this context.

Keywords: Eisenstein cohomology; square-integrable cohomology classes; construction of non-trivial classes; automorphic forms; Eisenstein series; split groups of rank two

MSC 2010: 11F75; 11F67; 11F70; 22E40; 22E55

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About the article


Received: 2019-01-31

Revised: 2019-03-15

Published Online: 2019-06-13

Published in Print: 2019-09-01


Funding Source: Hrvatska Zaklada za Znanost

Award identifier / Grant number: 3628

Award identifier / Grant number: 9364

The first named author was supported in part by the Croatian Science Foundation (projects 3628 and 9364) and by the University of Rijeka (research grant 13.14.1.2.02). Both authors acknowledge the support obtained within the frame work of the Croatian-Austrian Scientific agreement (HR 17/2014).


Citation Information: Forum Mathematicum, Volume 31, Issue 5, Pages 1225–1263, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2019-0029.

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