On Eisenstein series and the cohomology of arithmetic groups

Neven Grbac; Joachim Schwermer

doi:10.1016/j.crma.2010.04.007

Number Theory/Group Theory

On Eisenstein series and the cohomology of arithmetic groups
[Sur les séries d'Eisenstein et la cohomologie des groupes arithmétiques]

Présenté par : Jean-Pierre Serre

Neven Grbac ¹ ; Joachim Schwermer ^2,³

¹ Department of Mathematics, University of Rijeka, Omladinska 14, 51000 Rijeka, Croatia
² Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, A-1090 Vienna, Austria
³ Erwin Schrödinger International Institute for Mathematical Physics, Boltzmanngasse 9, A-1090 Vienna, Austria

Comptes Rendus. Mathématique, Volume 348 (2010) no. 11-12, pp. 597-600.

Abstract (VO)
Résumé

The automorphic cohomology of a reductive $Q$ -group G captures essential analytic aspects of the arithmetic subgroups of G. The subspace spanned by all possible residues and principal values of derivatives of Eisenstein series, attached to cuspidal automorphic forms π on the Levi factor of proper parabolic $Q$ -subgroups of G, forms the Eisenstein cohomology. We show that non-trivial classes can only arise if the point of evaluation features a “half-integral” property. Consequently, only the analytic behavior of the automorphic L-functions at half-integral arguments matters whether an Eisenstein series attached to a globally generic π gives rise to a residual class or not.

La cohomologie automorphe d'un $Q$ -groupe réductif G détecte des propriétés analytiques essentielles des sous-groupes arithmétiques de G. La cohomologie d'Eisenstein est le sous-espace engendré par tous les résidus ainsi que par les valeurs principales des dérivées des séries d'Eisenstein, attachées aux formes automorphes cuspidales π sur les facteurs de Levi des $Q$ -sous-groupes paraboliques propres de G. Nous montrons que les classes non triviales ne peuvent provenir que des évaluations aux points « demi-entiers ». Ainsi, savoir si une série d'Eisenstein attachée à une forme π générique donne lieu à une classe résiduelle ou non, ne dépend que du comportement analytique de fonctions L automorphes en des points demi-entiers.

Reçu le : 2009-12-10
Accepté le : 2010-04-06
Publié le : 2010-04-30

DOI : 10.1016/j.crma.2010.04.007

Affiliations des auteurs :

Neven Grbac ¹ ; Joachim Schwermer ^{2, 3}

¹ Department of Mathematics, University of Rijeka, Omladinska 14, 51000 Rijeka, Croatia
² Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, A-1090 Vienna, Austria
³ Erwin Schrödinger International Institute for Mathematical Physics, Boltzmanngasse 9, A-1090 Vienna, Austria

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     title = {On {Eisenstein} series and the cohomology of arithmetic groups},
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Neven Grbac; Joachim Schwermer. On Eisenstein series and the cohomology of arithmetic groups. Comptes Rendus. Mathématique, Volume 348 (2010) no. 11-12, pp. 597-600. doi : 10.1016/j.crma.2010.04.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.04.007/

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Neven Grbac Endoscopic transfer and automorphic $L$ -functions: the case of the general spin group and the twisted symmetric and exterior square $L$ -functions, Rad Hrvatske Akademije Znanosti i Umjetnosti. Matematičke Znanosti, Volume 558 (2024), pp. 71-91 | DOI:10.21857/m8vqrt3n19 | Zbl:1551.11046
Laurent Clozel On the Eisenstein functoriality in cohomology for maximal parabolic subgroups, Selecta Mathematica. New Series, Volume 28 (2022) no. 4, p. 26 (Id/No 80) | DOI:10.1007/s00029-022-00794-y | Zbl:1511.11055
Neven Grbac Eisenstein cohomology and automorphic $L$ -functions, Cohomology of arithmetic groups. On the occasion of Joachim Schwermer's 66th birthday, Bonn, Germany, June 2016, Cham: Springer, 2018, pp. 35-50 | DOI:10.1007/978-3-319-95549-0_2 | Zbl:1483.11102
Neven Grbac; Joachim Schwermer Eisenstein series, cohomology of arithmetic groups, and automorphic $L$ -functions at half integral arguments, Forum Mathematicum, Volume 26 (2014) no. 6, pp. 1635-1662 | DOI:10.1515/forum-2012-0050 | Zbl:1317.11055

Cité par 4 documents. Sources : zbMATH

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