Comptes Rendus
Number Theory/Group Theory
On Eisenstein series and the cohomology of arithmetic groups
Comptes Rendus. Mathématique, Volume 348 (2010) no. 11-12, pp. 597-600.

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.

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Published online:
DOI: 10.1016/j.crma.2010.04.007

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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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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