Comptes Rendus
no. 9
Number theory/Differential geometry
On the cohomology groups of certain quotients of products of upper half planes and upper half spaces
[Sur les groupes de cohomologie de certains quotients de produits de demi-plans et demi-espaces supérieurs]

Présenté par : the Editorial Board

Amod Agashe1 ; Lydia Eldredge1
1 Florida State University, Department of Mathematics, Tallahassee, FL, 32306, United States
Comptes Rendus. Mathématique, Volume 355 (2017) no. 9, pp. 937-941.

Un théorème de Matsushima et Shimura montre que l'espace des formes différentielles harmoniques sur un quotient compact d'un produit de demi-plans supérieurs sous l'action de certains groupes est la somme directe de deux sous-espaces appelés sous-espaces universel et cuspidal. Nous généralisons ce résultat aux quotients compacts d'un produit de demi-plans supérieurs et demi-espaces supérieurs (hyperboliques de dimension 3) sous l'action de certains groupes, obtenant une décomposition similaire.

A theorem of Matsushima–Shimura shows that the space of harmonic differential forms on the quotient of products of upper half planes under the action of certain groups, when the quotient is compact, is the direct sum of two subspaces called the universal and cuspidal subspaces. We generalize this result to compact quotients of products of upper half planes and upper half spaces (hyperbolic three spaces) under the action of certain groups to obtain a similar decomposition.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2017.04.016
Amod Agashe 1 ; Lydia Eldredge 1

1 Florida State University, Department of Mathematics, Tallahassee, FL, 32306, United States 
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Amod Agashe; Lydia Eldredge. On the cohomology groups of certain quotients of products of upper half planes and upper half spaces. Comptes Rendus. Mathématique, Volume 355 (2017) no. 9, pp. 937-941. doi : 10.1016/j.crma.2017.04.016. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.04.016/

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[3] H. Darmon Rational Points on Modular Elliptic Curves, CBMS Reg. Conf. Ser. Math., vol. 101, 2004 published for the Conference Board of the Mathematical Sciences, Washington, DC. MR 2020572 (2004k:11103)

[4] E. Freitag Hilbert Modular Forms, Springer-Verlag, Berlin, 1990 MR 1050763 (91c:11025)

[5] Y. Matsushima; G. Shimura On the cohomology groups attached to certain vector valued differential forms on the product of the upper half planes, Ann. of Math. (2), Volume 78 (1963), pp. 417-449 MR 0155340 (27 #5274)

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