Eisenstein cohomology and ratios of critical values of Rankin–Selberg <i>L</i>-functions

Günter Harder; A. Raghuram

doi:10.1016/j.crma.2011.06.013

Number Theory

Eisenstein cohomology and ratios of critical values of Rankin–Selberg L-functions
[Cohomologie dʼEisenstein et rapports de valeurs critiques des fonctions L de Rankin–Selberg]

Présenté par : Jean-Pierre Serre

Günter Harder ¹ ; A. Raghuram ²

¹ Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany
² Department of Mathematics, Oklahoma State University, 401 Mathematical Sciences, Stillwater, OK 74078, USA

Comptes Rendus. Mathématique, Volume 349 (2011) no. 13-14, pp. 719-724.

Résumé
Abstract

Cette Note annonce des résultats sur la cohomologie dʼEisenstein de rang 1 de ${GL}_{N}$ , avec $N ⩾ 3$ un entier impair, et donne des théorèmes dʼalgébricité pour les rapports de valeurs critiques successives de certaines fonctions L de Rankin–Selberg pour ${GL}_{n} \times {GL}_{n^{'}}$ lorsque n est pair et $n^{'}$ est impair.

This is an announcement of results on rank-one Eisenstein cohomology of ${GL}_{N}$ , with $N ⩾ 3$ an odd integer, and algebraicity theorems for ratios of successive critical values of certain Rankin–Selberg L-functions for ${GL}_{n} \times {GL}_{n^{'}}$ when n is even and $n^{'}$ is odd

Reçu le : 2011-05-03
Accepté le : 2011-06-10
Publié le : 2011-07-01

DOI : 10.1016/j.crma.2011.06.013

Affiliations des auteurs :

Günter Harder ¹ ; A. Raghuram ²

¹ Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany
² Department of Mathematics, Oklahoma State University, 401 Mathematical Sciences, Stillwater, OK 74078, USA

@article{CRMATH_2011__349_13-14_719_0,
     author = {G\"unter Harder and A. Raghuram},
     title = {Eisenstein cohomology and ratios of critical values of {Rankin{\textendash}Selberg} {\protect\emph{L}-functions}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {719--724},
     publisher = {Elsevier},
     volume = {349},
     number = {13-14},
     year = {2011},
     doi = {10.1016/j.crma.2011.06.013},
     language = {en},
}

TY  - JOUR
AU  - Günter Harder
AU  - A. Raghuram
TI  - Eisenstein cohomology and ratios of critical values of Rankin–Selberg L-functions
JO  - Comptes Rendus. Mathématique
PY  - 2011
SP  - 719
EP  - 724
VL  - 349
IS  - 13-14
PB  - Elsevier
DO  - 10.1016/j.crma.2011.06.013
LA  - en
ID  - CRMATH_2011__349_13-14_719_0
ER  -

%0 Journal Article
%A Günter Harder
%A A. Raghuram
%T Eisenstein cohomology and ratios of critical values of Rankin–Selberg L-functions
%J Comptes Rendus. Mathématique
%D 2011
%P 719-724
%V 349
%N 13-14
%I Elsevier
%R 10.1016/j.crma.2011.06.013
%G en
%F CRMATH_2011__349_13-14_719_0

Günter Harder; A. Raghuram. Eisenstein cohomology and ratios of critical values of Rankin–Selberg L-functions. Comptes Rendus. Mathématique, Volume 349 (2011) no. 13-14, pp. 719-724. doi : 10.1016/j.crma.2011.06.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.06.013/

Bibliographie
Cité par

[1] A. Borel; N. Wallach Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, Mathematical Surveys and Monographs, vol. 67, American Mathematical Society, Providence, RI, 2000 (xviii+260 pp)

[2] L. Clozel Motifs et formes automorphes : applications du principe de fonctorialité, Ann Arbor, MI, 1988 (Perspect. Math.), Volume vol. 10, Academic Press, Boston, MA (1990), pp. 77-159

[3] P. Deligne Valeurs de fonctions L et périodes dʼintégrales, Automorphic Forms, Representations and L-Functions, Proc. Sympos. Pure Math. (Proc. Sympos. Pure Math.), Volume vol. XXXIII, Part 2, Amer. Math. Soc., Providence, RI (1979), pp. 313-346 (with an appendix by N. Koblitz and A. Ogus, Oregon State Univ., Corvallis, OR, 1977)

[4] G. Harder Arithmetic aspects of rank one Eisenstein cohomology (V. Srinivas, ed.), Cycles, Motives, Shimura Varieties, Narosa Publishing, 2010 (with an appendix by Don Zagier, Tata Institute of Fundemental Research Studies in Mathematics)

[5] G. Harder Some results on the Eisenstein cohomology of arithmetic subgroups of ${GL}_{n}$ , Luminy-Marseille, 1989 (Lecture Notes in Math.), Volume vol. 1447, Springer, Berlin (1990), pp. 85-153

[6] G. Harder Cohomology of Arithmetic Groups http://www.math.uni-bonn.de/people/harder/Manuscripts/buch/ (book in preparation, preliminary version available at)

[7] Motives (Uwe Jannsen; S. Kleiman; J.-P. Serre, eds.), Proceedings of Symposia in Pure Mathematics, vol. 55, 1994

[8] C. Mœglin Representations of $GL (n)$ over the real field, Edinburgh, 1996 (Proc. Sympos. Pure Math.), Volume vol. 61, Amer. Math. Soc., Providence, RI (1997), pp. 157-166

[9] A. Raghuram On the special values of certain Rankin–Selberg L-functions and applications to odd symmetric power L-functions, International Mathematics Research Notices, IMRN, Volume 2 (2010), pp. 334-372

[10] A. Raghuram, Freydoon Shahidi, On certain period relations for cusp forms on ${GL}_{n}$ , International Mathematics Research Notices, IMRN (2008), Article ID rnn077, 23 pages.

[11] J. Schwermer Eisenstein series and cohomology of arithmetic groups: the generic case, Inventiones Mathematicae, Volume 116 (1994) no. 1–3, pp. 481-511

[12] F. Shahidi Local coefficients as Artin factors for real groups, Duke Math. J., Volume 52 (1985) no. 4, pp. 973-1007

Cité par Sources :

Commentaires - Politique