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

Harder, Günter; Raghuram, A.

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

Harder, Günter ¹ ; Raghuram, A.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, Tome 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 :

Harder, Günter ¹ ; Raghuram, A. 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 = {Harder, G\"unter and Raghuram, A.},
     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},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2011.06.013/}
}

TY  - JOUR
AU  - Harder, Günter
AU  - Raghuram, A.
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
UR  - http://www.numdam.org/articles/10.1016/j.crma.2011.06.013/
DO  - 10.1016/j.crma.2011.06.013
LA  - en
ID  - CRMATH_2011__349_13-14_719_0
ER  -

%0 Journal Article
%A Harder, Günter
%A Raghuram, A.
%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
%U http://www.numdam.org/articles/10.1016/j.crma.2011.06.013/
%R 10.1016/j.crma.2011.06.013
%G en
%F CRMATH_2011__349_13-14_719_0

Harder, Günter; Raghuram, A. Eisenstein cohomology and ratios of critical values of Rankin–Selberg L-functions. Comptes Rendus. Mathématique, Tome 349 (2011) no. 13-14, pp. 719-724. doi : 10.1016/j.crma.2011.06.013. http://www.numdam.org/articles/10.1016/j.crma.2011.06.013/

Bibliographie
Cité par

[1] Borel, A.; Wallach, N. 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] Clozel, L. 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] Deligne, P. 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] Harder, G. Arithmetic aspects of rank one Eisenstein cohomology (Srinivas, V., ed.), Cycles, Motives, Shimura Varieties, Narosa Publishing, 2010 (with an appendix by Don Zagier, Tata Institute of Fundemental Research Studies in Mathematics)

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

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

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

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

Cité par Sources :