Period relations for automorphic induction and applications, I

Lin, Jie

doi:10.1016/j.crma.2014.10.016

Number theory

Period relations for automorphic induction and applications, I
[Relations de périodes pour l'induction automorphe et applications, I]

Présenté par : Gérard Laumon

Lin, Jie ¹

¹ Institut de mathématiques de Jussieu, 4, place Jussieu, 75005 Paris, France

Comptes Rendus. Mathématique, Tome 353 (2015) no. 2, pp. 95-100.

Résumé
Abstract

Soit K un corps quadratique imaginaire. Soit Π (resp. $Π^{'}$ ) une représentation cuspidale régulière algébrique de ${GL}_{n} (A_{K})$ (resp. ${GL}_{n - 1} (A_{K})$ ), qui est, de plus, cohomologique et auto-duale. Si Π est une induction automorphe cyclique d'un caractère de Hecke χ sur un corps CM, on montre les relations entre les périodes automorphes de Π définies par Harris et celles de χ. Par conséquent, on affine une formule de Grobner et Harris pour les valeurs critiques de $L (s, Π \times Π^{'})$ , L étant la fonction de Rankin–Selberg. Cela complète la démonstration d'une version automorphe de la conjecture de Deligne dans certains cas.

Let K be a quadratic imaginary field. Let Π (resp. $Π^{'}$ ) be a regular algebraic cuspidal representation of ${GL}_{n} (A_{K})$ (resp. ${GL}_{n - 1} (A_{K})$ ), which is moreover cohomological and conjugate self-dual. When Π is a cyclic automorphic induction of a Hecke character χ over a CM field, we show relations between automorphic periods of Π defined by Harris and those of χ. Consequently, we refine a formula given by Grobner and Harris for critical values of the Rankin–Selberg L-function $L (s, Π \times Π^{'})$ . This completes the proof of an automorphic version of Deligne's conjecture in certain cases.

Reçu le : 2014-10-03
Accepté le : 2014-10-23
Publié le : 2014-11-11

DOI : 10.1016/j.crma.2014.10.016

Affiliations des auteurs :

Lin, Jie ¹

¹ Institut de mathématiques de Jussieu, 4, place Jussieu, 75005 Paris, France

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Lin, Jie. Period relations for automorphic induction and applications, I. Comptes Rendus. Mathématique, Tome 353 (2015) no. 2, pp. 95-100. doi : 10.1016/j.crma.2014.10.016. http://archive.numdam.org/articles/10.1016/j.crma.2014.10.016/

Bibliographie
Cité par

[1] Arthur, J. An introduction to the trace formula (Arthur, J.; Ellwood, D.; Kottwitz, R., eds.), Harmonic analysis, the trace formula and Shimura varieties, Clay Mathematics Proceedings, American Mathematical Society, Clay Mathematics Institute, 2003 (264 p)

[2] Clozel, L. Rerprésentations galoisiennes associées aux representations automorphes autoduales de Gl(n), Publ. Math. IHÉS, Volume 73 (1991), pp. 97-145

[3] Clozel, L.; Harris, M.; Taylor, R. Automorphy for some l-adic lifts of automorphic mod l Galois representations, Publ. Math. IHÉS, Volume 108 (2008), pp. 1-181

[4] Deligne, P. Valeurs de fonctions L et périodes d'intégrales (Borel, A.; Casselman, W., eds.), Automorphic Forms, Representations and L-Functions, Proceedings of the Symposium in Pure Mathematics of the American Mathematical Society, vol. 33, American Mathematical Society, 1979

[5] Grobner, H.; Harris, M. Whittaker periods, motivic periods, and special values of tensor product of L-functions, 23 August 2013 | arXiv

[6] Harris, M. L-functions of $2 \times 2$ unitary groups and factorization of periods of Hilbert modular forms, J. Amer. Math. Soc., Volume 6 (1993) no. 3, pp. 637-719

[7] Harris, M. L-functions and periods of polarized regular motives, J. Reine Angew. Math. (1997) no. 483, pp. 75-161

[8] Harris, M. The local Langland's conjecture for ${Gl}_{n}$ over a p-adic field, $n < p$ , Invent. Math. (1998) no. 134, pp. 177-210

[9] Harris, M.; Kudla, S.S. The central critical value of the triple product L-functions, Ann. Math. (2), Volume 133 (1991) no. 3, pp. 605-672

[10] Harris, M.; Labesse, J.-P. Conditional base change for unitary groups, Asian J. Math., Volume 8 (2004) no. 4, pp. 653-684

[11] Harris, M.; Taylor, R. The geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies, vol. 151, Princeton University Press, 2001

[12] Minguez, A. Unramified representations of unitary groups (Clozel, L.; Harris, M.; Labesse, J.-P.; Ngô, B.C., eds.), On the Stabilization of the Trace Formula, vol. 1, International Press, 2011

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