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://www.numdam.org/articles/10.1016/j.crma.2014.10.016/

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