Period relations for automorphic induction and applications, I

Jie Lin

doi:10.1016/j.crma.2014.10.016

Number theory

Period relations for automorphic induction and applications, I

Gérard Laumon

Jie Lin ¹

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

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

Abstract (Original language)
Résumé

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.

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.

Article information
Cite this article

Received: 2014-10-03
Accepted: 2014-10-23
Published online: 2014-11-11

DOI: 10.1016/j.crma.2014.10.016

Author's affiliations:

Jie Lin ¹

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

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

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