Diagonal cycles and Euler systems I:   A $p$-adic Gross-Zagier formula

Darmon, Henri; Rotger, Victor

doi:10.24033/asens.2227

Diagonal cycles and Euler systems I: A

p

-adic Gross-Zagier formula
[Cycles de Gross-Schoen et systèmes d'Euler I : une formule de Gross-Zagier

p

-adique]

Darmon, Henri ; Rotger, Victor

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 47 (2014) no. 4, pp. 779-832.

Résumé
Abstract

Cet article est le premier d'une série consacrée aux cycles de Gross-Kudla-Schoen généralisés appartenant aux groupes de Chow de produits de trois variétés de Kuga-Sato, et aux systèmes d'Euler qui leur sont associés. La série au complet repose sur une variante $p$ -adique de la formule de Gross-Zagier qui relie l'image des cycles de Gross-Kudla-Schoen par l'application d'Abel-Jacobi $p$ -adique aux valeurs spéciales de certaines fonctions $L$ $p$ -adiques attachées à la convolution de Garrett-Rankin de trois familles de Hida de formes modulaires cuspidales. L'objectif principal de cet article est de décrire et de démontrer cette variante.

This article is the first in a series devoted to studying generalised Gross-Kudla-Schoen diagonal cycles in the product of three Kuga-Sato varieties and the Euler system properties of the associated Selmer classes, with special emphasis on their application to the Birch-Swinnerton-Dyer conjecture and the theory of Stark-Heegner points. The basis for the entire study is a $p$ -adic formula of Gross-Zagier type which relates the images of these diagonal cycles under the $p$ -adic Abel-Jacobi map to special values of certain $p$ -adic $L$ -functions attached to the Garrett-Rankin triple convolution of three Hida families of modular forms. The main goal of this article is to describe and prove this formula.

Publié le : 2014-09-24

MR Zbl

DOI : 10.24033/asens.2227

Classification : 11F12, 11G05, 11G35, 11G40
Keywords: Gross-Kudla-Schoen cycle, Garrett-Rankin $p$-adic $L$-function, $p$-adic Abel-Jacobi map, Chow group, Coleman integration.
Mot clés : Cycle de Gross-Kudla-Schoen, fonction $L$ $p$-adique de Garrett-Rankin, application d'Abel-Jacobi $p$-adique, groupe de Chow, intégration de Coleman.

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     author = {Darmon, Henri and Rotger, Victor},
     title = {Diagonal cycles and {Euler} systems {I:}   {A} $p$-adic {Gross-Zagier} formula},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {779--832},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 47},
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     year = {2014},
     doi = {10.24033/asens.2227},
     mrnumber = {3250064},
     zbl = {1356.11039},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/asens.2227/}
}

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Darmon, Henri; Rotger, Victor. Diagonal cycles and Euler systems I:   A $p$-adic Gross-Zagier formula. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 47 (2014) no. 4, pp. 779-832. doi : 10.24033/asens.2227. http://www.numdam.org/articles/10.24033/asens.2227/

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