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

Darmon, Henri; Rotger, Victor

doi:10.24033/asens.2227

Geodesic

Parcourir par

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.

Voir la notice de l'article provenant de la source Numdam

Abstract (VO)
Résumé

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.

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.

Publié le : 2014-05-27

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.
Mots-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},
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     zbl = {1356.11039},
     language = {en},
     url = {http://geodesic.mathdoc.fr/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://geodesic.mathdoc.fr/articles/10.24033/asens.2227/

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