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

Abstract

This article is the first in a series devoted to studying generalised Gross-KudlaSchoen 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 Lfunctions 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 serie consacree aux cycles de Gross-Kudla-Schoen generalises appartenant aux groupes de Chow de produits de trois varietes de Kuga-Sato, et aux systemes d’Euler qui leur sont associes. La serie 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 speciales de certaines fonctions L p-adiques attachees a la convolution de Garrett-Rankin de trois familles de Hida de formes modulaires cuspidales. L’objectif principal de cet article est de decrire et de demontrer cette variante. MSC: 11F12, 11G05, 11G35, 11G40.