Darmon, Henri; Rotger, Victor 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.) (English. French summary) Zbl 1356.11039 Ann. Sci. Éc. Norm. Supér. (4) 47, No. 4, 779-832 (2014). Summary: 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. Cited in 5 ReviewsCited in 48 Documents MSC: 11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 11F85 \(p\)-adic theory, local fields 11G05 Elliptic curves over global fields 11G35 Varieties over global fields 14G25 Global ground fields in algebraic geometry 11S40 Zeta functions and \(L\)-functions Keywords:Gross-Kudla-Schoen cycle; Garrett-Rankin \(p\)-adic \(L\)-function; \(p\)-adic Abel-Jacobi map; Chow group; Coleman integration PDFBibTeX XMLCite \textit{H. Darmon} and \textit{V. Rotger}, Ann. Sci. Éc. Norm. Supér. (4) 47, No. 4, 779--832 (2014; Zbl 1356.11039) Full Text: DOI Link