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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
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.

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
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