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Generalized Heegner cycles and \(p\)-adic Rankin \(L\)-series. With an appendix by Brian Conrad. (English) Zbl 1302.11043

Summary: This article studies a distinguished collection of so-called generalized Heegner cycles in the product of a Kuga-Sato variety with a power of a CM elliptic curve. Its main result is a \(p\)-adic analogue of the Gross-Zagier formula which relates the images of generalized Heegner cycles under the \(p\)-adic Abel-Jacobi map to the special values of certain \(p\)-adic Rankin \(L\)-series at critical points that lie outside their range of classical interpolation.

MSC:

11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11G35 Varieties over global fields
11G15 Complex multiplication and moduli of abelian varieties
11G05 Elliptic curves over global fields
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