On the \(p\)-adic height of Heegner cycles
\textit{B. Gross} and \textit{D. Zagier} [Invent. Math. 84, 225-320 (1986; Zbl 0608.14019)]\ proved a formula which relates the first derivative of the \(L\)-function of a modular form \(f\) of weight 2 on \(\Gamma_0 (N)\) and the Néron-Tate height of a Heegner point on the \(f\)-part of the Jacobian \({\mathcal J}_0 (N)\). A \(p\)-adic version of this formula was later found by \textit{B. Perrin-Riou} [Invent. Math. 89, 455-510 (1987; Zbl 0645.14010)]. Now let \(f\) be a modular form of even weight \(2r> 2\). The author proves, under suitable hypotheses, a \(p\)-adic version of the Gross and Zagier formula in this context (Theorem A): the first derivative of a \(p\)-adic \(L\)-function at the central point is related to the \(p\)-adic height of a Heegner cycle. The proof of Theorem A closely follows Perrin-Riou's article. Some arguments are however different, mainly because an archimedean analogue of the Gross and Zagier formula for higher weight modular forms is still lacking. Ideas of \textit{J.-L. Brylinski}'s article [Duke Math. J. 59, 1-26 (1989; Zbl 0702.14016)]\ are of great importance here. The author uses Theorem A and his own generalization of Kolyvagin's method of Euler systems to modular forms of even weight to obtain a (weak) form of the conjecture of Beilinson and Bloch in this situation (Theorem B).
scientific article; zbMATH DE number 800583
scientific article