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On the \(p\)-adic height of Heegner cycles. (English) Zbl 0841.11025
B. Gross and 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 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 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).

MSC:
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
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