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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