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Derivatives of $$p$$-adic $$L$$-functions, Heegner cycles and monodromy modules attached to modular forms. (English) Zbl 1099.11032
Let $$f$$ be an elliptic newform of even weight $$k\geq 4$$ and square free level $$N$$; let $$p$$ be a prime number dividing $$N$$. Assume that $$f$$ corresponds to a modular form on a Shimura curve via the Jacquet-Langlands correspondence. Let $$K$$ be a quadratic imaginary field of class number one such that all prime divisors of $$N$$ inert in $$K$$.
It is known (due to the work of M. Bertolini, H. Darmon, A. Iovita and M. Spiess [Am. J. Math. 124, No. 2, 411–449 (2002; Zbl 1079.11036)]) that the anticyclotomic $$p$$-adic $$L$$-function $$L_p(f/K,s)$$ vanishes at $$s= k/2$$. The authors prove in Section 9 that the derivative of $$L_p(f/K,s)$$ at $$s= k/2$$ can be interpreted as the $$e_f$$-component of the image of a Heegner cycle under a $$p$$-adic Abel-Jacobi map (a $$p$$-adic Gross-Zagier type formula). Let us remark that J. Nikovář [Math. Ann. 302, No. 4, 609–686 (1995; Zbl 0841.11025)] has obtained a $$p$$-adic Gross-Zagier type formula involving $$p$$-adic heights for the cyclotomic $$p$$-adic $$L$$-function attached to $$f$$.
The proof uses $$p$$-adic Hodge theory: the authors explicitly describe the semistable Dieudonné module attached to $$V_p(f)$$ in terms of $$p$$-adic integration (Theorem 5.9). The main ingredient in the proof of this result are a comparison theorem between $$p$$-adic étale cohomology of semistable curves and log-crystalline cohomology (with coefficients) and the description of the log-crystalline cohomology groups.
As an application of Theorem 5.9 they also prove that the $$L_p$$-invariants of Fontaine-Mazur and Teitelbaum attached to $$f$$ are equal (Theorem 6.4).

##### MSC:
 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 11F11 Holomorphic modular forms of integral weight 11G18 Arithmetic aspects of modular and Shimura varieties 11M38 Zeta and $$L$$-functions in characteristic $$p$$
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