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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).

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 \(p\)
Full Text: DOI
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