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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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References:
[1] Berthelot, P.: Cohomologie rigide et cohomologie rigid à supports propre. Preprint · Zbl 0722.14008
[2] Besser, A.: CM cycles over Shimura curves. J. Algebr. Geom. 4, 659–691 (1995) · Zbl 0898.14004
[3] Bertolini, M., Darmon, H.: Heegner points on Mumford-Tate curves. Invent. Math. 126, 413–456 (1996) · Zbl 0882.11034
[4] Bertolini, M., Darmon, H.: Heegner points, p-adic L-functions and the Cerednik-Drinfeld uniformisation. Invent. Math. 131, 453–491 (1998) · Zbl 0899.11029
[5] Bertolini, M., Darmon, H., Iovita, A., Spiess, M.: Teitelbaum’s exceptional zero conjecture in the anticyclotomic setting. Am. J. Math. 124, 411–449 (2002) · Zbl 1079.11036
[6] Bloch, S., Kato, K.: L-functions and Tamagawa numbers of motives. In: The Grothendieck Festschrift I, 333–400, Progr. Math. 108. Boston: Birkhäuser 1993 · Zbl 0768.14001
[7] Boutot, J.F., Carayol, H.: Uniformisation p-adique des courbes de Shimura: les théorèmes de Čerednik et de Drinfeld. Courbes modulaires et courbes de Shimura (Orsay, 1987/1988). Astérisque 196–197 (1991), 7, 45–158 (1992)
[8] Čerednik, I.V.: Uniformisation of algebraic curves by discrete arithmetic subgroups of PGL2(kw) with compact quotient spaces. (Russian) Math. Sb. 100, 59–88 (1976)
[9] Coleman, R.: A p-adic Shimura Isomorphism and p-adic Periods of Modular forms. In: p-Adic Monodromy and the Birch and Swinnerton-Dyer Conjecture (Boston, 1991). Contemp. Math. 165, 21–51 (1994) · Zbl 0838.11033
[10] Coleman, R., Iovita, A.: Hidden structures on Semistable curves. Submitted for publication · Zbl 1251.11047
[11] Deligne, P.: Travaux de Shimura. Séminaire Bourbaki (1970/71), Exp. No. 389. Lect. Notes Math. 244, 123–165. Springer 1971
[12] Deligne, P.: Hodge cycles on abelian varieties. In: Hodge Cycles, Motives and Shimura Varieties (P. Deligne, J. Milne, A. Ogus, K. Shih). Lect. Notes Math. 900, 9–100. Springer 1982 · Zbl 0537.14006
[13] Deninger, C., Murre, J.: Motivic decomposition of abelian schemes and the Fourier transform. J. Reine Angew. Math. 422, 201–219 (1991) · Zbl 0745.14003
[14] Drinfeld, V.G.: Coverings of p-adic symmetric domains. (Russian) Funkts. Anal. Prilozh. 10, 29–40 (1976)
[15] Faltings, G.: Crystalline cohomology and p-adic Galois-representations. In: Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988). Johns Hopkins Univ. Press 1989, 25–80 · Zbl 0805.14008
[16] Faltings, G.: Cristalline cohomology of semistable curve–the \(\mathbb{Q}\)p-theory. J. Algebr. Geom. 6, 1–18 (1997) · Zbl 0883.14007
[17] Fontaine, J.M.: Les corps des périodes p-adiques. Périodes p-adiques. Séminaire de Bures (1988). Astérisque 223, 59–101 (1994)
[18] Fontaine, J.M.: Représentations p-adiques semi-stables. Périodes p-adiques. Séminaire de Bures (1988). Astérisque 223, 113–184 (1994)
[19] Hyodo, O.: H1 g(K,V)=H1 st(K,V). Unpublished manuscript
[20] Jannsen, U.: Continuous Étale Cohomology. Math. Ann. 280, 207–245 (1988) · Zbl 0649.14011
[21] Kato, K.: Logarithmic structures of Fontaine-Illusie. In: Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988). Johns Hopkins Univ. Press 1989, 191–224 · Zbl 0776.14004
[22] Kisin, M.: Potential semi-stability of p-adic étale cohomology. Isr. J. Math. 129, 157–173 (2002) · Zbl 0999.14005
[23] Künnemann, K.: On the Chow Motive of an Abelian Scheme. In: Motives (Seattle, WA, 1991). Proc. Symp. Pure Math. 55.1, 189–205 (1994) · Zbl 0823.14032
[24] Langer, A.: Local points of motives in semistable reduction. Compos. Math. 116, 189–217 (1999) · Zbl 1053.14506
[25] Mazur, B.: On monodromy invariants occurring in global arithmetic and Fontaine’s theory. In: p-Adic Monodromy and the Birch and Swinnerton-Dyer Conjecture (Boston, 1991). Contemp. Math. 165, 1–20 (1994) · Zbl 0846.11039
[26] Mazur, B., Tate, J., Teitelbaum, J.: On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer. Invent. Math. 84, 1–48 (1986) · Zbl 0699.14028
[27] Nekovar, J.: Kolyvagin’s method for Chow groups of Kuga-Sato varieties. Invent. Math. 107, 99–125 (1992) · Zbl 0741.14002
[28] Nekovar, J.: On p-adic height pairings. Séminaire de Théorie des Nombres, Paris, 1990–91, 127–202, Progr. Math. 108. Boston: Birkhäuser 1993
[29] Nekovar, J.: On the p-adic height of Heegner cycles. Math. Ann. 302, 609–686 (1995) · Zbl 0841.11025
[30] Nekovar, J.: p-adic Abel-Jacobi maps and p-adic heights. In: The arithmetic and geometry of algebraic cycles (Banff, AB, 1998). CRM Proc. Lect. Notes, 24, 367–379. Providence, RI: Am. Math. Soc. 2000
[31] Nekovar, J.: Syntomic cohomology and p-adic regulators. Preprint 1998
[32] Ogus, A.: F-Isocrystals and de Rham Cohomology II – Convergent Isocrystals. Duke Math. J. 51, 765–850 (1984) · Zbl 0584.14008
[33] Ribet, K.: Sur les variétés abéliennes à multiplications réelles. C. R. Acad. Sci., Paris, Sér. I, Math. 291, 121–123 (1980) · Zbl 0442.14014
[34] Ribet, K.: On l-adic representations attached to modular forms. II. Glasgow Math. J. 27, 185–194 (1985) · Zbl 0596.10027
[35] Rapoport, M., Zink, T.: Period spaces for p-divisible groups. Annals of Mathematics Studies 141. Princeton: Princeton University Press 1996 · Zbl 0873.14039
[36] Scholl, A.J.: Motives for modular forms. Invent. Math. 100, 419–430 (1990) · Zbl 0760.14002
[37] Scholl, A.J.: Classical motives. In: Motives (Seattle, WA, 1991). Proc. Symp. Pure Math. 55.1, 163–187 (1994) · Zbl 0814.14001
[38] de Shalit, E.: Eichler cohomology and periods of modular forms on p-adic Schottky groups. J. Reine Angew. Math. 400, 3–31 (1989) · Zbl 0674.14031
[39] de Shalit, E.: A formula for the cup product on Mumford curves. Séminaire de Théorie des Nombres, 1987–1988 (Talence, 1987–1988), Exp. No. 47, 10 pp. Univ. Bordeaux I
[40] de Shalit, E.: Differentials of the second kind on Mumford curves. Isr. J. Math. 71, 1–16 (1990) · Zbl 0715.14020
[41] Tsuji, T.: p-adic étale cohomology and crystalline in the semi-stable reduction case. Invent. Math. 137, 233–411 (1999) · Zbl 0945.14008
[42] Teitelbaum, J.: Values of p-adic L-functions and a p-adic Poisson kernel. Invent. Math. 101, 395–410 (1990) · Zbl 0731.11065
[43] Wortmann, S.: Motives attached to quaternionic automorphic forms. Preprint 2003 · Zbl 1052.14022
[44] Yasuda, S.: Euler systems on Shimura curves and finiteness of Ш associated to modular forms. Preprint
[45] Zink, T.: Catiertheorie commutativer formaler Gruppen. Teubner Texte zur Mathematik 68. Leipzig: Teubner 1984
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