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Exceptional zero formulae and a conjecture of Perrin-Riou

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Abstract

Let \(A/\mathbf{Q}\) be an elliptic curve with split multiplicative reduction at a prime p. We prove (an analogue of) a conjecture of Perrin-Riou, relating p-adic Beilinson–Kato elements to Heegner points in \(A(\mathbf{Q})\), and a large part of the rank-one case of the Mazur–Tate–Teitelbaum exceptional zero conjecture for the cyclotomic p-adic L-function of A. More generally, let f be the weight-two newform associated with A, let \(f_{\infty }\) be the Hida family of f, and let \(L_{p}(f_{\infty },k,s)\) be the Mazur–Kitagawa two-variable p-adic L-function attached to \(f_{\infty }\). We prove a p-adic Gross–Zagier formula, expressing the quadratic term of the Taylor expansion of \(L_{p}(f_{\infty },k,s)\) at \((k,s)=(2,1)\) as a non-zero rational multiple of the extended height-weight of a Heegner point in \(A(\mathbf{Q})\).

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Notes

  1. Théorème 7 of [22] proves these facts assuming that the residual Galois representation \(\overline{\rho }_{\mathbf {f}}\) of \(\mathbb {T}\) is absolutely irreducible. As pointed out to us by J. Nekovář, loc. cit. also requires \(\overline{\rho }_{\mathbf {f}}\) to be p-distinguished (see [9]). As \(\overline{\rho }_{\mathbf {f}}\cong {}A_{p}\) and \(p\not =2\), this hypothesis is automatically satisfied in our case, by Tate’s theory of p-adic uniformisation.

References

  1. Bertolini, M., Darmon, H.: Heegner points, \(p\)-adic L-functions and the Cerednik–Drinfeld uniformization. Invent. Math. 131, 453–491 (1998)

  2. Bertolini, M., Darmon, H.: Hida families and rational points on elliptic curves. Invent. Math. 168(2), 371–431 (2007)

  3. Bertolini, M., Darmon, H.: Kato’s Euler system and rational points on elliptic curves I: a \(p\)-adic Beilinson formula. Isr. J. Math. 199(1), 163–188 (2014)

  4. Bertolini, M., Darmon, H., Prasanna, K.: Generalized Heegner cycles and p-adic Rankin L-series. With an appendix by Brian Conrad. Duke Math. J. 162(6), 1033–1148 (2013)

  5. Bertrand, D.: Transcendence et lois de groupes algébriques. Séminaire Delange-Pisot-Poitou. Théorie de nombres. 18(1), 1–10 (1976–1977)

  6. Bloch, S., Kato, K.: \(L\)-functions and Tamagawa numbers of motives. In: Cartier, P., Illusie, L., Katz, N., Laumon, G. (eds.) The Grothendieck Festschrift. Modern Birkhäuser Classics (1990)

  7. Barré-Sirieix, K., Diaz, G., Gramain, F., Philibert, G.: Une Preuve de la Conjecture de Mahler-Manin. Invent. Math. 124, 1–9 (1996)

  8. Coleman, R.: Division values in local fields. Invent. Math. 53, 91–116 (1979)

  9. Emerton, M., Pollack, R., Weston, T.: Variation of Iwasawa invariants in Hida families. Invent. Math. 163, 523–580 (2006)

  10. Greenberg, R., Stevens, G.: \(p\)-Adic L-functions and \(p\)-adic periods of modular forms. Invent. Math. 4(111), 407–447 (1993)

  11. Greenberg, R., Vatsal, V.: On the Iwasawa invariants of elliptic curves. Invent. Math. 142, 17–63 (2000)

  12. Gross, B., Zagier, D.: Heegner points and derivatives of \(L\)-series. Invent. Math. 86(2), 225–320 (1986)

  13. Hida, H.: Galois representations into \({GL}_{{2}}(\mathbf{Z}_{p}[\!\![X]\!\!])\) attached to ordinary cusp forms. Invent. Math. 85(3), 545–613 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hida, H.: Iwasawa modules attached to congruences of cusp forms. Ann. Sci. Éc. Norm. Sup. 19(4), 231–273 (1986)

    MathSciNet  MATH  Google Scholar 

  15. Kato, K.: Lectures in the approach to Iwasawa theory for Hasse–Weil L-functions via \(B_{\rm dR}\). In: Arithmetic Algebraic Geometry (Trento 1991). Lecture Notes in Math., vol. 1553, pp. 50–163. Springer, New York (1993)

  16. Kato, K.: \(p\)-Adic Hodge theory and values of zeta functions of modular forms. Astérisque 295, 117–290 (2004)

  17. Kitagawa, K.: On standard \(p\)-adic \(L\)-functions of families of elliptic cusp forms. In: Mazur, B., Stevens, G. (eds.) \(p\)-Adic Monodromy and the Birch and Swinnerton-Dyer Conjecture. American Mathematical Society, Providence (1994)

    Google Scholar 

  18. Kobayashi, S.: The \(p\)-adic Gross–Zagier formula for elliptic curves at supersingular primes. Invent. Math. 191(3), 527–629 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kolyvagin, V.A.: Euler systems. In: The Grothendieck Festschrift, vol. II. Progr. Math., vol. 87. Birkhäuser, Boston (1990)

  20. Loeffler, D., Venjacob, O., Zerbes, S.L.: Local epsilon isomorphisms. Kyoto J. Math. 55(1), 63–127 (2015)

    Article  MathSciNet  Google Scholar 

  21. Mok, C.P.: Heegner points and p-adic L-functions for elliptic curves over certain totally real fields. Comment. Math. Helv. 86(4), 867–945 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mazur, B., Tilouine, J.: Représentations galoisiennes, différentielles de Kähler et conjectures principales. Publ. Math. de l’I.H.E.S. 71, 65–103 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  23. Mazur, B., Tate, J., Teitelbaum, J.: On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer. Invent. Math. 126(1), 1–48 (1986)

    Article  MathSciNet  Google Scholar 

  24. Nekovar, J.: On p-adic height pairings. In: Seminaire de Theorie des Nombres, Paris, 1990–1991. In: Progr. in Math., vol. 108. Birkhäuser, Boston (1993)

  25. Nekovar, J.: Selmer complexes. Astérisque 310 (2006)

  26. Nekovar, J., Plater, A.: On the parity of ranks of Selmer groups. Asian J. Math. 4(2), 437–498 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ochiai, T.: A generalization of the Coleman map for Hida deformations. Am. J. Math. 125(4), 849–892 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ochiai, T.: On the two-variable Iwasawa main conjecture. Compos. Math. 142, 1157–1200 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  29. Ohta, M.: On the \(p\)-adic Eichler–Shimura isomorphism for \(\Lambda \)-adic cusp forms. J. Reine Angew. Math. 463, 49–98 (1995)

    MathSciNet  MATH  Google Scholar 

  30. Perrin-Riou, B.: Points de Heegner et dérivées de fonctions L p-adiques. Invent. Math. 89(3), 455–510 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  31. Perrin-Riou, B.: Thèorie d’Iwasawa et hauteurs \(p\)-adiques. Invent. Math. 109(1), 137–185 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  32. Perrin-Riou, B.: Fonctions \(L\) \(p\)-adiques d’une courbe elliptique et points rationnels. Ann. Inst. Fourier, Grenoble 43(4), 945–995 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  33. Perrin-Riou, B.: Théorie d’Iwasawa des représentations p-adiques sur un corps local. With an appendix by Jean-Marc Fontaine. Invent. Math. 115(1), 81–161 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  34. Rohrlich, D.E.: On L-functions of elliptic curves and cyclotomic towers. Invent. Math. 75(3), 409–423 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  35. Rubin, K.: Abelian varieties, \(p\)-adic heights and derivatives. In: Frey, G., Ritter, J. (eds.) Algebra and Number Theory. de Gruyter (1994)

  36. Rubin, K.: Euler systems and modular elliptic curves. In: Scholl, A., Taylor, R. (eds.) Galois Representations in Arithmetic Algebraic Geometry. London Math. Soc. Lect. Notes, vol. 254. Cambridge University Press, Cambridge (1998)

  37. Rubin, K.: Euler systems. In: Annals of Mathematics Studies, vol. 147. Hermann Weyl Lectures. The Institute for Advanced Study. Princeton University Press, Princeton (2000)

  38. Schneider, P.: \(p\)-Adic height pairings I. Invent. Math. 69(3), 401–409 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  39. Serre, J.P.: Local class field theory. In: Cassels, J.W.S., Frohlich, A. (eds.) Algebraic Number Theory. Academic Press, New York (1967)

  40. Silverman, J.: The Arithmetic of Elliptic Curves. Springer, New York (1986)

  41. Silverman, J.: Advanced Topics in the Arithmetic of Elliptic Curves. Springer, New York (1994)

  42. Venerucci, R.: \(p\)-Adic regulators and \(p\)-adic analytic families of modular forms. Ph.D. thesis, University of Milan (2013)

  43. Venerucci, R.: \(p\)-adic regulators and \(p\)-adic families of modular forms. (2015, preprint)

  44. Wiles, A.: Higher explicit reciprocity laws. Ann. Math. (2) 107(2), 235–254 (1978)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

Much of the work on this article was carried out during my Ph.D. at the University of Milan. It is a pleasure to express my sincere gratitude to my supervisor, Prof. Massimo Bertolini, who constantly encouraged and motivated my work. Every meeting with him has been a source of ideas and enthusiasm; this paper surely originated from and grew up through these meetings. I would like to thank Marco Seveso for a careful reading of the paper and for many interesting discussions related to this work. I am also grateful to the anonymous referee; the current version of the article is greatly inspired by his/her corrections and valuable comments, which helped me to significantly clarify and improve the exposition.

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  1. Fakultät für Mathematik, Mathematikcarrée, Universität Duisburg-Essen, Thea-Leymann-Straße 9, 45127, Essen, Germany

    Rodolfo Venerucci

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Venerucci, R. Exceptional zero formulae and a conjecture of Perrin-Riou. Invent. math. 203, 923–972 (2016). https://doi.org/10.1007/s00222-015-0606-8

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