Mathematische Annalen

, Volume 356, Issue 4, pp 1247–1282 | Cite as

Heegner cycles and higher weight specializations of big Heegner points

  • Francesc Castella


Let \({\mathbf{{f}}}\) be a \(p\)-ordinary Hida family of tame level \(N\), and let \(K\) be an imaginary quadratic field satisfying the Heegner hypothesis relative to \(N\). By taking a compatible sequence of twisted Kummer images of CM points over the tower of modular curves of level \(\Gamma _0(N)\cap \Gamma _1(p^s)\), Howard has constructed a canonical class \(\mathfrak{Z }\) in the cohomology of a self-dual twist of the big Galois representation associated to \({\mathbf{{f}}}\). If a \(p\)-ordinary eigenform \(f\) on \(\Gamma _0(N)\) of weight \(k>2\) is the specialization of \({\mathbf{{f}}}\) at \(\nu \), one thus obtains from \(\mathfrak{Z }_{\nu }\) a higher weight generalization of the Kummer images of Heegner points. In this paper we relate the classes \(\mathfrak{Z }_{\nu }\) to the étale Abel-Jacobi images of Heegner cycles when \(p\) splits in \(K\).


Elliptic Curve Absolute Galois Group Heegner Point Imaginary Quadratic Field Selmer Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



It is a pleasure to thank my advisor, Prof. Henri Darmon, for suggesting that I work on this problem, and for sharing with me some of his wonderful mathematical insights. I thank both him and Adrian Iovita for critically listening to me while the results in this paper were being developed, and also Jan Nekovář and Victor Rotger for encouragement and helpful correspondence. It is a pleasure to acknowledge the debt that this work owes to Ben Howard, especially for pointing out an error in an early version of this paper, and for providing several helpful comments and corrections. Finally, I am very thankful to an anonymous referee whose valuable comments and suggestions had a considerable impact on the final form of this paper.


  1. 1.
    Bertolini, Massimo, Darmon, Henri, Prasanna, Kartik: Generalised Heegner cycles and \(p\)-adic Rankin \(L\)-series. to appear in Duke Math. Journal.Google Scholar
  2. 2.
    Breuil, Christophe, Emerton, Matthew: Représentations \(p\)-adiques ordinaires de \({\rm GL}_2({\bf Q}_p)\) et compatibilité local-global. Astérisque 331, 255–315 (2010)MathSciNetGoogle Scholar
  3. 3.
    Bloch, Spencer, Kato, Kazuya: \(L\)-functions and Tamagawa numbers of motives. In: The Grothendieck Festschrift, Vol. I, volume 86 of Progr. Math., pages 333–400. Birkhäuser Boston, Boston, MA, (1990)Google Scholar
  4. 4.
    Buzzard, Kevin: Analytic continuation of overconvergent eigenforms. J. Amer. Math. Soc. 16(1): 29–55 (electronic) (2003)Google Scholar
  5. 5.
    Castella, Francesc: \(p\)-adic \(L\)-functions and the \(p\)-adic variation of Heegner points. preprint, (2012)Google Scholar
  6. 6.
    Coleman, Robert F.: Reciprocity laws on curves. Compositio Math. 72(2), 205–235 (1989)MathSciNetMATHGoogle Scholar
  7. 7.
    Coleman, Robert F.: A \(p\)-adic inner product on elliptic modular forms. In: Barsotti Symposium in Algebraic Geometry (Abano Terme, 1991), volume 15 of Perspect. Math., pages 125–151. Academic Press, San Diego, CA, (1994)Google Scholar
  8. 8.
    Coleman, Robert F.: A \(p\)-adic Shimura isomorphism and \(p\)-adic periods of modular forms. In: \(p\)-adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, MA, 1991), volume 165 of Contemp. Math., pages 21–51. Amer. Math. Soc., Providence, RI, (1994)Google Scholar
  9. 9.
    Coleman, Robert F.: Classical and overconvergent modular forms. Invent. Math. 124(1–3), 215–241 (1996)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Coleman, Robert F.: Classical and overconvergent modular forms of higher level. J. Théor. Nombres Bordeaux 9(2), 395–403 (1997)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Coleman, Robert F.: \(p\)-adic Banach spaces and families of modular forms. Invent. Math. 127(3), 417–479 (1997)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Faltings, Gerd: Crystalline cohomology and \(p\)-adic Galois-representations. In Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), pages 25–80. Johns Hopkins Univ. Press, Baltimore, MD, (1989)Google Scholar
  13. 13.
    Faltings, Gerd: Almost étale extensions. Astérisque, (279):185–270, 2002. Cohomologies \(p\)-adiques et applications arithmétiques, IIGoogle Scholar
  14. 14.
    Gouvêa, Fernando Q.: Arithmetic of \(p\) -adic modular forms, volume 1304 of Lecture Notes in Mathematics. Springer-Verlag, Berlin (1988)Google Scholar
  15. 15.
    Gross, Benedict H.: Heegner points on \(X_0(N)\). In: Modular forms (Durham, 1983), Ellis Horwood Ser. Math. Appl.: Statist. Oper. Res., pages 87–105. Horwood, Chichester, (1984)Google Scholar
  16. 16.
    Hida, Haruzo: Iwasawa modules attached to congruences of cusp forms. Ann. Sci. École Norm. Sup. (4) 19(2), 231–273 (1986)MathSciNetMATHGoogle Scholar
  17. 17.
    Hyodo, Osamu., Kato, Kazuya.: Semi-stable reduction and crystalline cohomology with logarithmic poles. Astérisque, (223):221–268, 1994. Périodes \(p\)-adiques (Bures-sur-Yvette, 1988)Google Scholar
  18. 18.
    Howard, Benjamin: Central derivatives of \(L\)-functions in Hida families. Math. Ann. 339(4), 803–818 (2007)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Howard, Benjamin: Variation of Heegner points in Hida families. Invent. Math. 167(1), 91–128 (2007)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Howard, Benjamin: Twisted Gross-Zagier theorems. Canad. J. Math. 61(4), 828–887 (2009)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Katz, Nicholas M.: \(p\)-adic properties of modular schemes and modular forms. In: Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), pages 69–190. Lecture Notes in Mathematics, Vol. 350. Springer, Berlin, (1973)Google Scholar
  22. 22.
    Katz, Nicholas M., Mazur, Barry: Arithmetic moduli of elliptic curves, volume 108 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ (1985)Google Scholar
  23. 23.
    Mazur, B., Tilouine, J.: Représentations galoisiennes, différentelles de Käller at “conjectures principales”. Publ. Math. Inst. Hautes Études Sci. 71, 65–103 (1990)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Mazur, B., Wiles, A.: On \(p\)-adic analytic families of Galois representations. Compositio Math. 59(2), 231–264 (1986)MathSciNetMATHGoogle Scholar
  25. 25.
    Nekovář, Jan: Kolyvagin’s method for Chow groups of Kuga-Sato varieties. Invent. Math. 107(1), 99–125 (1992)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Nekovář, Jan.: On \(p\)-adic height pairings. In: Séminaire de Théorie des Nombres, Paris, 1990–91, volume 108 of Progr. Math., pages 127–202. Birkhäuser Boston, Boston, MA, (1993)Google Scholar
  27. 27.
    Nekovář, Jan: On the \(p\)-adic height of Heegner cycles. Math. Ann. 302(4), 609–686 (1995)Google Scholar
  28. 28.
    Nekovář, Jan.: \(p\)-adic Abel-Jacobi maps and \(p\)-adic heights. In The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), volume 24 of CRM Proc. Lecture Notes, pages 367–379. Amer. Math. Soc., Providence, RI, (2000)Google Scholar
  29. 29.
    Nekovář, Jan, Plater, Andrew: On the parity of ranks of Selmer groups. Asian J. Math. 4(2), 437–497 (2000)Google Scholar
  30. 30.
    Ochiai, Tadashi: A generalization of the Coleman map for Hida deformations. Amer. J. Math. 125(4), 849–892 (2003)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Ochiai, Tadashi: On the two-variable Iwasawa main conjecture. Compos. Math. 142(5), 1157–1200 (2006)MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Rubin, Karl.: Euler systems, volume 147 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 2000. Hermann Weyl Lectures. The Institute for Advanced StudyGoogle Scholar
  33. 33.
    Tsuji, Takeshi: \(p\)-adic étale cohomology and crystalline cohomology in the semi-stable reduction case. Invent. Math. 137(2), 233–411 (1999)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Wiles, A.: On ordinary \(\lambda \)-adic representations associated to modular forms. Invent.Math. 94(3), 529–573 (1988)MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Zhang, Shouwu: Heights of Heegner cycles and derivatives of \(L\)-series. Invent. Math. 130(1), 99–152 (1997)MathSciNetMATHCrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMcGill UniversityMontrealCanada

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