Mathematische Annalen

, Volume 370, Issue 1–2, pp 567–628 | Cite as

Heegner cycles and p-adic L-functions

  • Francesc CastellaEmail author
  • Ming-Lun Hsieh


In this paper, we deduce the vanishing of Selmer groups for the Rankin–Selberg convolution of a cusp form with a theta series of higher weight from the nonvanishing of the associated L-value, thus establishing the rank 0 case of the Bloch–Kato conjecture in these cases. Our methods are based on the connection between Heegner cycles and p-adic L-functions, building upon recent work of Bertolini, Darmon and Prasanna, and on an extension of Kolyvagin’s method of Euler systems to the anticyclotomic setting. In the course of the proof, we also obtain a higher weight analogue of Mazur’s conjecture (as proven in weight 2 by Cornut–Vatsal), and as a consequence of our results, we deduce from Nekovář’s work a proof of the parity conjecture in this setting.



Fundamental parts of this paper were written during the visits of the first-named author to the second-named author in Taipei during February 2014 and August 2014; it is a pleasure to thank NCTS and the National Taiwan University for their hospitality and financial support. We would also like to thank Ben Howard, Shinichi Kobayashi and David Loeffler for their comments and enlightening conversations related to this work.


  1. 1.
    Bertolini, M., Darmon, H.: Kolyvagin’s descent and Mordell–Weil groups over ring class fields. J. Reine Angew. Math. 412, 63–74 (1990)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bertolini, M., Darmon, H., Prasanna, K.: Generalized Heegner cycles and \(p\)-adic Rankin \(L\)-series. Duke Math. J. 162(6), 1033–1148 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bloch, S., Kato, K.: \(L\)-functions and Tamagawa numbers of motives. In: The Grothendieck Festschrift, vol. I. Progr. Math. vol. 86, pp. 333–400. Birkhäuser Boston, Boston (1990)Google Scholar
  4. 4.
    Brakočević, M.: Anticyclotomic \(p\)-adic \(L\)-function of central critical Rankin–Selberg \(L\)-value. Int. Math. Res. Not. 2011(21), 4967–5018 (2011)Google Scholar
  5. 5.
    Castella, F.: Heegner cycles and higher weight specializations of big Heegner points. Math. Ann. 356(4), 1247–1282 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Castella, F.: On the \(p\)-adic Variation of Heegner Points. Preprint. arXiv:1410.6591 (2014)
  7. 7.
    Deligne, P.: Formes modulaires et représentations \(l\)-adiques. In: Séminaire Bourbaki. Vol. 1968/69: Exposés 347–363. Lecture Notes in Mathematics, vol. 175, Exp. No. 355, pp. 139–172. Springer, Berlin (1971)Google Scholar
  8. 8.
    Deligne, P.: La conjecture de Weil. II. Inst. Hautes Études Sci. Publ. Math. 52, 137–252 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    de Shalit, E.: Iwasawa theory of elliptic curves with complex multiplication: \(p\)-adic \(L\) functions. In: Perspectives in Mathematics, vol. 3. Academic Press, Inc., Boston (1987)Google Scholar
  10. 10.
    Faltings, G., Chai, C.-L.: Degeneration of abelian varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 22. Springer, Berlin. (1990). With an appendix by David MumfordGoogle Scholar
  11. 11.
    Fontaine, J.-M.: Représentations \(p\)-adiques semi-stables. Astérisque, 223, 113–184 (1994). Périodes \(p\)-adiques (Bures-sur-Yvette, 1988). With an appendix by Pierre ColmezGoogle Scholar
  12. 12.
    Fontaine, J.-M., Perrin-Riou, B.: Autour des conjectures de Bloch et Kato: cohomologie galoisienne et valeurs de fonctions \(L\). In: Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math, vol. 55, pp. 599–706. American Mathematical Society, Providence (1994)Google Scholar
  13. 13.
    Hida, H.: Elementary theory of \(L\)-functions and Eisenstein series. In: London Mathematical Society Student Texts, vol. 26. Cambridge University Press, Cambridge (1993)Google Scholar
  14. 14.
    Hida, H.: \(p\)-adic automorphic forms on Shimura varieties. In: Springer Monographs in Mathematics. Springer, New York (2004)Google Scholar
  15. 15.
    Howard, B.: Variation of Heegner points in Hida families. Invent. Math. 167(1), 91–128 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hsieh, M.-L.: Special values of anticyclotomic Rankin–Selberg \(L\)-functions. Doc. Math. 19, 709–767 (2014)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Jacquet, H.: Automorphic forms on \({\rm GL}(2)\). Part II. In: Lecture Notes in Mathematics, vol. 278, Springer, Berlin (1972)Google Scholar
  18. 18.
    Jacquet, H., Langlands, R.P.: Automorphic forms on \({\rm GL}(2)\). In: Lecture Notes in Mathematics, vol. 114. Springer, Berlin (1970)Google Scholar
  19. 19.
    Katz, N.M.: \(p\)-adic properties of modular schemes and modular forms. Modular functions of one variable, III. In: Proceedings of the International Summer School, University of Antwerp, Antwerp, 1972. Lecture Notes in Mathematics, vol. 350, pp. 69–190. Springer, Berlin (1973)Google Scholar
  20. 20.
    Katz, N.: \(p\)-adic \(L\)-functions for CM fields. Invent. Math. 49(3), 199–297 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Katz, N.: Serre–Tate local moduli, Algebraic surfaces (Orsay, 1976–78). In: Lecture Notes in Math, vol. 868, pp. 138–202. Springer, Berlin (1981)Google Scholar
  22. 22.
    Loeffler, D., Zerbes, S.L.: Iwasawa theory and \(p\)-adic \(L\)-functions over \(\mathbb{Z}_p^2\)-extensions. Int. J. Number Theory 10(8), 2045–2095 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Loeffler, D., Zerbes, S.L.: Rankin–Eisenstein classes in Coleman families. Res. Math. Sci. 3(29), 53 (2016)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Mazur, B.: Modular curves and arithmetic. In: Proceedings of the International Congress of Mathematicians (Warsaw, 1983), vol. 1, no 2, pp. 185–211. PWN, Warsaw (1984)Google Scholar
  25. 25.
    Mumford, D.: Abelian varieties. In: Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Published for the Tata Institute of Fundamental Research, Bombay, 2008. With appendices by C. P. Ramanujam and Yuri Manin, Corrected reprint of the second edition (1974)Google Scholar
  26. 26.
    Nekovář, J.: Kolyvagin’s method for Chow groups of Kuga–Sato varieties. Invent. Math. 107(1), 99–125 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Nekovář, J.: On the \(p\)-adic height of Heegner cycles. Math. Ann. 302(4), 609–686 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Nekovář, J.: \(p\)-adic Abel–Jacobi maps and \(p\)-adic heights. In: The Arithmetic and Geometry of Algebraic Cycles (Banff, AB, 1998). CRM Proceedings and Lecture Notes, vol. 24, American Mathematical Society, Providence, pp. 367–379 (2000)Google Scholar
  29. 29.
    Nekovář, J.: Selmer complexes, Astérisque, no. 310, viii+559 (2006)Google Scholar
  30. 30.
    Nekovář, J.: On the parity of ranks of Selmer groups. III. Doc. Math. 12, 243–274 (2007)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Nekovář, J.: Erratum for “On the parity of ranks of Selmer groups. III” cf. Documenta Math. 12: 243–274 [mr2350290]. Doc. Math. 14(2009), 191–194 (2007)Google Scholar
  32. 32.
    Niziol, W.: On the image of \(p\)-adic regulators. Invent. Math. 127(2), 375–400 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Perrin-Riou, B.: Théorie d’Iwasawa des représentations \(p\)-adiques sur un corps local. Invent. Math. 115(1), 81–161 (1994). With an appendix by Jean-Marc FontaineGoogle Scholar
  34. 34.
    Rubin, K.: Elliptic curves with complex multiplication and the conjecture of Birch and Swinnerton-Dyer. Invent. Math. 64(3), 455–470 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Rubin, K.: Euler systems. In: Annals of Mathematics Studies, Hermann Weyl Lectures, vol. 147. The Institute for Advanced Study, Princeton University Press, Princeton (2000)Google Scholar
  36. 36.
    Scholl, A.J.: Motives for modular forms. Invent. Math. 100(2), 419–430 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Schmidt, R.: Some remarks on local newforms for \({\rm GL}(2)\). J. Ramanujan Math. Soc. 17(2), 115–147 (2002)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Shimura, G.: Abelian Varieties with Complex Multiplication and Modular Functions, Princeton Mathematical Series, vol. 46. Princeton University Press, Princeton, NJ (1998)CrossRefzbMATHGoogle Scholar
  39. 39.
    Shnidman, A.: \(p\)-adic heights of generalized Heegner cycles. Ann. Inst. Fourier (Grenoble) 66(3), 1117–1174 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Serre, J.-P., Tate, J.: Good reduction of abelian varieties. Ann. Math. (2) 88, 492–517 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Tate, J.: Number theoretic background, Automorphic forms, representations and \(L\)-functions. In: Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII. American Mathamatical Society, Providence, pp. 3–26 (1979)Google Scholar
  42. 42.
    Tsuji, T.: \(p\)-adic étale cohomology and crystalline cohomology in the semi-stable reduction case. Invent. Math. 137(2), 233–411 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Wiles, A.: On ordinary \(\lambda \)-adic representations associated to modular forms. Invent. Math. 94(3), 529–573 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Yang, T.: On CM abelian varieties over imaginary quadratic fields. Math. Ann. 329(1), 87–117 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Zhang, S.: Heights of Heegner cycles and derivatives of \(L\)-series. Invent. Math. 130(1), 99–152 (1997)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Mathematics DepartmentPrinceton UniversityPrincetonUSA
  2. 2.Institute of Mathematics, Academia SinicaTaipeiTaiwan
  3. 3.National Center for Theoretic SciencesTaipeiTaiwan
  4. 4.Department of MathematicsNational Taiwan UniversityTaipeiTaiwan

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