p-converse to a theorem of Gross–Zagier, Kolyvagin and Rubin

  • Ashay A. BurungaleEmail author
  • Ye Tian


Let E be a CM elliptic curve over the rationals and \(p>3\) a good ordinary prime for E. We show that
$$\begin{aligned} {\mathrm {corank}}_{{\mathbb {Z}}_{p}} {\mathrm {Sel}}_{p^{\infty }}(E_{/{\mathbb {Q}}})=1 \implies {\mathrm {ord}}_{s=1}L(s,E_{/{\mathbb {Q}}})=1 \end{aligned}$$
for the \(p^{\infty }\)-Selmer group \({\mathrm {Sel}}_{p^{\infty }}(E_{/{\mathbb {Q}}})\) and the complex L-function \(L(s,E_{/{\mathbb {Q}}})\). In particular, the Tate–Shafarevich group \(\hbox {X}(E_{/{\mathbb {Q}}})\) is finite whenever \({\mathrm {corank}}_{{\mathbb {Z}}_{p}} {\mathrm {Sel}}_{p^{\infty }}(E_{/{\mathbb {Q}}})=1\). We also prove an analogous p-converse for CM abelian varieties arising from weight two elliptic CM modular forms with trivial central character. For non-CM elliptic curves over the rationals, first general results towards such a p-converse theorem are independently due to Skinner (A converse to a theorem of Gross, Zagier and Kolyvagin, arXiv:1405.7294, 2014) and Zhang (Camb J Math 2(2):191–253, 2014).



We are grateful to Karl Rubin, Chris Skinner and Wei Zhang for inspiring conversations and encouragement. We are also grateful to Francesc Castella, Laurent Clozel, Haruzo Hida, Chandrashekhar Khare and Peter Sarnak for insightful conversations. We thank Adebisi Agboola, Ben Howard and Xin Wan for helpful correspondence. We also thank Li Cai, John Coates, Henri Darmon, Daniel Disegni, Ralph Greenberg, Yukako Kezuka, Shinichi Kobayashi, Chao Li, Richard Taylor and Shou-Wu Zhang for instructive conversations about the topic. We are grateful to organisers of the program ‘Euler Systems and Special Values of L-functions’ held at CIB Lausanne during July–December 2017 for stimulating atmosphere. Part of this work was done while the authors were visiting CIB during an early part of the program. The first named author is also grateful to MCM Beijing for persistent warm hospitality. The article was conceived in Beijing during the summer of 2017. Finally, we are indebted to the referee. The current form of the article owes much to the perceptive comments and incisive suggestions.


  1. 1.
    Aflalo, E., Nekovář, J.: Non-triviality of CM points in ring class field towers, With an appendix by Christophe Cornut. Israel J. Math. 175, 225–284 (2010)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Agboola, A., Howard, B.: Anticyclotomic Iwasawa theory of CM elliptic curves. Ann. Inst. Fourier (Grenoble) 56(4), 1001–1048 (2006)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Arnold, T.: Anticyclotomic main conjectures for CM modular forms. J. Reine Angew. Math. 606, 41–78 (2007)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bertolini, M., Darmon, H., Prasanna, K.: \(p\)-adic Rankin L-series and rational points on CM elliptic curves. Pac. J. Math. 260(2), 261–303 (2012)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bertolini, M., Darmon, H., Prasanna, K.: Generalized Heegner cycles and \(p\)-adic Rankin L-series. Duke Math. J. 162(6), 1033–1148 (2013)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bump, D., Friedberg, S., Hoffstein, J.: Nonvanishing theorems for L-functions of modular forms and their derivatives. Invent. Math. 102(3), 543–618 (1990)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Burungale, A.: On the \(\mu \)-invariant of the cyclotomic derivative of a Katz \(p\)-adic L-function. J. Inst. Math. Jussieu 14(1), 131–148 (2015)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Burungale, A.: Non-triviality of Generalised Heegner Cycles Over Anticyclotomic Towers: A Survey, p-Adic Aspects Of Modular Forms, pp. 279–306. World Scientific Publishing, Hackensack (2016)zbMATHGoogle Scholar
  9. 9.
    Burungale, A.: On the non-triviality of the \(p\)-adic Abel-Jacobi image of generalised Heegner cycles modulo \(p\), II: Shimura curves. J. Inst. Math. Jussieu 16(1), 189–222 (2017)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Burungale, A.: On the non-triviality of the \(p\)-adic Abel-Jacobi image of generalised Heegner cycles modulo \(p\), I: modular curves. J. Alg. Geom. (to appear, 2020)Google Scholar
  11. 11.
    Burungale, A., Disegni, D.: On the non-vanishing of p-adic heights on CM abelian varieties, and the arithmetic of Katz p-adic L-functions (2018). preprint, arXiv:1803.09268
  12. 12.
    Burungale, A., Castella, F., Kim, C.-H.: A proof of Perrin-Riou’s Heegner point main conjecture (2019). preprint, arXiv:1908.09512
  13. 13.
    Cai, L., Shu, J., Tian, Y.: Explicit Gross–Zagier and Waldspurger formulae. Algebra Number Theory 8(10), 2523–2572 (2014)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Castella, F.: \(p\)-adic heights of Heegner points and Beilinson–Flach classes. J. Lond. Math. Soc. 96(1), 156–180 (2017)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Castella, F., Wan, X.: Perrin-Riou’s main conjecture for elliptic curves at supersingular primes (2016). preprint, arXiv:1607.02019
  16. 16.
    Coates, J., Wiles, A.: On the conjecture of Birch and Swinnerton-Dyer. Invent. Math. 39(3), 223–251 (1977)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Cornut, C.: Mazur’s conjecture on higher Heegner points. Invent. Math. 148(3), 495–523 (2002)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Disegni, D.: The \(p\)-adic Gross-Zagier formula on Shimura curves. Compos. Math. 153(10), 1987–2074 (2017)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Gross, B., Zagier, D.: Heegner points and derivatives of L-series. Invent. Math. 84(2), 225–320 (1986)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Hida, H., Tilouine, J.: Anticyclotomic Katz \(p\)-adic L-functions and congruence modules. Ann. Sci. Ecole Norm. Sup. (4) 26(2), 189–259 (1993)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Hida, H.: Hilbert Modular Forms and Iwasawa Theory. Oxford University Press, Oxford (2006)zbMATHGoogle Scholar
  22. 22.
    Hida, H.: The Iwasawa \(\mu \)-invariant of \(p\)-adic Hecke L-functions. Ann. of Math. (2) 172(1), 41–137 (2010)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Howard, B.: The Heegner point Kolyvagin system. Compos. Math. 140(6), 1439–1472 (2004)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Hsieh, M.-L.: On the \(\mu \)-invariant of anticyclotomic \(p\)-adic L-functions for CM fields. J. Reine Angew. Math. 688, 67–100 (2014)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Kato, K.: \(p\)-adic Hodge theory and values of zeta functions of modular forms. Cohomologies \(p\)-adiques et applications arithmétiques. III. Astérisque 295, 117–209 (2004)zbMATHGoogle Scholar
  26. 26.
    Katz, N.M.: \(p\)-adic L-functions for CM fields. Invent. Math. 49(3), 199–297 (1978)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Kobayashi, S.: The \(p\)-adic Gross-Zagier formula for elliptic curves at supersingular primes. Invent. Math. 191(3), 527–629 (2013)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Euler systems, The Grothendieck Festschrift, Vol. II, 435–483, Progress in Mathathematics, 87. Birkhser Boston, Boston (1990)Google Scholar
  29. 29.
    Li, Y., Liu, Y., Tian, Y.: On the Birch and Swinnerton–Dyer conjecture for CM elliptic curves over \({{\mathbb{Q}}}\) (2016). preprint, arXiv:1605.01481
  30. 30.
    Mazur, B., Rubin, K., Silverberg, A.: Twisting commutative algebraic groups. J. Algebra 314(1), 419–438 (2007)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Nekovář, J.: On the parity of ranks of Selmer groups. II. C. R. Acad. Sci. Paris S I Math. 332(2), 99–104 (2001)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Nekovář, J.: \(p\)-adic Abel-Jacobi Maps and \(p\)-adic Heights, The Arithmetic And Geometry Of Algebraic Cycles (Banff, AB, 1998), 367–379, CRM Proc. Lecture Notes, 24, Amer. Math. Soc., Providence, RI (2000)Google Scholar
  33. 33.
    Nekovář, J.: Selmer complexes. Astérisque No. 310, viii+559 (2006)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Nekovář, J.: The Euler system method for CM points on Shimura curves, L-functions and Galois representations, 471–547, London Math. Soc. Lecture Note Ser., 320, Cambridge University Press, Cambridge, (2007)Google Scholar
  35. 35.
    Perrin-Riou, B.: Fonctions \(L\) \(p\)-adiques, théorie d’Iwasawa et points de Heegner. Bull. Soc. Math. France 115(4), 399–456 (1987)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Rohrlich, D.: On L-functions of elliptic curves and anticyclotomic towers. Invent. Math. 75(3), 383–408 (1984)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Rubin, K.: Tate-Shafarevich groups and L-functions of elliptic curves with complex multiplication. Invent. Math. 89(3), 527–559 (1987)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Rubin, K.: The “main conjectures” of Iwasawa theory for imaginary quadratic fields. Invent. Math. 103(1), 25–68 (1991)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Rubin, K.: \(p\)-adic L-functions and rational points on elliptic curves with complex multiplication. Invent. Math. 107(2), 323–350 (1992)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Rubin, K.: \(p \)-adic variants of the Birch and Swinnerton-Dyer conjecture for elliptic curves with complex multiplication, \(p\)-adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, MA, 1991), 71–80, Contemp. Math., 165, Amer. Math. Soc., Providence, RI (1994)Google Scholar
  41. 41.
    Saito, H.: On Tunnell’s formula for characters of \({{\rm GL}}(2)\). Compositio Math. 85(1), 99–108 (1993)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Skinner, C., Urban, E.: The Iwasawa main conjectures for \({{\rm GL}}_2\). Invent. Math. 195(1), 1–277 (2014)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Skinner, C.: A converse to a theorem of Gross, Zagier and Kolyvagin (2014). preprint, arXiv:1405.7294
  44. 44.
    Tian, Y.: Congruent numbers and Heegner points. Camb. J. Math. 2(1), 117–161 (2014)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Tunnell, J.: Local \(\epsilon \)-factors and characters of \({{\rm GL}}(2)\). Am. J. Math. 105(6), 1277–1307 (1983)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Vatsal, V.: Special values of anticyclotomic L-functions. Duke Math J. 116, 219–261 (2003)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Wan, X.: Iwasawa main conjecture for Rankin–Selberg \(p\)-adic L-functions (2014). preprint, arXiv:1408.4044
  48. 48.
    Wan, X.: Heegner point Kolyvagin system and Iwasawa main conjecture.
  49. 49.
    Yuan, X., Zhang, S.-W., Zhang, W.: The Gross–Zagier formula on Shimura curves. Ann. Math. Stud. 184, viii+272 (2013)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Zhang, W.: Selmer groups and the indivisibility of Heegner points. Camb. J. Math. 2(2), 191–253 (2014)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.California Institute of TechnologyPasadenaUSA
  2. 2.School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijingChina
  3. 3.MCM, HLM, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina

Personalised recommendations