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Heegner cycles and higher weight specializations of big Heegner points

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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 \(K\).

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Notes

  1. As defined in [29, (1.3.7)].

  2. Notice the effect of the Tate twist on the filtrations.

  3. Perhaps most commonly denoted by \(W_{2r-2}\); cf. [35] and [27], for example.

  4. Notice that our indices differ from those in [1].

  5. So that \(\mathbb{T }^*\otimes _{\mathbb{I }}F_\nu \cong V_{{\mathbf{{f}}}_\nu }\) for every \(\nu \in \mathcal{X }_\mathrm{arith}(\mathbb{I })\).

  6. As opposed to \(\omega ^{\frac{p-1}{2}}.\)

  7. That \(\Omega _{\nu }^{(\eta )}\), which a priori just lies in \(F_\nu \), is indeed a unit is shown in [31, Prop. 6.4].

References

  1. Bertolini, Massimo, Darmon, Henri, Prasanna, Kartik: Generalised Heegner cycles and \(p\)-adic Rankin \(L\)-series. to appear in Duke Math. Journal.

  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)

    MathSciNet  Google Scholar 

  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)

  4. Buzzard, Kevin: Analytic continuation of overconvergent eigenforms. J. Amer. Math. Soc. 16(1): 29–55 (electronic) (2003)

    Google Scholar 

  5. Castella, Francesc: \(p\)-adic \(L\)-functions and the \(p\)-adic variation of Heegner points. preprint, (2012)

  6. Coleman, Robert F.: Reciprocity laws on curves. Compositio Math. 72(2), 205–235 (1989)

    MathSciNet  MATH  Google Scholar 

  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)

  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)

  9. Coleman, Robert F.: Classical and overconvergent modular forms. Invent. Math. 124(1–3), 215–241 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  10. Coleman, Robert F.: Classical and overconvergent modular forms of higher level. J. Théor. Nombres Bordeaux 9(2), 395–403 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  11. Coleman, Robert F.: \(p\)-adic Banach spaces and families of modular forms. Invent. Math. 127(3), 417–479 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  13. Faltings, Gerd: Almost étale extensions. Astérisque, (279):185–270, 2002. Cohomologies \(p\)-adiques et applications arithmétiques, II

  14. Gouvêa, Fernando Q.: Arithmetic of \(p\) -adic modular forms, volume 1304 of Lecture Notes in Mathematics. Springer-Verlag, Berlin (1988)

  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)

  16. Hida, Haruzo: Iwasawa modules attached to congruences of cusp forms. Ann. Sci. École Norm. Sup. (4) 19(2), 231–273 (1986)

    MathSciNet  MATH  Google Scholar 

  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)

  18. Howard, Benjamin: Central derivatives of \(L\)-functions in Hida families. Math. Ann. 339(4), 803–818 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Howard, Benjamin: Variation of Heegner points in Hida families. Invent. Math. 167(1), 91–128 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Howard, Benjamin: Twisted Gross-Zagier theorems. Canad. J. Math. 61(4), 828–887 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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. Mazur, B., Tilouine, J.: Représentations galoisiennes, différentelles de Käller at “conjectures principales”. Publ. Math. Inst. Hautes Études Sci. 71, 65–103 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  24. Mazur, B., Wiles, A.: On \(p\)-adic analytic families of Galois representations. Compositio Math. 59(2), 231–264 (1986)

    MathSciNet  MATH  Google Scholar 

  25. Nekovář, Jan: Kolyvagin’s method for Chow groups of Kuga-Sato varieties. Invent. Math. 107(1), 99–125 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  27. Nekovář, Jan: On the \(p\)-adic height of Heegner cycles. Math. Ann. 302(4), 609–686 (1995)

  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)

  29. Nekovář, Jan, Plater, Andrew: On the parity of ranks of Selmer groups. Asian J. Math. 4(2), 437–497 (2000)

  30. Ochiai, Tadashi: A generalization of the Coleman map for Hida deformations. Amer. J. Math. 125(4), 849–892 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  31. Ochiai, Tadashi: On the two-variable Iwasawa main conjecture. Compos. Math. 142(5), 1157–1200 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  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 Study

  33. Tsuji, Takeshi: \(p\)-adic étale cohomology and crystalline cohomology in the semi-stable reduction case. Invent. Math. 137(2), 233–411 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  34. Wiles, A.: On ordinary \(\lambda \)-adic representations associated to modular forms. Invent.Math. 94(3), 529–573 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  35. Zhang, Shouwu: Heights of Heegner cycles and derivatives of \(L\)-series. Invent. Math. 130(1), 99–152 (1997)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

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.

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  1. Department of Mathematics and Statistics, McGill University, Montreal, Canada

    Francesc Castella

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Castella, F. Heegner cycles and higher weight specializations of big Heegner points. Math. Ann. 356, 1247–1282 (2013). https://doi.org/10.1007/s00208-012-0871-4

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