Skip to main content
SpringerLink
Log in

On the Selmer group attached to a modular form and an algebraic Hecke character

  • Published:
The Ramanujan Journal Aims and scope Submit manuscript

Abstract

We construct an Euler system of generalized Heegner cycles to bound the Selmer group associated to a modular form and an algebraic Hecke character. The main argument is based on Kolyvagin’s method adapted by Bertolini and Darmon (J Reine Angew Math 412:63–74, 1990) and by Nekovář (Invent Math 107(1):99–125, 1992), while the key object of the Euler system, the generalized Heegner cycles were first considered by Bertolini et al. (Duke Math J 162(6):1033–1148, 2013).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Beĭlinson, A.A.: Height pairing between algebraic cycles. In: \(K\)-Theory, Arithmetic and Geometry (Moscow, 1984–1986). Lecture Notes in Mathematics, vol. 1289, pp. 1–25. Springer, Berlin (1987)

  2. Bertolini, M., Darmon, H.: Kolyvagin’s descent and Mordell–Weil groups over ring class fields. J. Reine Angew. Math. 412, 63–74 (1990)

    MathSciNet  MATH  Google Scholar 

  3. Bertolini, M., Darmon, H., Prasanna, K.: Chow-Heegner points on CM elliptic curves and values of \(p\)-adic \(L\)-functions. Int. Math. Res. Not. 2014(3), 745–793 (2014)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Bloch, S., Kato, K.: \(L\)-functions and Tamagawa numbers of motives. In: The Grothendieck Festschrift, vol. I. Progress in Mathematics, vol. 86, pp. 333–400. Birkhäuser, Boston (1990)

  6. Colmez, P.: Fonctions \({L}\) \(p\)-adiques. Séminaire Bourbaki. 41:21–58 (1998–1999)

  7. Deligne, P.: Formes Modulaires et Representations de GL(2). In: Modular Functions of One Variable II. Lecture Notes in Mathematics, vol. 349, pp. 55–105. Springer, Berlin (1973)

  8. Elias, Y.: Kolyvagin’s method for Chow groups of Kuga-Sato varieties over ring class fields. Ann. Math. Qué. 39(2), 147–167 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gross, B.H.: Arithmetic on Elliptic Curves with Complex Multiplication. PhD Thesis, Harvard University (1980)

  10. Gross, B.H.: Heegner points on \(X_0(N)\). In: Modular forms (Durham, 1983). Ellis Horwood Series in Mathematics and its Applications: Statistics, Operational Research, pp. 87–105. Horwood, Chichester (1984)

  11. Gross, B.H.: Kolyvagin’s work on modular elliptic curves. In: \(L\)-Functions and Arithmetic (Durham, 1989). London Mathematical Society Lecture Note Series, vol. 153, pp. 235–256. Cambridge University Press, Cambridge (1991)

  12. Gross, B.H., Zagier, D.B.: Heegner points and derivatives of L-series. Invent. Math. 84(2), 225–320 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  13. Jannsen, U.: Algebraic cycles, K-theory, and extension classes. In: Mixed Motives and Algebraic K-Theory. Lecture Notes in Mathematics, vol. 1400, pp. 57–188. Springer, Berlin (1990)

  14. Kolyvagin, V.A.: Euler Systems. The Grothendieck Festschrift, vol. 2. Birkhäuser, Boston (1990)

    MATH  Google Scholar 

  15. Lang, S.: Fundamentals of Diophantine Geometry. Springer, Berlin (1983)

    Book  MATH  Google Scholar 

  16. Milne, J.S.: Arithmetic Duality Theorems. Perspectives in Mathematics. Academic Press, New York (1986)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. Nekovář, J.: On the p-adic height of Heegner cycles. Math. Ann. 302(1), 609–686 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  19. 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 

  20. Schneider, P.: Introduction to the Beilinson conjectures. Perspect. Math. 4, 1–36 (1988)

    MathSciNet  MATH  Google Scholar 

  21. Schoen, C.: On the computation of the cycle class map for nullhomologous cycles over the algebraic closure of a finite field. Ann. Sci.éc. Norm. Supér. 28(1), 1–50 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  22. Scholl, A.J.: Motives for modular forms. Invent. Math. 100(1), 419–430 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  23. Shnidman, A.: p-Adic heights of generalized Heegner cycles. arXiv:1407.0785v2, pp. 1–34 (2014)

  24. Suzuki, M.: Group Theory. Grundlehren der mathematischen Wissenschaften, vol. 1. Springer, Berlin (1982)

    MATH  Google Scholar 

Download references

Acknowledgements

It is a pleasure to thank my advisor Henri Darmon for numerous discussions of the subject of this article as well as for his suggestions, corrections, and valuable feedback on the writing of this monograph. I am grateful to Olivier Fouquet, Eyal Goren, Ariel Shnidman, and the anonymous referees for many corrections and suggestions. This work was finalized at the Max Planck Institute for Mathematics, and I am thankful for its hospitality.

Author information

Authors and Affiliations

  1. Department of Mathematics, McGill University, Montreal, QC, Canada

    Yara Elias

Authors
  1. Yara Elias
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yara Elias.

Additional information

Supported by a doctoral scholarship of the Fonds Québécois de la Recherche sur la Nature et les Technologies.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Elias, Y. On the Selmer group attached to a modular form and an algebraic Hecke character. Ramanujan J 45, 141–169 (2018). https://doi.org/10.1007/s11139-016-9866-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-016-9866-1

Keywords

Mathematics Subject Classification

Search

Navigation