Skip to main content
SpringerLink
Log in
Book cover

Number Theory, Analysis and Geometry pp 215–241Cite as

On the local divisibility of Heegner points

  • Chapter
  • First Online:

Abstract

We relate the local ℓ-divisibility of a Heegner point on an elliptic curve of conductor N, at a prime p which is inert in the imaginary quadratic field, to the first ℓ-descent on a related abelian variety of level Np.

”To Serge Lang”

Mathematics Subject Classification (2010): 11G05, 11607, 11G10

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor, On the modularity of elliptic curves over \(\mathbb{Q}\) : wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001), no. 4, 843–939 (electronic).

    Google Scholar 

  2. M. Bertolini and H. Darmon, Euler systems and Jochnowitz congruences, Amer. J. Math. 121 (1999), no. 2, 259–281.

    Article  MathSciNet  MATH  Google Scholar 

  3. M. Bertolini and H. Darmon, Iwasawa’s main conjecture for elliptic curves over anticyclotomic Z p -extensions, Annals of Mathematics 162 (2005), 1–64.

    Article  MathSciNet  MATH  Google Scholar 

  4. B. J. Birch and W. Kuyk (eds.), Modular functions of one variable. IV, Springer-Verlag, Berlin, 1975, Lecture Notes in Mathematics, Vol. 476.

    Google Scholar 

  5. Siegfried Bosch and Werner Lütkebohmert, Degenerating abelian varieties, Topology 30 (1991), no. 4, 653–698.

    Article  MathSciNet  MATH  Google Scholar 

  6. B. J. Birch and H. P. F. Swinnerton-Dyer, Elliptic curves and modular functions, Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Springer, Berlin, 1975, pp. 2–32. Lecture Notes in Math., Vol. 476.

    Google Scholar 

  7. Henri Carayol, Sur les représentations l-adiques associées aux formes modulaires de Hilbert, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 3, 409–468.

    Google Scholar 

  8. Henri Darmon, Fred Diamond, and Richard Taylor, Fermat’s last theorem, Current developments in mathematics, 1995 (Cambridge, MA), Internat. Press, Cambridge, MA, 1994, pp. 1–154.

    Google Scholar 

  9. Fred Diamond, The refined conjecture of Serre, Elliptic curves, modular forms, Fermat’s last theorem (Hong Kong, 1993), Ser. Number Theory, I, Internat. Press, Cambridge, MA, 1995, pp. 22–37.

    Google Scholar 

  10. P. Deligne, Valeurs de fonctions L et périodes d’intégrales, Automorphic forms, and representations, and L-functions, Proceedings of Symposia in Pure Mathematics, volume 33, part 2 (1979) 313–346.

    Google Scholar 

  11. P. Deligne and M. Rapoport, Les schémas de modules de courbes elliptiques, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Springer, Berlin, 1973, pp. 143–316. Lecture Notes in Math., Vol. 349.

    Google Scholar 

  12. G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), no. 3, 349–366.

    Article  MathSciNet  MATH  Google Scholar 

  13. Benedict H. Gross, Kolyvagin’s work on modular elliptic curves, L-functions and arithmetic (Durham, 1989), London Math. Soc. Lecture Note Ser., vol. 153, Cambridge Univ. Press, Cambridge, pp. 235–256.

    Google Scholar 

  14. to3em______________________________________________________________________ , Heegner points on X 0 (N), Modular forms (Durham, 1983), Ellis Horwood Ser. Math. Appl.: Statist. Oper. Res., Horwood, Chichester, 1984, pp. 87–105.

    Google Scholar 

  15. Benedict H. Gross and Don B. Zagier, Heegner points and derivatives of L-series, Invent. Math. 84 (1986), no. 2, 225–320.

    Article  MathSciNet  MATH  Google Scholar 

  16. Serge Lang, Algebraic groups over finite fields, Amer. J. Math. 78 (1956), 555–563.

    Article  MathSciNet  MATH  Google Scholar 

  17. Barry Mazur, Rational points of abelian varieties with values in towers of number fields, Invent. Math. 18 (1972), 183–266.

    Article  MathSciNet  MATH  Google Scholar 

  18. James S. Milne, Étale cohomology, Princeton Mathematical Series, vol. 33, Princeton University Press, Princeton, N.J., 1980.

    Google Scholar 

  19. J. S. Milne, Arithmetic duality theorems, Perspectives in Mathematics, vol. 1, Academic Press Inc., Boston, MA, 1986.

    Google Scholar 

  20. Michel Raynaud, Schémas en groupes de type \((p,\ldots,p)\), Bull. Soc. Math. France 102 (1974), 241–280.

    MathSciNet  MATH  Google Scholar 

  21. Kenneth A. Ribet, Raising the levels of modular representations, Séminaire de Théorie des Nombres, Paris 1987–88, Progr. Math., vol. 81, Birkhäuser Boston, Boston, MA, 1990, pp. 259–271.

    Google Scholar 

  22. Jean-Pierre Serre, Sur les représentations modulaires de degré 2 de \(\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\), Duke Math. J. 54 (1987), no. 1, 179–230.

    Article  MathSciNet  MATH  Google Scholar 

  23. Schémas en groupes. I: Propriétés générales des schémas en groupes, Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3). Dirigé par M. Demazure et A. Grothendieck. Lecture Notes in Mathematics, Vol. 151, Springer-Verlag, Berlin, 1970.

    Google Scholar 

  24. Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, vol. 11, Princeton University Press, Princeton, NJ, 1994, Reprint of the 1971 original, Kanô Memorial Lectures, 1.

    Google Scholar 

  25. ___ , Abelian varieties with complex multiplication and modular functions, Princeton Mathematical Series, vol. 46, Princeton University Press, Princeton, NJ, 1998.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Harvard University, Cambridge, MA, 02138, USA

    Benedict H. Gross

  2. Department of Mathematics, Hood College, 401 Rosemont Avenue, Frederick, Maryland, 21701, USA

    James A. Parson

Authors
  1. Benedict H. Gross
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. James A. Parson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Benedict H. Gross .

Editor information

Editors and Affiliations

  1. , Department of Mathematics, Columbia University, Broadway 2990, New York, 10027, New York, USA

    Dorian Goldfeld

  2. , Department of Mathematics, City College of New York, New York, 10031, New York, USA

    Jay Jorgenson

  3. , Department of Mathematics, Yale University, Hillhouse Ave. 10, New Haven, 06520, Connecticut, USA

    Peter Jones

  4. , Department of Mathematics, California Institute of Technology, E. California Blvd. 1200, Pasadena, 91125, California, USA

    Dinakar Ramakrishnan

  5. , Department of Mathematics, University of California at Berkeley, Evans Hall 925, Berkeley, 94720-3840, California, USA

    Kenneth Ribet

  6. , Department of Mathematics, Harvard University, Science Center/One Oxford Street, Cambridge, 02138, Massachusetts, USA

    John Tate

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Gross, B.H., Parson, J.A. (2012). On the local divisibility of Heegner points. In: Goldfeld, D., Jorgenson, J., Jones, P., Ramakrishnan, D., Ribet, K., Tate, J. (eds) Number Theory, Analysis and Geometry. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-1260-1_11

Download citation

Publish with us

Policies and ethics

Search

Navigation