Skip to main content
SpringerLink
Log in

On CM abelian varieties over imaginary quadratic fields

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract.

In this paper, we associate canonically to every imaginary quadratic field K= one or two isogenous classes of CM (complex multiplication) abelian varieties over K, depending on whether D is odd or even (D≠4). These abelian varieties are characterized as of smallest dimension and smallest conductor, and such that the abelian varieties themselves descend to . When D is odd or divisible by 8, they are the scalar restriction of ‘canonical’ elliptic curves first studied by Gross and Rohrlich. We prove that these abelian varieties have the striking property that the vanishing order of their L-function at the center is dictated by the root number of the associated Hecke character. We also prove that the smallest dimension of a CM abelian variety over K is exactly the ideal class number of K and classify when a CM abelian variety over K has the smallest dimension.

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. Gross, B.: Arithmetic on elliptic curves with complex multiplication info LNM 776. Springer-Verlag, 1980

  2. Gross, B.: Minimal models for elliptic curves with complex multiplication. Compos. Math. 45, 155–164 (1982)

    MathSciNet  MATH  Google Scholar 

  3. Gross, B., Zagier, D.: Heegner points and derivative of L-series. Invent. Math. 84, 225–320 (1986)

    MathSciNet  MATH  Google Scholar 

  4. Kolyvagin, V.A., Yu.Logachev, D.: Finiteness of the Shafarevich-Tate group and the group of rational points for some modular abelian varieties (Russian). Algebra i Analiz 1(5), 171–196 (1989); Translation in Leningrad Math. J. 1(5), 1229–1253 (1990)

    MATH  Google Scholar 

  5. Chunlei Liu, Lanju Xu: The critical order of certain Hecke L-functions of imaginary quadratic fields. Preprint, 2002

  6. Montgomery, H., Rohrlich, D.: On the L-functions of canonical Hecke characters of imaginary quadratic fields II. Duke Math. J. 49, 937–942 (1982)

    MathSciNet  MATH  Google Scholar 

  7. Miller, S., Yang, T.H.: Non-vanishing of the central derivative of canonical Hecke L-functions. Math. Res. Lett. 7, 263–277 (2000)

    MathSciNet  MATH  Google Scholar 

  8. Nakamura, T.: On abelian varieties associated with elliptic curves with complex multiplication. Acta Arith. 97, 379–385 (2001)

    MathSciNet  MATH  Google Scholar 

  9. Rohrlich, D.: Galois conjugacy of unramified twists of Hecke characters. Duke Math. J. 47, 695–703 (1980)

    MathSciNet  MATH  Google Scholar 

  10. Rohrlich, D.: Root numbers of Hecke L-functions of CM fields. Am. J. Math. 104, 517–543 (1982)

    MathSciNet  MATH  Google Scholar 

  11. Rohrlich, D.: The non-vanishing of certain Hecke L-functions at the center critical strip. Duke Math. J 47, 223–232 (1980)

    MathSciNet  MATH  Google Scholar 

  12. Rohrlich, D.: On the L-functions of canonical Hecke characters of imaginary quadratic fields. Duke Math. J 47, 547–557 (1980)

    MathSciNet  MATH  Google Scholar 

  13. Shimura, G.: On the zeta-function of an abelian variety with complex multiplication. Ann. Math. (2) 94, 504–533 (1971)

    Google Scholar 

  14. Shimura, G.: Abelian varieties with complex multiplication and modular functions. Princeton Univ. Press, 1998

  15. Yang, T.H.: Eigenfunctions of the Weil representation of unitary groups of one variable. Trans. AMS 350, 2393–2407 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  16. Yang, T.H.: Local densities of 2-adic quadratic forms. To appear in J. number theory, 2003, p. 58

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Wisconsin Madison, WI 53706, USA

    Tonghai Yang

Authors
  1. Tonghai Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tonghai Yang.

Additional information

Mathematics Subject Classification (1991): 11G05, 11M20, 14H52

Partially supported by a NSF grant DMS-0302043

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yang, T. On CM abelian varieties over imaginary quadratic fields. Math. Ann. 329, 87–117 (2004). https://doi.org/10.1007/s00208-004-0511-8

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-004-0511-8

Keywords

Search

Navigation