Duke Mathematical Journal

Stark-Heegner points on elliptic curves defined over imaginary quadratic fields

Mak Trifković

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Let E be an elliptic curve defined over an imaginary quadratic field F of class number 1. No systematic construction of global points on such an E is known. In this article, we present a p-adic analytic construction of points on E, which we conjecture to be global, defined over ring class fields of a suitable relative quadratic extension K/F. The construction follows ideas of Darmon to produce an analog of Heegner points, which is especially interesting since none of the geometry of modular parametrizations extends to this setting. We present some computational evidence for our construction

Article information

Source
Duke Math. J., Volume 135, Number 3 (2006), 415-453.

Dates
First available in Project Euclid: 10 November 2006

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1163170198

Digital Object Identifier
doi:10.1215/S0012-7094-06-13531-7

Mathematical Reviews number (MathSciNet)
MR2272972

Zentralblatt MATH identifier
1111.14025

Subjects
Primary: 14H52: Elliptic curves [See also 11G05, 11G07, 14Kxx] 14Q05: Curves
Secondary: 11R37: Class field theory 11G15: Complex multiplication and moduli of abelian varieties [See also 14K22]

Citation

Trifković, Mak. Stark-Heegner points on elliptic curves defined over imaginary quadratic fields. Duke Math. J. 135 (2006), no. 3, 415--453. doi:10.1215/S0012-7094-06-13531-7. https://projecteuclid.org/euclid.dmj/1163170198


Export citation

References

  • M. Bertolini and H. Darmon, The rationality of Stark-Heegner points over genus fields of real quadratic fields, to appear in Ann. of Math. (2).
  • C. Breuil, B. Conrad, F. Diamond, and R. Taylor, On the modularity of elliptic curves over Q: Wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001), 843--939.
  • H. Cohen, A Course in Computational Algebraic Number Theory, Grad. Texts in Math. 138, Springer, Berlin, 1993.
  • J. Cremona, Hyperbolic tessellations, modular symbols, and elliptic curves over complex quadratic fields, Compositio Math. 51 (1984), 275--324.
  • J. Cremona and E. Whitley, Periods of cusp forms and elliptic curves over imaginary quadratic fields, Math. Comp. 62 (1994), 407--429.
  • H. Darmon, Integration on $\cal H_p\,\times\,\cal H$ and arithmetic applications, Ann. of Math. (2) 154 (2001), 589--639.
  • H. Darmon and R. Pollack, The efficient calculation of Stark-Heegner points via overconvergent modular symbols, to appear in Israel J. Math.
  • M. Greenberg, Lifting modular symbols of non-critical slope, preprint, 2005.
  • R. Greenberg and G. Stevens, ``On the conjecture of Mazur, Tate, and Teitelbaum'' in $p$-adic Monodromy and the Birch and Swinnerton-Dyer Conjecture, (Boston, 1991), Contemp. Math. 165, Amer. Math. Soc., Providence, 1994, 183, --211.
  • B. H. Gross and D. B. Zagier, Heegner points and derivatives of $L$-series, Invent. Math. 84 (1986), 225, --320.
  • V. A. Kolyvagin, Finiteness of $E(Q)$ and \sha$(E,Q)$ for a subclass of Weil curves (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 52, no. 3 (1998), 522--540.; English translation in Math. USSR-Izv. 32 (1989), 523--541.
  • P. KurčAnov, The cohomology of discrete groups and Dirichlet series that are related to Jacquet-Langlands cusp forms (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 42, no. 3 (1978), 588--601.; English translation in Math. USSR-Izv. 12 (1978), 543--555.
  • D. Lemelin, Mazur-Tate type conjectures for elliptic curves defined over quadratic imaginary fields, Ph.D. dissertation, McGill University, Montréal, 2001.
  • D. E. Rohrlich, Galois theory, elliptic curves, and root numbers, Compositio Math. 100 (1996), 311--349.
  • D. Simon, PARI, www.math.unicaen.fr/$\sim$simon/
  • G. Stevens, Overconvergent modular symbols, unpublished notes.
  • R. Taylor and A. Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), 553--572.
  • J. T. Teitelbaum, Values of $p$-adic $L$-functions and a $p$-adic Poisson kernel, Invent. Math. 101 (1990), 395--410.
  • A. Wiles, Modular elliptic curves and Fermat's last theorem, Ann. of Math. (2) 141 (1995), 443--551.
  • S. Zhang, Heights of Heegner points on Shimura curves, Ann. of Math. (2) 153 (2001), 27--147.