Skip to main content
SpringerLink
Log in

On the elliptic curve y 2 = x 3 −2rDx and factoring integers

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

Let D = pq be the product of two distinct odd primes. Assuming the parity conjecture, we construct infinitely many r ⩾ 1 such that E 2rD : y 2 = x 3 −2rDx has conjectural rank one and v p (x([k]Q)) ≠ v q (x([k]Q)) for any odd integer k, where Q is the generator of the free part of E(ℚ). Furthermore, under the generalized Riemann hypothesis, the minimal value of r is less than c log4 D for some absolute constant c. As a corollary, one can factor D by computing the generator Q.

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. Bach E, Sorenson J. Explicit bounds for primes in residue classes. Math Comp, 1996, 65: 1717–1735

    Article  MATH  MathSciNet  Google Scholar 

  2. Birch B J, Stephens N M. The parity of the rank of the Mordell-Weil group. Topology, 1966, 5: 295–299

    Article  MATH  MathSciNet  Google Scholar 

  3. Bremner A, Silverman J H, Tzanakis N. Integral points in arithmetic progression on y 2 = x(x 2 − n 2). J Number Theory, 2000, 80: 187–208

    Article  MATH  MathSciNet  Google Scholar 

  4. Burhanuddin I A. Some computational problems motivated by the Birch and Swinnerton-Dyer conjecture. PhD. Thesis. California: University of Southern California, 2007

    Google Scholar 

  5. Dokchitser T. Notes on the parity conjecture. In: Hilbert Modular Forms and Galois Deformations Advanced Courses in Mathematics-CRM Barcelona. Basel: Springer, 2013, 201–249

    Google Scholar 

  6. Edixhoven B. Rational elliptic curves are modular. Asérisque, 2002, 276: 161–188

    MathSciNet  Google Scholar 

  7. Elkies N. Heegner point computations. In: Algorithmic Number Theory. Lecture Notes in Computer Science, vol. 877. Berlin: Springer, 1994, 122–133

    Chapter  Google Scholar 

  8. Gross B, Zagier D. Heegner points and derivatives of L-series. Invent Math, 1986, 84: 225–320

    Article  MATH  MathSciNet  Google Scholar 

  9. Lang S. Conjectured Diophantine estimates on elliptic curves. Progr Math, 1983, 35: 155–171

    Google Scholar 

  10. Siksek S. Infinite descent on elliptic curves. Rocky Mountain J Math, 1995, 25: 1501–1538

    Article  MATH  MathSciNet  Google Scholar 

  11. Silverman J H. The Arithmetic of Elliptic Curves. New York: Springer-Verlag, 1986

    Book  MATH  Google Scholar 

  12. Silverman J H. Advances Topics in the Arithmetic of Elliptic Curves. New York: Springer-Verlag, 1994

    Book  Google Scholar 

  13. Silverman J H. Computing rational points on rank 1 elliptic curves via L-series and canonical heights. Math Comp, 1999, 68: 835–858

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

    XiuMei Li & JinXiang Zeng

Authors
  1. XiuMei Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. JinXiang Zeng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to XiuMei Li.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Li, X., Zeng, J. On the elliptic curve y 2 = x 3 −2rDx and factoring integers. Sci. China Math. 57, 719–728 (2014). https://doi.org/10.1007/s11425-014-4769-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-014-4769-3

Keywords

MSC(2010)

Search

Navigation