Advertisement

Manuscripta Mathematica

, Volume 142, Issue 3–4, pp 383–390 | Cite as

Elliptic curves of rank zero satisfying the p-part of the Birch and Swinnerton-Dyer conjecture

  • Dongho ByeonEmail author
  • Nayoung Kim
Article
  • 181 Downloads

Abstract

Let \({p \in \{3,5,7\}}\) and \({E/\mathbb{Q}}\) an elliptic curve with a rational point P of order p. Let D be a square-free integer and E D the D-quadratic twist of E. Vatsal (Duke Math J 98:397–419, 1999) found some conditions such that E D has (analytic) rank zero and Frey (Can J Math 40:649–665, 1988) found some conditions such that the p-Selmer group of E D is trivial. In this paper, we will consider a family of E D satisfying both of the conditions of Vatsal and Frey and show that the p-part of the Birch and Swinnerton-Dyer conjecture is true for these elliptic curves E D . As a corollary we will show that there are infinitely many elliptic curves \({E/\mathbb{Q}}\) such that for a positive portion of D, E D has rank zero and satisfies the 3-part of the Birch and Swinnerton-Dyer conjecture. Previously only a finite number of such curves were known, due to James (J Number Theory 15:199–202, 1982).

Mathematics Subject Classification

11G05 11G40 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Antoniadis J.A., Bungert M., Frey G.: Properties of twists of elliptic curves. J. Reine Angew. Math. 405, 1–28 (1990)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Brüdern J., Kawada K., Wooley T.D.: Additive representation in thin sequence II: the binary Goldbach problem. Mathematica 47, 117–125 (2000)zbMATHGoogle Scholar
  3. 3.
    Byeon D., Jeon D., Kim C.H.: Rank-one quadratic twists of an infinite family of elliptic curves. J. Reine Angew. Math. 633, 67–76 (2009)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Cassels J.W.S.: Arithmetic on curves of genus 1 VIII. On conjectures of Birch and Swinnerton-Dyer. J. Reine Angew. Math. 217, 180–199 (1965)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Davenport H., Heilbronn H.: On the density of disciminants of cubic fields II. Proc. Roy. Soc. Lond. Ser. A 322, 405–420 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dummigan N.: Rational torsion on optimal curves. Int. J. Number Theory 1, 513–531 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Frey G.: On the Selmer group of twists of elliptic curves with \({\mathbb{Q}}\) -rational torsion points. Can. J. Math. 40, 649–665 (1988)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Greenberg R., Vatsal V.: On the Iwasawa invariants of elliptic curves. Invent. Math. 142, 17–63 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hadano T.: Elliptic curves with a torsion point. Nagoya Math. J. 66, 99–108 (1977)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Iwaniec, H.: Primes represented by quadratic polynomials in two variables. Collection of articles dedicated to Carl Ludwig Siegel on the occasion of his seventy-fifth birthday, V. Acta Arith. 24, 435–459 (1973/74)Google Scholar
  11. 11.
    James K.: Elliptic curves satisfying the Birch and Swinnerton-Dyer conjecture mod 3. J. Number Theory 76, 16–21 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    James K., Ono K.: Selmer groups of quadratic twists of elliptic curves. Math. Ann. 314, 1–17 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kenku M.: On the number of \({\mathbb{Q}}\) -isomorphism classes of elliptic curves in each \({\mathbb{Q}}\) -isogeny class. J. Number Theory 15, 199–202 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Milne J.S.: Elliptic curves. Book Surge Publishers, Charleston, SC (2006)zbMATHGoogle Scholar
  15. 15.
    Nakagawa J., Horie K.: Elliptic curves with no torsion points. Proc. Am. Math. Soc. 104, 20–25 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Pal V.: Periods of quadratic twists of elliptic curves. Proc. Am. Math. Soc. 140, 1513–1525 (2012)CrossRefzbMATHGoogle Scholar
  17. 17.
    Perelli A.: Goldbach numbers represented by polynomials. Rev. Mat. Iberoamericana 12, 477–490 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Silverman, J.H.: The arithmetic of elliptic curves. Graduate Texts in Mathematics 106. Springer, New York (1985)Google Scholar
  19. 19.
    Tate, J.: Algorithm for determining the type of a singular fiber in an elliptic pencil. In: Modular Functions of One Variable IV, Lecture Notes in Mathematics 476, 33–52. Springer, New York (1975)Google Scholar
  20. 20.
    Vatsal V.: Canonical periods and congruence formulae. Duke Math. J. 98, 397–419 (1999)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of MathematicsSeoul National UniversitySeoulRepublic of Korea

Personalised recommendations