Rank Frequencies for Quadratic Twists of Elliptic Curves

@article{Rubin2001RankFF,
  title={Rank Frequencies for Quadratic Twists of Elliptic Curves},
  author={Karl Rubin and Alice Silverberg},
  journal={Experimental Mathematics},
  year={2001},
  volume={10},
  pages={559-569}
}
We give explicit examples of infinite families of elliptic curves E over Q with (nonconstant) quadratic twists over Q(t) of rank at least 2 and 3. We recover some results announced by Mestre, as well as some additional families. Suppose D is a squarefree integer and let rE(D) denote the rank of the quadratic twist of E by D. We apply results of Stewart and Top to our examples to obtain results of the form #{D : |D| < x, rE(D) ≥ 2} x #{D : |D| < x, rE(D) ≥ 3} x for all sufficiently large x. 
Highly Cited
This paper has 23 citations. REVIEW CITATIONS
14 Citations
9 References
Similar Papers

References

Publications referenced by this paper.
Showing 1-9 of 9 references

Rang de courbes elliptiques d’invariant donné

  • J-F. Mestre
  • C. R. Acad. Sci. Paris 314
  • 1992
Highly Influential
12 Excerpts

How many rational points can a curve have

  • L. Caporaso, J. Harris, B. Mazur
  • The moduli space of curves
  • 1994
Highly Influential
4 Excerpts

Rang de certaines familles de courbes elliptiques d’invariant donné

  • J-F. Mestre
  • C. R. Acad. Sci. Paris 327
  • 1998
3 Excerpts

Top , On ranks of twists of elliptic curves and powerfree values of binary forms

  • J.
  • J . Amer . Math . Soc .
  • 1995

Heights and the " The square - free sieve and the rank of elliptic curves " , specialization map for families of abelian varieties "

  • B. Mazur, J. H. Silverman
  • Math .
  • 1994

Variation of the root number in families of elliptic curves

  • D. Rohrlich
  • Compositio Math. 87
  • 1993
1 Excerpt

Heights and the specialization map for families of abelian varieties

  • J. Silverman
  • J. Reine Angew. Math. 342
  • 1983
1 Excerpt

Similar Papers

Loading similar papers…