On the ranks of elliptic curves in families of quadratic twists over number fields

Jung-Jo Lee

Czechoslovak Mathematical Journal (2014)

  • Volume: 64, Issue: 4, page 1003-1018
  • ISSN: 0011-4642

Abstract

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A conjecture due to Honda predicts that given any abelian variety over a number field K , all of its quadratic twists (or twists of a fixed order in general) have bounded Mordell-Weil rank. About 15 years ago, Rubin and Silverberg obtained an analytic criterion for Honda’s conjecture for a family of quadratic twists of an elliptic curve defined over the field of rational numbers. In this paper, we consider this problem over number fields. We will prove that the existence of a uniform upper bound for the ranks of elliptic curves in this family is equivalent to the convergence of a certain infinite series. This result extends the work of Rubin and Silverberg over ℚ .

How to cite

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Lee, Jung-Jo. "On the ranks of elliptic curves in families of quadratic twists over number fields." Czechoslovak Mathematical Journal 64.4 (2014): 1003-1018. <http://eudml.org/doc/269844>.

@article{Lee2014,
abstract = {A conjecture due to Honda predicts that given any abelian variety over a number field $K$, all of its quadratic twists (or twists of a fixed order in general) have bounded Mordell-Weil rank. About 15 years ago, Rubin and Silverberg obtained an analytic criterion for Honda’s conjecture for a family of quadratic twists of an elliptic curve defined over the field of rational numbers. In this paper, we consider this problem over number fields. We will prove that the existence of a uniform upper bound for the ranks of elliptic curves in this family is equivalent to the convergence of a certain infinite series. This result extends the work of Rubin and Silverberg over $\mathbb \{Q\}$.},
author = {Lee, Jung-Jo},
journal = {Czechoslovak Mathematical Journal},
keywords = {elliptic curve; rank; quadratic twist; rank of elliptic curves; quadratic twists; canonical height},
language = {eng},
number = {4},
pages = {1003-1018},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the ranks of elliptic curves in families of quadratic twists over number fields},
url = {http://eudml.org/doc/269844},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Lee, Jung-Jo
TI - On the ranks of elliptic curves in families of quadratic twists over number fields
JO - Czechoslovak Mathematical Journal
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 4
SP - 1003
EP - 1018
AB - A conjecture due to Honda predicts that given any abelian variety over a number field $K$, all of its quadratic twists (or twists of a fixed order in general) have bounded Mordell-Weil rank. About 15 years ago, Rubin and Silverberg obtained an analytic criterion for Honda’s conjecture for a family of quadratic twists of an elliptic curve defined over the field of rational numbers. In this paper, we consider this problem over number fields. We will prove that the existence of a uniform upper bound for the ranks of elliptic curves in this family is equivalent to the convergence of a certain infinite series. This result extends the work of Rubin and Silverberg over $\mathbb {Q}$.
LA - eng
KW - elliptic curve; rank; quadratic twist; rank of elliptic curves; quadratic twists; canonical height
UR - http://eudml.org/doc/269844
ER -

References

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  10. Rubin, K., Silverberg, A., 10.1090/S0273-0979-02-00952-7, Bull. Am. Math. Soc., New Ser. 39 (2002), 455-474. (2002) Zbl1052.11039MR1920278DOI10.1090/S0273-0979-02-00952-7
  11. Rubin, K., Silverberg, A., 10.1080/10586458.2000.10504661, Exp. Math. 9 (2000), 583-590. (2000) Zbl0959.11023MR1806293DOI10.1080/10586458.2000.10504661
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  13. Silverman, J. H., 10.1007/978-0-387-09494-6, Graduate Texts in Mathematics 106 Springer, New York (2009). (2009) MR2514094DOI10.1007/978-0-387-09494-6
  14. Silverman, J. H., 10.1016/j.jnt.2003.07.001, J. Number Theory 104 (2004), 353-372. (2004) Zbl1053.11052MR2029512DOI10.1016/j.jnt.2003.07.001
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