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

Czechoslovak Mathematical Journal (2014)

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

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topLee, 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 -

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