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

Title:

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

Creator:

Lee, Jung-Jo

Identifier:

https://cdk.lib.cas.cz/client/handle/uuid:ffaf9c58-bc0e-42c7-9f69-a01432102756
uuid:ffaf9c58-bc0e-42c7-9f69-a01432102756

Subject:

elliptic curve, rank, and quadratic twist

Type:

model:article and TEXT

Format:

bez média and svazek

Description:

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 $\mathbb Q$.

Language:

English

Rights:

http://creativecommons.org/publicdomain/mark/1.0/
policy:public

Coverage:

1003-1018

Source:

Czechoslovak Mathematical Journal | 2014 Volume:64 | Number:4

Harvested from:

CDK

Metadata only:

false