Geodesic
On the ranks of elliptic curves in families of quadratic twists over number fields
Czechoslovak Mathematical Journal, Tome 64 (2014) no. 4, pp. 1003-1018
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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 $\mathbb Q$.
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 $\mathbb Q$.
DOI : 10.1007/s10587-014-0149-y
Classification : 11G05
Keywords: elliptic curve; rank; quadratic twist 
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Lee, Jung-Jo. On the ranks of elliptic curves in families of quadratic twists over number fields. Czechoslovak Mathematical Journal, Tome 64 (2014) no. 4, pp. 1003-1018. doi: 10.1007/s10587-014-0149-y

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