Elliptic curves with all quadratic twists of positive rank

TL;DR

This paper classifies elliptic curves over number fields whose all quadratic twists have positive rank, revealing implications for Goldfeld's conjecture and connecting local behavior with Galois actions.

Contribution

It provides a classification of such elliptic curves based on local properties and Galois actions, highlighting their unique rank behavior.

Findings

Existence of elliptic curves with all quadratic twists of positive rank under BSD conjecture

Classification in terms of local behavior and Galois action

Implication that Goldfeld's conjecture does not extend to number fields

Abstract

We observe that there are elliptic curves over number fields all of whose quadratic twists must have positive rank, assuming the Birch-Swinnerton-Dyer conjecture. We give a classification of such curves in terms of their local behaviour, and characterise them in terms of the Galois action on the Tate module. In particular, their existence shows that Goldfeld's conjecture does not extend directly to elliptic curves over number fields.