Elliptic curves with all quadratic twists of positive rank

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.

Minor corrections; final version, to appear in Acta Arithmetica; 4 pages

Cited by in corpus (3)

• Ranks of twists of elliptic curves and Hilbert's Tenth Problem
• Root numbers and parity of ranks of elliptic curves
• Disparity in Selmer ranks of quadratic twists of elliptic curves