Elliptic curves with all quadratic twists of positive rank

Details Elliptic curves with all quadratic twists of positive rank Elliptic curves with all quadratic twists of positive rank

2008-02-27

arxiv.org/abs/0802.4027v3

Mathematics

Number Theory

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

Scientific paper

10.4064/aa137-2-7

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.

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