Elliptic curves with all quadratic twists of positive rank

Tim Dokchitser; Vladimir Dokchitser

Displaying similar documents to “Elliptic curves with all quadratic twists of positive rank”

Quadratic twists of elliptic curves with small Selmer rank

Sungkon Chang (2010)

Acta Arithmetica

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Elliptic Curves With Complex Multiplication and the Conjecture of Birch and Swinnerton-Dyer.

Karl Rubin (1981)

Inventiones mathematicae

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Ranks of quadratic twists of elliptic curves

Mark Watkins, Stephen Donnelly, Noam D. Elkies, Tom Fisher, Andrew Granville, Nicholas F. Rogers (2014)

Publications mathématiques de Besançon

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We report on a large-scale project to investigate the ranks of elliptic curves in a quadratic twist family, focussing on the congruent number curve. Our methods to exclude candidate curves include 2-Selmer, 4-Selmer, and 8-Selmer tests, the use of the Guinand-Weil explicit formula, and even 3-descent in a couple of cases. We find that rank 6 quadratic twists are reasonably common (though still quite difficult to find), while rank 7 twists seem much more rare. We also describe our inability...

Ranks of elliptic curves in families of quadratic twists.

Rubin, Karl, Silverberg, Alice (2000)

Experimental Mathematics

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Experimental study of the rank of families of elliptic curves over $ℚ$ . (Étude expérimentale du rang de familles de courbes elliptiques sur $ℚ$ .)

Fermigier, Stéfane (1996)

Experimental Mathematics

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The average rank of elliptic curves I. With Appendix "The Explicit formula for elliptic curves over function fields" by Oisín McGuinness.

Armand Brumer, Oisín McGuinness (1992)

Inventiones mathematicae

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Lang-Trotter and Sato-Tate distributions in single and double parametric families of elliptic curves

Min Sha, Igor E. Shparlinski (2015)

Acta Arithmetica

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We obtain new results concerning the Lang-Trotter conjectures on Frobenius traces and Frobenius fields over single and double parametric families of elliptic curves. We also obtain similar results with respect to the Sato-Tate conjecture. In particular, we improve a result of A. C. Cojocaru and the second author (2008) towards the Lang-Trotter conjecture on average for polynomially parameterised families of elliptic curves when the parameter runs through a set of rational numbers of...