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arXiv · 1307.7030

The distribution of 2-Selmer ranks of quadratic twists of elliptic curves with partial two-torsion

Zev Klagsbrun·Robert J. Lemke Oliver
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Abstract

This paper presents a new result concerning the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve over an arbitrary number field K with a single point of order two that does not have a cyclic 4-isogeny defined over its two-division field. We prove that at least half of all the quadratic twists of such an elliptic curve have arbitrarily large 2-Selmer rank, showing that the distribution of 2-Selmer ranks in the quadratic twist family of such an elliptic curve differs from the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve having either no rational two-torsion or full rational two-torsion.

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Zev Klagsbrun, Robert J. Lemke Oliver. 2013-07-26. The distribution of 2-Selmer ranks of quadratic twists of elliptic curves with partial two-torsion. https://doi.org/10.1112/s0025579315000121

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