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arXiv · 1111.2321

Disparity in Selmer ranks of quadratic twists of elliptic curves

Zev Klagsbrun·Barry Mazur·Karl Rubin
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Abstract

We study the parity of 2-Selmer ranks in the family of quadratic twists of an arbitrary elliptic curve E over an arbitrary number field K. We prove that the fraction of twists (of a given elliptic curve over a fixed number field) having even 2-Selmer rank exists as a stable limit over the family of twists, and we compute this fraction as an explicit product of local factors. We give an example of an elliptic curve E such that as K varies, these fractions are dense in [0, 1]. More generally, our results also apply to p-Selmer ranks of twists of 2-dimensional self-dual F_p-representations of the absolute Galois group of K by characters of order p.

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Zev Klagsbrun, Barry Mazur, Karl Rubin. 2011-11-09. Disparity in Selmer ranks of quadratic twists of elliptic curves. https://doi.org/10.4007/annals.2013.178.1.5

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