Disparity in Selmer ranks of quadratic twists of elliptic curves

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 $\mathbf{F}_p$-representations of the absolute Galois group of $K$ by characters of order $p$.

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Authors

Zev Klagsbrun

Department of Mathematics
University of Wisconsin - Madison
480 Lincoln Dr.
Madison, WI 53706

Barry Mazur

Department of Mathematics
Harvard University
One Oxford Street
Cambridge, MA 02138-2901

Karl Rubin

Department of Mathematics
University of California Irvine
Irvine, CA 92697-3875