A Markov model for Selmer ranks in families of twists (Q2874604)

In this paper, the authors study the distribution of 2-Selmer ranks in the family of quadratic twists of an elliptic curve \(E\) over an arbitrary number field \(K\). Under the assumption that \(\mathrm{Gal}(K(E[2])/K) \cong S_3\), they show that the density (counted in an appropriate way) of twists with Selmer rank \(r\) exists for all positive integers \(r\), and is given via an equilibrium distribution, depending only on a single parameter (the disparity, i.e., the difference between \(1/2\) and the density of the twists with even 2-Selmer rank), of a certain Markov process that is itself independent of \(E\) and \(K\).