A Markov model for Selmer ranks in families of twists

TL;DR

This paper models the distribution of 2-Selmer ranks in quadratic twists of elliptic curves over number fields using a Markov process, revealing a universal distribution dependent on a single parameter under specific Galois conditions.

Contribution

It introduces a Markov model to describe Selmer rank distributions in quadratic twists, generalizing to p-Selmer ranks and establishing a universal distribution framework.

Findings

Density of twists with a given Selmer rank exists for all positive integers.

The distribution depends only on a single parameter called the 'disparity'.

Results are independent of the specific elliptic curve or number field under certain Galois assumptions.

Abstract

We 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 Gal(K(E[2])/K) = S_3 we show that the density (counted in a non-standard 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'), of a certain Markov process that is itself independent of E and K. 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.