Distribution of the 2-Selmer rank under twisting

Consider the elliptic curve y 2 = x 3 - x (which is associated to congruent numbers). Heath-Brown described the 2 -Selmer rank distribution of the quadratic twists of this curve over the set of odd squarefree integers, using methods involving computing the moments of the 2-Selmer rank via an analysis of quadratic residue symbols (these appear in a Monsky matrix). Swinnerton-Dyer then investigated the distribution of the 2-Selmer rank under twisting for a wider family (elliptic curves with full 2-torsion and no 4-torsion), using a Markov chain analysis to show the expected distribution. However, his result used an unnatural ordering of the integers, namely by the number of prime factors. This was remedied by Kane, who showed the result under the natural ordering via methodology similar to Heath-Brown’s. More recently, Smith (as part of a larger work) has shown the same result using Swinnerton-Dyer’s method, essentially showing that the input (involving quadratic residue symbols of the prime divisors of the twist factor) to his Markov chain analysis can be shown to have the expected equi-distribution under the natural ordering. We give an exposition of Smith’s work, with an explicit (and effective) error bound. We also discuss the related problem of the 4-rank of quadratic class groups, initially done by Fouvry and Klüners.