The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point

TL;DR

This paper proves that the average size of the 2-Selmer group of Jacobians of hyperelliptic curves with a rational Weierstrass point is 3, implying bounds on the average rank and rational points for most such curves.

Contribution

It establishes the average size of the 2-Selmer group for these curves and derives consequences for their Mordell-Weil rank and rational points, using new methods.

Findings

Average 2-Selmer group size is 3

Average Mordell-Weil rank is at most 1.5

Most hyperelliptic curves of genus ≥ 3 have fewer than 20 rational points

Abstract

We prove that when all hyperelliptic curves of genus $n\geq 1$ having a rational Weierstrass point are ordered by height, the average size of the 2-Selmer group of their Jacobians is equal to 3. It follows that (the limsup of) the average rank of the Mordell-Weil group of their Jacobians is at most 3/2. The method of Chabauty can then be used to obtain an effective bound on the number of rational points on most of these hyperelliptic curves; for example, we show that a majority of hyperelliptic curves of genus $n\geq 3$ with a rational Weierstrass point have fewer than 20 rational points.