On the average number of 2-Selmer elements of elliptic curves over $\mathbb{F}_q(X)$ with two marked points

TL;DR

This paper investigates the average size of 2-Selmer groups of elliptic curves over function fields with two marked points, establishing that the average exists and equals 12, with equidistribution results for associated torsors.

Contribution

It proves the existence and exact value of the average 2-Selmer group size for a specific family of elliptic curves over function fields, and analyzes the distribution of related torsors.

Findings

Average 2-Selmer group size is 12.

2-Selmer elements become equidistributed in the moduli space.

Results hold for a certain subfamily of elliptic curves.

Abstract

We consider elliptic curves over global fields of positive characteristic with two distinct marked non-trivial rational points. Restricting to a certain subfamily of the universal one, we show that the average size of the 2-Selmer groups of these curves exists, in a natural sense, and equals 12. Along the way, we consider a map from these 2-Selmer groups to the moduli space of $G$-torsors over an algebraic curve, where $G$ is isogenous to $\mathrm{SL}_2^4$, and show that the images of 2-Selmer elements under this map become equidistributed in the limit.