On the Average Number of 2-Selmer Elements of Elliptic Curves over ${\mathbb{F}}_q(X)$ with Two Marked Points

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.