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We consider elliptic curves over global fields of positive characteristicwith two distinct marked non-trivial rational points. Restricting to a certainsubfamily of the universal one, we show that the average size of the 2-Selmergroups of these curves exists, in a natural sense, and equals 12. Along theway, 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 mapbecome equidistributed in the limit.

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