# On the Average Number of 2-Selmer Elements of Elliptic Curves over $\mathbb F_q(X)$ with Two Marked Points

### Jack A. Thorne

Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, United Kingdom

## 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.

## Cite this article

Jack A. Thorne, On the Average Number of 2-Selmer Elements of Elliptic Curves over $\mathbb F_q(X)$ with Two Marked Points. Doc. Math. 24 (2019), pp. 1179–1223

DOI 10.4171/DM/702