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

##### Doc. Math. 24, 1179-1223 (2019)
DOI: 10.25537/dm.2019v24.1179-1223

### Summary

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.

11G05, 14H60

### Keywords/Phrases

elliptic curves, rational points, Selmer groups

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### Affiliation

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