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arXiv · 1607.00997

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

Jack A. Thorne
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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.

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Jack A. Thorne. 2016-07-04. On the average number of 2-Selmer elements of elliptic curves over $\mathbb{F}_q(X)$ with two marked points. https://arxiv.org/abs/1607.00997

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