标题： 在平均情况下，椭圆曲线在$\mathbb{F}_q(X)$上带有两个标记点时的2-Selmer元素数量 显示英文标题

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

摘要： 我们考虑在正特征的全局域上的椭圆曲线，这些曲线有两个不同的非平凡有理点。 限制到普遍族的一个特定子族，我们证明这些曲线的2-Selmer群的平均大小在自然的意义下存在，并且等于12。 在此过程中，我们考虑从这些2-Selmer群到代数曲线上的$G$-torsors 模空间的映射，其中$G$与$\mathrm{SL}_2^4$是同源的，并且证明在该映射下2-Selmer元素的像在极限情况下趋于均匀分布。 显示英文摘要

摘要： 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.