We show, in the large $q$ limit, that the average size of $n$-Selmer groups of elliptic curves of bounded height over ${\mathbb{𝔽}}_q\left(t\right)$ is the sum of the divisors of $n$. As a corollary, again in the large $q$ limit, we deduce that $100\%$ of elliptic curves of bounded height over ${\mathbb{𝔽}}_q\left(t\right)$ have rank $0$ or $1$.

中文翻译：

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Selmer 群在函数域上的几何平均大小

我们展示，在大 $q$ 限制，即平均大小 $n$-Selmer 群的有界高度的椭圆曲线 ${\mathbb{𝔽}}_q\left(吨\right)$ 是除数的总和 $n$. 作为推论，再次在大$q$ 极限，我们推断 $100\%$ 有界高度的椭圆曲线 ${\mathbb{𝔽}}_q\left(吨\right)$ 有等级 $0$ 或者 $1$.