The geometric average size of Selmer groups over function fields

Abstract

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 $1$.