The Average Size of 2-Selmer Groups of Elliptic Curves in Characteristic 2

Niven Achenjang

Published 2023-10-12Version 1

Let $K$ be the function field of a smooth curve $B$ over a finite field $k$ of arbitrary characteristic. We prove that the average size of the $2$-Selmer groups of elliptic curves $E/K$ is at most $1+2\zeta_B(2)\zeta_B(10)$, where $\zeta_B$ is the zeta function of the curve $B/k$. In particular, in the limit as $q=\#k\to\infty$ (with the genus $g(B)$ fixed), we see that the average size of 2-Selmer is bounded above by $3$, even in ``bad'' characteristics. Along the way, we also produce new bounds on 2-Selmer groups of elliptic curves over characteristic $2$ global function fields.