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

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 $E\mathrm{/}K$ is at most $1+2{\zeta }_B(2){\zeta }_B(10)$, where ${\zeta }_B$ is the zeta function of the curve $B$. In particular, in the limit as $q=\mathrm{\#}k\to \mathrm{\infty }$ (with the genus $g(B)$ fixed), we see that the average size of 2-Selmer is bounded above by $3$, even in “bad” characteristics. This completes the proof that the average rank of elliptic curves, over $\mathit{\text{any}}$ fixed global field, is finite. Handling the case of characteristic $2$ requires us to develop a new theory of integral models of 2-Selmer elements, dubbed “hyper-Weierstrass curves.”