Average rank of elliptic curves over function fields

Abstract

Let $q$ be a prime with $q\ge 5$. We show that the average rank of elliptic curves over a function field ${\mathbb{F}}_q(t)$, when ordered by naive height, is bounded above by $25/14\approx 1.8$. Our result improves the previous upper bound of $2.3$ proven by Brumer. The upper bound obtained is less than $2$, which shows that a positive proportion of elliptic curves has either rank $0$ or $1$. The proof adapts the work of Young, which shows that under the assumption of the General Riemann Hypothesis for $L$-functions of elliptic curves, the average rank for the family of elliptic curves over the rational numbers is bounded above by $25/14\approx 1.8$.

18 pages