Average rank of elliptic curves over function fields

TL;DR

This paper establishes an improved upper bound of approximately 1.8 for the average rank of elliptic curves over function fields, indicating many have rank 0 or 1, using methods adapted from number field cases.

Contribution

The paper provides a tighter upper bound on the average rank of elliptic curves over function fields, improving previous results and adapting techniques from number field research.

Findings

Average rank bounded above by 25/14 (~1.8)

Positive proportion of curves have rank 0 or 1

Improves previous upper bound of 2.3

Abstract

Let $q$ be a prime with $q \geq 5$. 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$.