Average rank of elliptic curves over function fields
Irmak Balçık
TL;DR
The paper investigates the average Mordell-Weil rank of elliptic curves over the function field $\mathbb{F}_{q}(t)$ with $q\ge 5$, ordering curves by naive height. Building on Young’s strategy and leveraging the function-field analogue of the Riemann Hypothesis for $L$-functions (Weil), it derives an explicit upper bound $\limsup_{d\to\infty} \frac{1}{|\mathcal{D}(d)|}\sum_{E\in\mathcal{D}(d)} \mathrm{rk}(E) \le \frac{25}{14}$, improving Brumer’s previous bound of $2.3$. The approach relies on a detailed harmonic-analysis framework over $\mathbb{F}_{q}[t]$, including Poisson summation, Gauss sums, and Lindelöf-type bounds for Dirichlet $L$-functions, and culminates in showing a positive proportion of function-field elliptic curves have rank $0$ or $1$. This advances understanding of rank distributions in the function-field setting and mirrors, in this setting, the conditional results known over $\mathbb{Q}$ assuming GRH for elliptic $L$-functions, highlighting a sharp contrast with the number-field case where unconditional bounds are harder to reach.
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$.