Average rank of elliptic curves over function fields

Irmak Balçık

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arXiv · 2510.25630

Average rank of elliptic curves over function fields

Irmak Balçık

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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$.

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Irmak Balçık. 2025-10-29. Average rank of elliptic curves over function fields. https://arxiv.org/abs/2510.25630

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