Average rank of elliptic curves over function fields
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909876486144000 |
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| author | Balçık, Irmak |
| author_facet | Balçık, Irmak |
| contents | 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$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_25630 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Average rank of elliptic curves over function fields Balçık, Irmak Number Theory 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$. |
| title | Average rank of elliptic curves over function fields |
| topic | Number Theory |
| url | https://arxiv.org/abs/2510.25630 |