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arXiv · 2508.11048

On a remark of Serre

Katie Ahrens·Jon Grantham
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Abstract

Inspired by a remark of Serre, we extend the search for primes $p$ such that the maximum Hasse bound for the number of points on an elliptic curve over $\mathbb{F}_{p^5}$ is not achieved. We then give a list of all $q<10^{70}$ such that the Hasse bound is not achieved over $\mathbb{F}_{q}$. We explore the heuristics for how many such numbers should exist in each case. Finally, look at similar criteria for genus $2$ and $3$ curves.

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Katie Ahrens, Jon Grantham. 2025-08-14. On a remark of Serre. https://arxiv.org/abs/2508.11048

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