On a remark of Serre

Summary · qwen2.5:32b

Researchers extend Serre's work to identify primes $p$ where the maximum Hasse bound for points on an elliptic curve over $\mathbb{F}_{p^5}$ is not met, listing all such instances up to $10^{70}$. This exploration includes heuristics and similar analyses for curves of genus 2 and 3.

Excerpt

arXiv:2508.11048v3 Announce Type: replace 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.