On a remark of Serre

TL;DR

This paper investigates primes and finite fields where elliptic and higher genus curves do not reach the Hasse bound, providing explicit lists and heuristic analyses.

Contribution

It extends Serre's remark by identifying specific primes and fields where the Hasse bound is not achieved for elliptic and higher genus curves, with comprehensive data and heuristics.

Findings

List of all q<10^{70} where the Hasse bound is not achieved for elliptic curves.

Heuristic analysis on the distribution of such primes and fields.

Extension of criteria to genus 2 and 3 curves.

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.