On a remark of Serre

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.