Differential operators and hyperelliptic curves over finite fields

TL;DR

This paper extends the characterization of elliptic curves over finite fields to hyperelliptic curves of higher genus, showing that all ordinary hyperelliptic curves of genus at least 2 have level 2, with numerous examples and a conjecture.

Contribution

It generalizes the level characterization from elliptic to hyperelliptic curves of arbitrary genus, establishing that all ordinary hyperelliptic curves of genus ≥ 2 have level 2.

Findings

All ordinary hyperelliptic curves of genus ≥ 2 have level 2.

Provided numerous examples of hyperelliptic curves.

Formulated a conjecture based on observed patterns.

Abstract

Boix, De Stefani and Vanzo have characterized ordinary/supersingular elliptic curves over $\mathbb{F}_p$ in terms of the level of the defining cubic homogenous polynomial. We extend their study to arbitrary genus, in particular we prove that every ordinary hyperelliptic curve $\mathcal{C}$ of genus $g\geq 2$ has level $2$. We provide a good number of examples and raise a conjecture.