Differential operators and hyperelliptic curves over finite fields

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 $g\geq 2$haslevel$2$. We provide a good number of examples and raise a conjecture.