Differential operators and hyperelliptic curves over finite fields

Boix, De Stefani and Vanzo have characterised ordinary/supersingular elliptic curves over Fp in terms of the level of the defining cubic homogeneous polynomial. We extend their study to arbitrary genus, in particular we prove that every ordinary hyperelliptic curve C of genus g≥2 has level 2. We provide a good number of examples and raise a conjecture.