The supersingularity of Hurwitz curves

Abstract

We study when Hurwitz curves are supersingular. Specifically, we show that the curve $H_{n,\ell }:X^n{Y}^{\ell }+Y^n{Z}^{\ell }+Z^n{X}^{\ell }=0$, with $n$ and $\ell$ relatively prime, is supersingular over the finite field ${\mathbb{F}}_p$ if and only if there exists an integer $i$ such that $p^i\equiv -1mod\left(n^2-n\ell +{\ell }^2\right)$. If this holds, we prove that it is also true that the curve is maximal over ${\mathbb{F}}_{p^{2i}}$. Further, we provide a complete table of supersingular Hurwitz curves of genus less than 5 for characteristic less than 37.