The Hurwitz curve over a finite field and its Weierstrass points for the morphism of lines

Abstract

For any smooth Hurwitz curve ${\mathcal{H}}_n:X{Y}^n+Y{Z}^n+X^{n}Z=0$ over the finite field ${\mathbb{F}}_p$, an explict description of its Weierstrass points for the morphism of lines is presented. As a consequence, the full automorphism group $\mathrm{Aut}({\mathcal{H}}_n)$, as well as the genera of all Galois subcovers of ${\mathcal{H}}_n$, with $n\ne 3,p^r$, are computed. Finally, a question by F. Torres on plane non nonsingular maximal curves is answered.