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

Abstract

For any smooth Hurwitz curve Hn:XYn+YZn+XnZ=0 over the finite field Fp, an explicit description of its Weierstrass points for the morphism of lines is presented. As a consequence, bounds on the number of rational points of Hn are obtained via Stöhr-Voloch Theory. Further, the full automorphism group Aut(Hn), as well as the genera of all Galois subcovers of Hn, with n≠3,pr, are computed. Finally, a question by F. Torres on plane nonsingular maximal curves is answered.