Let M be the Artin-Mumford curve over the finite prime field F p with p > 2. By a result of Valentini and Madan, AutWe prove that if X is an algebraic curve of genus g = (p−1) 2 such that Aut Fp (X ) contains a subgroup isomorphic to H then X is birationally equivalent over F p to the Artin-Mumford curve M.

Let X be an irreducible algebraic curve defined over a finite field Fq of characteristic p > 2.Assume that the Fq-automorphism group of X admits as an automorphism group the direct product of two cyclic groups Cm and Cn of orders m and n prime to p such that both quotient curves X /Cn and X /Cm are rational. In this paper, we provide a complete classification of such curves, as well as a characterization of their full automorphism groups.

For Fermat curves F : aX n + bY n = Z n defined over Fq, we establish necessary and sufficient conditions for F to be Fq-Frobenius nonclassical with respect to the linear system of plane cubics.In the Fq-Frobenius nonclassical cases, we determine explicit formulas for the number Nq(F) of Fqrational points on F. For the remaining Fermat curves, nice upper bounds for Nq(F) are immediately given by the Stöhr-Voloch Theory.

Let X be a projective irreducible nonsingular algebraic curve defined over a finite field Fq. This paper presents a variation of the Stöhr-Voloch theory and sets new bounds to the number of Fqrrational points on X . In certain cases, where comparison is possible, the results are shown to improve other bounds such as Weil's, Stöhr-Voloch's and Ihara's.

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