ON THE NUMBER OF RATIONAL POINTS OF GENERALIZED FERMAT CURVES OVER FINITE FIELDS

Abstract

The Stöhr–Voloch approach has been largely used to deal with the classical problem of estimating the number of rational points of a Fermat curve over a finite field. The same method actually applies to any curve admitting as an automorphism group the direct product of two cyclic groups C1 and C2 of the same size k, and such that the quotient curves with respect to both C1 and C2 are rational. In this paper such a curve is called a generalized Fermat curve. Our main achievement is that of extending some known results on Fermat curves to generalized Fermat curves.