Reciprocal polynomials and curves with many points over a finite field

Abstract

Let ${\mathbb{F}}_{q^2}$ be the finite field with $q^2$ elements. We provide a simple and effective method, using reciprocal polynomials, for the construction of algebraic curves over ${\mathbb{F}}_{q^2}$ with many rational points. The curves constructed are Kummer covers or fibre products of Kummer covers of the projective line. Further, we compute the exact number of rational points for some of the curves.