Reciprocal polynomials and curves with many points over a finite field

TL;DR

This paper introduces a straightforward method using reciprocal polynomials to construct algebraic curves over finite fields with many rational points, specifically focusing on Kummer covers and their fiber products.

Contribution

It presents a new, simple approach for constructing algebraic curves with many points over finite fields using reciprocal polynomials and computes exact point counts for some cases.

Findings

Constructed curves with many rational points over finite fields.

Provided explicit formulas for the number of rational points on certain curves.

Demonstrated the effectiveness of reciprocal polynomials in curve construction.

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.