Rational and singular points of a family of curves

Abstract

This paper explores the properties of a family of absolutely irreducible projective plane curves, denoted 𝐶𝑎,𝑏, which are defined over a finite field F𝑚 of characteristic 2. The curves are explicitly given by the homogeneous equation 𝑌 𝑎𝑍𝑏−𝑎+𝑌 𝑍𝑏−1+𝑋𝑏 = 0, where 𝑎 and 𝑏 are natural numbers satisfying the conditions 𝑎 ≥ 2 and 𝑏 ≥ 𝑎. A primary objective of the paper is to determine the number of rational points on these curves. The work also includes a detailed analysis of the singular points of the curves, providing a classification of these points based on the parameters 𝑎 and 𝑏. Furthermore, the relationship between the number of rational points and the genus of the curves is investigated, with specific computations carried out for curves defined over the finite field F2 4 . In particular, the paper presents explicit calculations of the number of rational points for curves of the form 𝐶2,𝑏 and 𝐶3,𝑏 over F2 4 , illustrating the connection between these counts and the genus of the curves. This comprehensive analysis contributes to a deeper understanding of the arithmetic geometry of this family of curves over finite fields.