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 $C_{a,b}$, which are defined over a finite field $F_m$ of characteristic 2. The curves are explicitly given by the homogeneous equation $Y^a{Z}^{b-a}+Y{Z}^{b-1}+X^b=0$, where $a$ and $b$ are natural numbers satisfying the conditions $a\ge 2$ and $b\ge a$. 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 $F_{2^4}$. In particular, the paper presents explicit calculations of the number of rational points for curves of the form $C_{2,b}$ and $C_{3,b}$ over $F_{2^4}$, 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.