On the maximum number of rational points on singular curves over finite fields

TL;DR

This paper introduces a construction method for singular curves over finite fields that allows for establishing bounds on the maximum number of rational points on such curves, advancing understanding in algebraic geometry over finite fields.

Contribution

It provides a new construction technique for singular curves with many rational points, enabling improved bounds on their maximum number over finite fields.

Findings

Constructed singular curves with many rational points.

Proved results on maximum rational points for curves with given genus.

Enhanced bounds for algebraic curves over finite fields.

Abstract

We give a construction of singular curves with many rational points over finite fields. This construction enables us to prove some results on the maximum number of rational points on an absolutely irreducible projective algebraic curve defined over Fq of geometric genus g and arithmetic genus $\pi$.