Explicit formulas and exact values for the number of rational points on singular curves over finite fields

TL;DR

This paper develops explicit formulas to accurately determine the maximum number of rational points on singular curves over finite fields, extending previous results and providing exact values for specific cases.

Contribution

It introduces new explicit formulas for bounding and calculating the maximum rational points on singular curves, including special cases for genus 2.

Findings

Derived explicit formulas for rational point bounds on singular curves

Calculated exact maximum rational points for specific genus and genus pairs

Extended previous work to genus 2 cases

Abstract

We provide new explicit formulas for bounding the number of rational points on singular curves over finite fields. This enables us to obtain exact values of N q (g, $\pi$) which is defined as the maximum number of rational points over F q on a curve of geometric genus g and arithmetic genus $\pi$. We also give special attention to the case g = 2 in order to extend the work of Aubry and Iezzi on N q (0, $\pi$) and N q (1, $\pi$).