Optimal and maximal singular curves

TL;DR

This paper establishes new bounds on the number of degree-2 points on algebraic curves over finite fields, providing conditions for the existence or non-existence of curves with specified genera and rational points.

Contribution

It introduces a novel Euclidean approach to bound degree-2 points and characterizes the existence of maximal curves over finite fields with given genus parameters.

Findings

New upper bound for degree-2 points on curves

Explicit non-existence conditions for certain curves

Characterization of maximal curves over square finite fields

Abstract

Using an Euclidean approach, we prove a new upper bound for the number of closed points of degree 2 on a smooth absolutely irreducible projective algebraic curve defined over the finite field $\mathbb F\_q$.This bound enables us to provide explicit conditions on $q, g$ and $\pi$ for the non-existence of absolutely irreducible projective algebraic curves defined over $\mathbb F\_q$ of geometric genus $g$, arithmetic genus $\pi$ and with $N\_q(g)+\pi-g$ rational points.Moreover, for $q$ a square, we study the set of pairs $(g,\pi)$ for which there exists a maximal absolutely irreducible projective algebraic curve defined over $\mathbb F\_q$ of geometric genus $g$ and arithmetic genus $\pi$, i.e. with $q+1+2g\sqrt{q}+\pi-g$ rational points.