Classification of quadratic forms over finite fields with maximal and minimal Artin-Schreier curves

Abstract

This paper explores quadratic forms over finite fields with associated Artin-Schreier curves. Specifically, we investigate quadratic forms of ${\mathbb{F}}_{q^n}/{\mathbb{F}}_q$ represented by polynomials over ${\mathbb{F}}_{q^n}$ with $q$ odd, characterizing them using certain matrices defined by coefficients of the polynomials. In particular, a comprehensive treatment will be given for those polynomials whose coefficients all lie in ${\mathbb{F}}_q$. Afterwards, the results on quadratic forms will be applied to get maximal and minimal Artin-Schreier curves explicitly.