More on quadratic functions and maximal Artin–Schreier curves

Abstract

For an odd prime \(p\) and an even integer \(n\) with \(\gcd (n,p) > 1\), we consider quadratic functions from \({\mathbb F}_{p^{n}}\) to \({\mathbb F}_p\) of codimension \(k\). For various values of \(k\), we obtain classes of quadratic functions giving rise to maximal and minimal Artin–Schreier curves over \({\mathbb F}_{p^{n}}\). We completely classify all maximal and minimal curves obtained from quadratic functions of codimension \(2\) and coefficients in the prime field \({\mathbb F}_p\). The results complement our results obtained earlier for the case \(\gcd (n,p) = 1\). The arguments are more involved than for the case \(\gcd (n,p) = 1\).