More on quadratic functions and maximal Artin-Schreier curves
Consider the Artin-Schreier cover of the \(\mathbb F_{p^n}\)-projective line given by \[ \mathcal{X}: y^p - y = \sum_{i=0}^{\lfloor n/2\rfloor} a_ix^{p^i+1} \] with \(a_i\in\mathbb F_{p^n}\). Let \(Q:\mathbb F_{p^n}\to\mathbb F_{p}\) be the function defined by \(Q(x) = \mathrm{Tr}_n\left(\sum_{i=0}^{\lfloor n/2\rfloor}a_ix^{p^i+1}\right)\), where \(\mathrm{Tr}_n(z)\)is the absolute trace of \(z\in\mathbb F_{p^n}\). The number of rational points of \(\mathcal X\) is\(N(\mathcal X) = 1+pN_0(Q)\) where \(N_0(Q)\) is the number of solutions of \(Q(x) = 0\). For an odd prime \(p\) and an even integer \(n\) with \(\gcd(n,p)>1\), the authors investigate quadratic functions \(Q\) of codimension \(k\) for which \(N(\mathcal X)\) attains the Hasse-Weil bound. They show that \(N_0(Q)\) is related to the Walsh coefficient of \(Q(x)\) at 0. Using this approach, they completely classify all minimal and maximal Artin-Schreier curves obtained from quadratic functions of codimension \(2\) with coefficients in \(\mathbb F_p\). This extends their work in [\textit{N. Anbar} and \textit{W. Meidl}, Finite Fields Appl. 30, 49--71 (2014; Zbl 1296.11074)] where they consider the case \(\gcd(n,p)=1\).
scientific article; zbMATH DE number 6489836
scientific article