Criteria of maximality and minimality of van der Geer-van der Vlugt curves

TL;DR

This paper establishes criteria for van der Geer-van der Vlugt curves to be maximal or minimal over finite fields and explores their L-polynomials, identifying new maximal or minimal curves.

Contribution

It provides explicit criteria for maximality and minimality of these curves and analyzes their L-polynomials, extending understanding of their properties.

Findings

Criteria for maximality and minimality established

Identification of new maximal/minimal curves among generalizations

Explicit formulas for L-polynomials used in proofs

Abstract

The van der Geer-van der Vlugt curves are Artin-Schreier coverings of the affine line defined by linearized polynomials over finite fields. We give several criteria for them to be maximal or minimal, i.e. attaining the upper or lower bound in the Hasse-Weil inequalities. We also study the $L$-polynomials of certain generalizations of van der Geer-van der Vlugt curves. As applications, we find several maximal (or minimal) curves among them. Our proof is based on an explicit formula of $L$-polynomials recently obtained by Takeuchi and the third author.