Zeta functions of a class of Artin-Schreier curves with many automorphisms

TL;DR

This paper studies a specific class of Artin-Schreier curves with large automorphism groups, enabling explicit computation of their zeta functions and leading to new examples of maximal curves in odd characteristic.

Contribution

It generalizes previous results to odd characteristic and explicitly computes zeta functions for curves with large automorphism groups, revealing new maximal curves.

Findings

Explicit zeta functions for the class of curves.

Identification of large automorphism subgroups.

Construction of new maximal curves.

Abstract

This paper describes a class of Artin-Schreier curves, generalizing results of Van der Geer and Van der Vlugt to odd characteristic. The automorphism group of these curves contains a large extraspecial group as a subgroup. Precise knowledge of this subgroup makes it possible to compute the zeta functions of the curves in the class over the field of definition of all automorphisms in the subgroup. As a consequence, we obtain new examples of maximal curves.