Zeta Functions of a Class of Artin–Schreier Curves with Many Automorphisms

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 function of the curves in this class over the field of definition of all automorphisms in the subgroup.

New examples of maximal curves with low genus

“…As the degree of the function field extension F p (x, y) : F p (θ, ρ) is clearly 2, we have that F ix α 3 = F p (θ, ρ). By Equation (10),…”

“…In this section we extend to the positive characteristic case a characterization of curves with equation (10) in terms of their automorphism group, provided by Shaska in [70].…”

“…Theorem 1.1 has already been used e.g. in [2,10,47,48,[51][52][53][54]66] to describe examples of curves with many points with respect to their genus, including maximal curves. For some curves investigated in this work, Theorem 1.1 gives a decomposition of their Jacobian into elliptic curves.…”

“…In Section 3 we deal with a family of curves with genus 5, given by Equation (10). In characteristic zero, these curves were characterized by Shaska [70] as the genus 5 hyperelliptic curves with automorphism group isomorphic to the direct product of A 4 and a group of order 2; Paulhus later observed that their Jacobian decomposes in the product of three isogenous elliptic curves and a hyperelliptic curve of genus 2; see [67,Proposition 4].…”

Divisibility of L-polynomials for a family of Artin–Schreier curves

“…It is also well known that c 0 = 1 and c 2g = q g . 1 We wish to consider the question of divisibility of L-polynomials. In previous papers [2], [3], we have studied conditions on the curves under which the L-polynomial of one curve divides the L-polynomial of another curve.…”