Genus 3 curves with many involutions and application to maximal curves in characteristic 2

TL;DR

This paper constructs explicit genus 3 curves with many involutions over characteristic 2 fields, characterizes their elliptic quotients, and applies these results to produce new maximal curves with many rational points.

Contribution

It provides explicit models and equations for genus 3 curves with many involutions and characterizes elliptic curve triples admitting Artin-Schreier covers, advancing the construction of maximal curves.

Findings

Explicit models for genus 3 curves with many involutions

Characterization of elliptic triples admitting Artin-Schreier covers

Construction of infinitely many maximal curves over finite fields

Abstract

Let k=F_q be a finite field of characteristic 2. A genus 3 curve C/k has many involutions if the group of k-automorphisms admits a C_2\times C_2 subgroup H (not containing the hyperelliptic involution if C is hyperelliptic). Then C is an Artin-Schreier cover of the three elliptic curves obtained as the quotient of C by the nontrivial involutions of H, and the Jacobian of C is k-isogenous to the product of these three elliptic curves. In this paper we exhibit explicit models for genus 3 curves with many involutions, and we compute explicit equations for the elliptic quotients. We then characterize when a triple (E_1,E_2,E_3) of elliptic curves admits an Artin-Schreier cover by a genus 3 curve, and we apply this result to the construction of maximal curves. As a consequence, when q is nonsquare and m=\lfloor 2 sqrt(q) \rfloor = 1,5,7 mod 8, we obtain that N_q(3)=1+q+3m. We also show that this occurs for an infinite number of values of q nonsquare.