Representation of non-special curves of genus 5 as plane sextic curves and its application to finding curves with many rational points

TL;DR

This paper develops an effective parametrization for certain genus-5 curves and creates an algorithm to find non-special curves with many rational points, demonstrated over finite fields using MAGMA.

Contribution

It introduces a new parametrization for non-special genus-5 curves and an algorithm for enumerating those with many rational points, with practical implementation over finite fields.

Findings

Effective parametrization for genus-5 curves established.

Algorithm successfully enumerates non-special curves with many rational points.

Implementation in MAGMA demonstrates practical applicability.

Abstract

In algebraic geometry, it is important to provide effective parametrizations for families of curves, both in theory and in practice. In this paper, we present such an effective parametrization for the moduli of genus-$5$ curves that are neither hyperelliptic nor trigonal. Subsequently, we construct an algorithm for a complete enumeration of non-special genus-$5$ curves having more rational points than a specified bound, where ``non-special curve'' means that the curve is non-hyperelliptic and non-trigonal with mild singularities of the associated sextic model that we propose. As a practical application, we implement this algorithm using the computer algebra system MAGMA, specifically for curves over the prime field of characteristic $3$.