Picard curves over Q with good reduction away from 3

TL;DR

This paper presents an algorithm to classify Picard curves over Q with good reduction outside 3, establishing a correspondence with specific binary forms and providing an exhaustive list of models, with applications to number theory questions.

Contribution

It introduces a novel algorithm for classifying Picard curves with specified reduction properties and links their isomorphism classes to particular binary forms.

Findings

Complete list of Picard curves with good reduction outside 3.

Established correspondence between curves and binary forms.

Application to a question of Ihara.

Abstract

Inspired by methods of N. P. Smart, we describe an algorithm to determine all Picard curves over Q with good reduction away from 3, up to Q-isomorphism. A correspondence between the isomorphism classes of such curves and certain quintic binary forms possessing a rational linear factor is established. An exhaustive list of integral models is determined, and an application to a question of Ihara is discussed.