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Picard curves over $\mathbb{Q}$ with good reduction away from 3

Part of: Forms and linear algebraic groups Diophantine approximation, transcendental number theory Computational aspects in algebraic geometry Computational number theory Curves Arithmetic algebraic geometry

Published online by Cambridge University Press:  01 March 2017

Beth Malmskog  and
Christopher Rasmussen
Beth Malmskog
Affiliation:
Department of Mathematics and Statistics, Villanova University, 800 Lancaster Avenue, Villanova PA 19085, USA email beth.malmskog@gmail.com
Christopher Rasmussen
Affiliation:
Department of Mathematics and Computer Science, Wesleyan University, 265 Church Street, Exley Tower, 6th Floor, Middletown CT 06549, USA email crasmussen@wesleyan.edu

Abstract

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Inspired by methods of N. P. Smart, we describe an algorithm to determine all Picard curves over $\mathbb{Q}$ with good reduction away from 3, up to $\mathbb{Q}$-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.

MSC classification

Primary: 11E76: Forms of degree higher than two 11G30: Curves of arbitrary genus or genus $ne 1$ over global fields
Secondary: 11J86: Linear forms in logarithms; Baker's method 11Y40: Algebraic number theory computations 14Q05: Curves 14H30: Coverings, fundamental group 11G32: Dessins d'enfants, Belyĭ theory
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 19 , Issue 2 , 2016 , pp. 382 - 408
Copyright
© The Author(s) 2017 

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