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LINEAR INDEPENDENCE OF POWERS OF SINGULAR MODULI OF DEGREE THREE

Part of: Diophantine approximation, transcendental number theory Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  12 September 2018

FLORIAN LUCA Open the ORCID record for FLORIAN LUCA [Opens in a new window]  and
ANTONIN RIFFAUT Open the ORCID record for ANTONIN RIFFAUT [Opens in a new window]
FLORIAN LUCA
Affiliation:
School of Mathematics, University of the Witwatersrand, Private Bag X3, Wits 2050, Johannesburg, South Africa Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany Department of Mathematics, Faculty of Sciences, University of Ostrava, 30 dubna 22, 701 03 Ostrava 1, Czech Republic email Florian.Luca@wits.ac.za
ANTONIN RIFFAUT*
Affiliation:
Institut de Mathématiques de Bordeaux, Université de Bordeaux, A33, 351 Cours de la Libération, 33400 Talence, France email antonin.riffaut@math.u-bordeaux.fr
*
✉antonin.riffaut@math.u-bordeaux.fr
Rights & Permissions [Opens in a new window]

Abstract

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We show that two distinct singular moduli $j(\unicode[STIX]{x1D70F}),j(\unicode[STIX]{x1D70F}^{\prime })$, such that for some positive integers $m$ and $n$ the numbers $1,j(\unicode[STIX]{x1D70F})^{m}$ and $j(\unicode[STIX]{x1D70F}^{\prime })^{n}$ are linearly dependent over $\mathbb{Q}$, generate the same number field of degree at most two. This completes a result of Riffaut [‘Equations with powers of singular moduli’, Int. J. Number Theory, to appear], who proved the above theorem except for two explicit pairs of exceptions consisting of numbers of degree three. The purpose of this article is to treat these two remaining cases.

Keywords

linear independencesingular modulilinear forms in two logarithms

MSC classification

Primary: 11J20: Inhomogeneous linear forms
Secondary: 11F03: Modular and automorphic functions 11J61: Approximation in non-Archimedean valuations 11J86: Linear forms in logarithms; Baker's method
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 99 , Issue 1 , February 2019 , pp. 42 - 50
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

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