Equations with powers of singular moduli

Abstract : We treat two different equations involving powers of singular moduli. On the one hand, we show that, with two possible (explicitly specified) exceptions, two distinct singular moduli j(τ), j(τ ′) such that the numbers 1, j(τ) m and j(τ ′) n are linearly dependent over Q for some positive integers m, n, must be of degree at most 2. This partially generalizes a result of Allombert, Bilu and Pizarro-Madariaga, who studied CM-points belonging to straight lines in C 2 defined over Q. On the other hand, we show that, with " obvious " exceptions, the product of any two powers of singular moduli cannot be a non-zero rational number. This generalizes a result of Bilu, Luca and Pizarro-Madariaga, who studied CM-points belonging to an hyperbola xy = A, where A ∈ Q.
Type de document :
Pré-publication, Document de travail
2017
Liste complète des métadonnées

Littérature citée [17 références]  Voir  Masquer  Télécharger

https://hal.archives-ouvertes.fr/hal-01630363
Contributeur : Antonin Riffaut <>
Soumis le : mardi 7 novembre 2017 - 15:15:37
Dernière modification le : jeudi 11 janvier 2018 - 06:21:23
Document(s) archivé(s) le : jeudi 8 février 2018 - 13:52:31

Identifiants

  • HAL Id : hal-01630363, version 1

Collections

Citation

Antonin Riffaut. Equations with powers of singular moduli. 2017. 〈hal-01630363〉

Partager

Métriques

Consultations de la notice

53

Téléchargements de fichiers

28