Multiplicative independence of modular functions

Guy Fowler

doi:10.5802/jtnb.1167

Guy Fowler ¹

¹ Mathematical Institute University of Oxford Oxford, OX2 6GG, United Kingdom

Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 2, pp. 459-509.

Abstract
Résumé

We provide a new, elementary proof of the multiplicative independence of pairwise distinct ${GL}_{2}^{+} (ℚ)$ -translates of the modular $j$ -function, a result due originally to Pila and Tsimerman. We are thereby able to generalise this result to a wider class of modular functions. We show that this class includes a set comprising modular functions which arise naturally as Borcherds lifts of certain weakly holomorphic modular forms. For $f$ a modular function belonging to this class, we deduce, for each $n \geq 1$ , the finiteness of $n$ -tuples of distinct $f$ -special points that are multiplicatively dependent and minimal for this property. This generalises a theorem of Pila and Tsimerman on singular moduli. We then show how these results relate to the Zilber–Pink conjecture for subvarieties of the mixed Shimura variety $Y {(1)}^{n} \times 𝔾_{m}^{n}$ and prove some special cases of this conjecture.

Nous offrons une nouvelle preuve élémentaire de l’indépendance multiplicative de ${GL}_{2}^{+} (ℚ)$ -translatées, deux à deux distinctes, de la fonction modulaire $j$ , un résultat dû initialement à Pila et Tsimerman. Nous sommes ainsi en mesure de généraliser ce résultat à une classe de fonctions modulaires plus large. Nous montrons que cette classe contient un ensemble composé de fonctions modulaires qui apparaissent naturellement comme des relèvements de Borcherds de certaines formes modulaires faiblement holomorphes. Pour une fonction modulaire $f$ appartenant à cette classe, nous déduisons que pour chaque $n \geq 1,$ le nombre de $n$ -uplets de points $f$ -spéciaux distincts qui sont multiplicativement dépendants et minimaux pour cette propriété est fini. Cela généralise un théorème de Pila et Tsimerman sur les invariants modulaires singuliers. Nous montrons ensuite comment ces résultats sont liés à la conjecture de Zilber–Pink pour les sous-variétés algébriques de la variété de Shimura mixte $Y {(1)}^{n} \times 𝔾_{m}^{n}$ , et nous prouvons quelques cas particuliers de cette conjecture.

Received: 2020-05-27
Revised: 2021-03-16
Accepted: 2021-04-19
Published online: 2021-09-10

DOI: 10.5802/jtnb.1167

Classification: 11F03, 11G15
Keywords: Modular functions, multiplicative independence, Zilber–Pink conjecture

Author's affiliations:

Guy Fowler ¹

¹ Mathematical Institute University of Oxford Oxford, OX2 6GG, United Kingdom

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {Multiplicative independence of modular functions},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {459--509},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
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Guy Fowler. Multiplicative independence of modular functions. Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 2, pp. 459-509. doi : 10.5802/jtnb.1167. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1167/

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