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Multiplicative independence of modular functions

Research output: Contribution to journalArticleResearchpeer review

Authors

  • Guy Fowler

External Research Organisations

  • University of Oxford

Details

Original languageEnglish
Pages (from-to)459-509
Number of pages51
JournalJournal de Theorie des Nombres de Bordeaux
Volume33
Issue number2
Publication statusPublished - 10 Sept 2021
Externally publishedYes

Abstract

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 ≥ 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 × G nm and prove some special cases of this conjecture.

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Cite this

Multiplicative independence of modular functions. / Fowler, Guy.
In: Journal de Theorie des Nombres de Bordeaux, Vol. 33, No. 2, 10.09.2021, p. 459-509.

Research output: Contribution to journalArticleResearchpeer review

Fowler G. Multiplicative independence of modular functions. Journal de Theorie des Nombres de Bordeaux. 2021 Sept 10;33(2):459-509. doi: 10.5802/jtnb.1167
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