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Preprint Number 1791

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1791. Guy Fowler
Multiplicative independence of modular functions

Submission date: 27 May 2020


We provide two new proofs 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 extend their result to a wider class of modular functions. For modular functions f in \overlineℚ(j) belonging to this class, we deduce, for each n ≥ 1, the finiteness of n-tuples of 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_m^n and prove some special cases of this conjecture.

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Full text arXiv 2005.13328: pdf, ps.

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