Multiplicative independence of modular functions

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) x G(m)(n) and prove some special cases of this conjecture.