We construct a natural family of rational functions Ψ̃m on a Hilbert modular surface from the classical j-invariant and its Hecke translates. These functions are obtained by means of a multiplicative analogue of the Doi-Naganuma lifting and can be viewed as twisted Borcherds products. We then study when the value of Ψ̃m at a CM point associated to a non-biquadratic quartic CM field generates the ...