  We construct a natural family of rational functions $\tilde\Psi_m$ 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 $\tilde\Psi_m$ at a CM point associated to a
non-biquadratic quartic CM field generates the `CM class field' of the reflex
field. For the real quadratic field $\Q(\sqrt{5})$, we factorize the norm of
some of these CM values to $\Q(\sqrt 5)$ numerically.
