On singular moduli for arbitrary discriminants

Let $d_1$ and $d_2$ be discriminants of distinct quadratic imaginary orders ${{\cal O}}_{d_1}$ and ${{\cal O}}_{d_2}$ and let $J(d_1,d_2)$ denote the product of differences of CM $j$-invariants with discriminants $d_1$ and $d_2$. In 1985, Gross and Zagier gave an elegant formula for the factorization of the integer $J(d_1,d_2)$ in the case that $d_1$ and $d_2$ are relatively prime and discriminant...