On Singular Moduli for Arbitrary Discriminants

Let d1 and d2 be discriminants of distinct quadratic imaginary orders and let J(d1, d2) denote the product of differences of CM j-invariants with discriminants d1 and d2. In 1985, Gross and Zagier gave an elegant formula for the factorization of the integer J(d1, d2) in the case that d1 and d2 are relatively prime and discriminants of maximal orders. We generalize their methods and give a complete factorization in the case that d1 is squarefree and d2 is the discriminant of any quadratic imaginary order. We also give a partial factorization in all other cases, and give a conjectural formula when the conductors of d1 and d2 are relatively prime.