A conjectural extension of the Gross-Zagier formula on singular moduli
The formula in the title is for the resultant \(J(d_1,d_2)\) of the class polynomials for the orders of discriminant \(d_1,d_2\) of relatively prime fundamental discriminants of imaginary quadratics, due to \textit{B. H. Gross} and \textit{D. B. Zagier} [J. Reine Angew. Math. 355, 191-220 (1985; Zbl 0545.10015)]. The formula is interesting because it presents a highly composite resultant with factors \(F(m)\) equal to powers of primes of dividing \(m\) as \(m\) runs over positive integers \((d_1d_2-x^2)/4\). The author has conjectural extensions for the cases where \(d_1,d_2\) need not be fundamental and may have \((d_1,d_2)\) a prime power. There is an extended calculation serving as verification.
fundamental discriminants of imaginary quadratics
scientific article; zbMATH DE number 1207098