Around the support problem for Hilbert class polynomials

Let H_{D}(T) denote the Hilbert class polynomial of the imaginary quadratic order of discriminant D . We study the rate of growth of the greatest common divisor of H_{D}(a) and H_{D}(b) as |D| \to \infty for a and b belonging to various Dedekind domains. We also study the modular support problem: if for all but finitely many D every prime ideal dividing H_{D}(a) also divides H_{D}(b) , what can we say about a and b ? If we replace H_{D}(T) by T^{n}-1 and the Dedekind domain is a ring of S -integers in some number field, then these are classical questions that have been investigated by Bugeaud–Corvaja–Zannier, Corvaja–Zannier, and Corrales-Rodrigáñez–Schoof.