Around the support problem for Hilbert class polynomials

Abstract

Let HD(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 HD(a) and HD(b) as |D| → ∞ 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 HD(a) also divides HD(b), what can we say about a and b? If we replace HD(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.