Around the support problem for Hilbert class polynomials

TL;DR

This paper investigates the growth of the gcd of Hilbert class polynomial evaluations and explores the modular support problem related to divisibility properties of these polynomials over various Dedekind domains.

Contribution

It extends classical divisibility questions to Hilbert class polynomials, analyzing gcd growth and divisibility patterns in the context of algebraic number theory.

Findings

GCD of H_D(a) and H_D(b) exhibits specific growth behavior as |D| increases.

Conditions under which prime divisors of H_D(a) also divide H_D(b) are characterized.

Connections to classical support problems for cyclotomic polynomials are established.

Abstract

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-Rodrig\'a\~{n}ez-Schoof.