Hilbert's Irreducibility for ${\mathbb{G}}_m$

Abstract

Let $K$ be a number field and $S$ a finite set of non-archimedean places. Write ${\mathcal{O}}_S$ for the ring of $S$-integers of $K$ and ${\mathcal{O}}_S^{\times }$ for its unit group. Let $\pi :X\to {\mathbb{P}}^1$ be a morphism of (irreducible) curves defined over $K$, and denote by $\mathrm{Red}(\pi )$ the set of $\alpha \in {\mathbb{P}}^1(K)$ such that the fibre ${\pi }^{-1}(\alpha )$ is reducible (i.e. the Galois action on the fibre is intransitive). Hilbert's Irreducibility Theorem asserts that $\mathrm{Red}(\pi )$ is contained in a thin subset of ${\mathbb{P}}^1(K)$. In this paper we give an explicit description of ${\mathcal{O}}_S^{\times }\cap \mathrm{Red}(\pi )$. As an application we prove the following result inspired by a classical theorem of Pólya and Siegel. Let $p_1,\dots ,p_s$ be rational primes. Let $f\in \mathbb{Q}[x]$.Then the following are equivalent: - There are infinitely many tuples $(e_1,\dots ,e_s)\in {\mathbb{N}}^s$ such that the polynomial $f(x)-p_1^{e_1}\cdots p_s^{e_s}$ is reducible. - $f=p_1^{a_1}\cdots p_s^{a_s}{g}^{\ell }$ (with $\ell$ prime) or $f=-4p_1^{a_1}\cdots p_s^{a_s}{g}^4$ for some $g\in \mathbb{Q}[x]$ and some integers $a_1,\dots ,a_s$.