An equivariant isomorphism theorem for mod $\mathfrak{p}$ reductions of arboreal Galois representations

Abstract

Let $\varphi$ be a quadratic, monic polynomial with coefficients in ${\mathcal{O}}_{F,D}[t]$, where ${\mathcal{O}}_{F,D}$ is a localization of a number ring ${\mathcal{O}}_F$. In this paper, we first prove that if $\varphi$ is non-square and non-isotrivial, then there exists an absolute, effective constant $N_{\varphi }$ with the following property: for all primes $\mathfrak{p}\subseteq {\mathcal{O}}_{F,D}$ such that the reduced polynomial ${\varphi }_{\mathfrak{p}}\in ({\mathcal{O}}_{F,D}/\mathfrak{p})[t][x]$ is non-square and non-isotrivial, the squarefree Zsigmondy set of ${\varphi }_{\mathfrak{p}}$ is bounded by $N_{\varphi }$. Using this result, we prove that if $\varphi$ is non-isotrivial and geometrically stable then outside a finite, effective set of primes of ${\mathcal{O}}_{F,D}$ the geometric part of the arboreal representation of ${\varphi }_{\mathfrak{p}}$ is isomorphic to that of $\varphi$. As an application of our results we prove R. Jones' conjecture on the arboreal Galois representation attached to the polynomial $x^2+t$.

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