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

Let $ϕ$ 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 $ϕ$ is non-square and non-isotrivial, then there exists an absolute, effective constant $N_ϕ$ with the following property: for all primes $\mathfrak p\subseteq\mathcal O_{F,D}$ such that the reduced polynomial $ϕ_\mathfrak p\in (\mathcal O_{F,D}/\mathfrak p)[t][x]$ is non-square and non-isotrivial, the squarefree Zsigmondy set of $ϕ_{\mathfrak p}$ is bounded by $N_ϕ$. Using this result, we prove that if $ϕ$ 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 $ϕ_{\mathfrak p}$ is isomorphic to that of $ϕ$. 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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