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

Abstract:

Let $\phi$ 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 $\phi$ is non-square and non-isotrivial, then there exists an absolute, effective constant $N_\phi$ with the following property: for all primes $\mathfrak {p}\subseteq \mathcal {O}_{F,D}$ such that the reduced polynomial $\phi _\mathfrak {p}\in (\mathcal {O}_{F,D}/\mathfrak {p})[t][x]$ is non-square and non-isotrivial, the squarefree Zsigmondy set of $\phi _\mathfrak {p}$ is bounded by $N_\phi$. Using this result, we prove that if $\phi$ 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 $\phi _\mathfrak {p}$ is isomorphic to that of $\phi$. As an application of our results we prove R. Jones’ conjecture on the arboreal Galois representation attached to the polynomial $x^2+t$.