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

Let φ be a quadratic, monic polynomial with coefficients in OF,D[t], where OF,D is a localization of a number ring OF. In this paper, we first prove that if φ is non-square and non-isotrivial, then there exists an absolute, effective constant Nφ with the following property: for all primes p ⊆ OF,D such that the reduced polynomial φp ∈ (OF,D/p)[t][x] is non-square and non-isotrivial, the squarefree Zsigmondy set of φp is bounded by Nφ. Using this result, we prove that if φ is non-isotrivial and geometrically stable, then outside a finite, effective set of primes of OF,D the geometric part of the arboreal representation of φp 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 x2 + t.