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

## TL;DR

This paper establishes an effective bound on prime reductions of quadratic polynomials and demonstrates an isomorphism between arboreal Galois representations outside a finite set of primes, confirming a conjecture by R. Jones.

## Contribution

It proves an effective uniform bound on the Zsigmondy set for reductions of quadratic polynomials and confirms Jones' conjecture on arboreal Galois representations for specific polynomials.

## Key findings

- Bounded Zsigmondy set for reductions of quadratic polynomials
- Isomorphism of arboreal Galois representations outside finite primes
- Confirmation of R. Jones' conjecture for x^2+t

## 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 $x^2+t$.