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Journal Article

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


Ferraguti,  Andrea
Max Planck Institute for Mathematics, Max Planck Society;

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Ferraguti, A., & Micheli, G. (2020). An equivariant isomorphism theorem for mod $\mathfrak p$ reductions of arboreal Galois representations. Transactions of the American Mathematical Society, 373(12), 8525-8542. doi:10.1090/tran/8247.

Cite as: http://hdl.handle.net/21.11116/0000-0007-8603-E
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 $x^2+t$.