An equivariant isomorphism theorem for mod $\mathfrak {p}$ reductions of arboreal Galois representations
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- by Andrea Ferraguti and Giacomo Micheli PDF
- Trans. Amer. Math. Soc. 373 (2020), 8525-8542 Request permission
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 $x^2+t$.References
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Additional Information
- Andrea Ferraguti
- Affiliation: Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany
- MR Author ID: 1156160
- Email: and.ferraguti@gmail.com
- Giacomo Micheli
- Affiliation: University of South Florida, 4202 E Fowler Ave, Tampa, Florida 33620
- MR Author ID: 1078793
- Email: gmicheli@usf.edu
- Received by editor(s): February 16, 2020
- Published electronically: October 5, 2020
- Additional Notes: The first author is grateful to the Max Planck Institute for Mathematics in Bonn for its hospitality and financial support, and would also like to thank the EPFL, where most of the ideas of this paper were generated, for the hospitality and financial support.
The second author was partially supported by the Swiss National Science Foundation grant number 171249. - © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 8525-8542
- MSC (2010): Primary 37P05, 37P15; Secondary 11G05, 20E08
- DOI: https://doi.org/10.1090/tran/8247
- MathSciNet review: 4177267