Odoni’s conjecture on arboreal Galois representations is false

Dittmann, Philip; Kadets, Borys

doi:10.1090/proc/15920

Odoni’s conjecture on arboreal Galois representations is false
HTML articles powered by AMS MathViewer

by Philip Dittmann and Borys Kadets

Proc. Amer. Math. Soc. 150 (2022), 3335-3343

DOI: https://doi.org/10.1090/proc/15920

Published electronically: April 1, 2022

PDF | Request permission

Abstract:

Suppose $f \in K[x]$ is a polynomial. The absolute Galois group of $K$ acts on the preimage tree $\mathrm {T}$ of $0$ under $f$. The resulting homomorphism $\phi _f\colon \operatorname {Gal}_K \to \operatorname {Aut} \mathrm {T}$ is called the arboreal Galois representation. Odoni conjectured that for all Hilbertian fields $K$ there exists a polynomial $f$ for which $\phi _f$ is surjective. We show that this conjecture is false.

References

Lior Bary-Soroker, On the characterization of Hilbertian fields, Int. Math. Res. Not. IMRN , posted on (2008), Art. ID rnn 089, 10. MR 2439555, DOI 10.1093/imrn/rnn089
Robert L. Benedetto and Jamie Juul, Odoni’s conjecture for number fields, Bull. Lond. Math. Soc. 51 (2019), no. 2, 237–250. MR 3937585, DOI 10.1112/blms.12225
David A. Cox, Galois theory, 2nd ed., Pure and Applied Mathematics (Hoboken), John Wiley & Sons, Inc., Hoboken, NJ, 2012. MR 2919975, DOI 10.1002/9781118218457
Michael D. Fried and Moshe Jarden, Field arithmetic, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 11, Springer-Verlag, Berlin, 2008. Revised by Jarden. MR 2445111
Wilfrid Hodges, A shorter model theory, Cambridge University Press, Cambridge, 1997. MR 1462612
Moshe Jarden, Algebraic patching, Springer Monographs in Mathematics, Springer, Heidelberg, 2011. MR 2768285, DOI 10.1007/978-3-642-15128-6
Rafe Jones, Galois representations from pre-image trees: an arboreal survey, Actes de la Conférence “Théorie des Nombres et Applications”, Publ. Math. Besançon Algèbre Théorie Nr., vol. 2013, Presses Univ. Franche-Comté, Besançon, 2013, pp. 107–136 (English, with English and French summaries). MR 3220023
Borys Kadets, Large arboreal Galois representations, J. Number Theory 210 (2020), 416–430. MR 4057534, DOI 10.1016/j.jnt.2019.09.021
David Marker, Model theory, Graduate Texts in Mathematics, vol. 217, Springer-Verlag, New York, 2002. An introduction. MR 1924282
Jürgen Neukirch, Alexander Schmidt, and Kay Wingberg, Cohomology of number fields, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 323, Springer-Verlag, Berlin, 2008. MR 2392026, DOI 10.1007/978-3-540-37889-1
R. W. K. Odoni, The Galois theory of iterates and composites of polynomials, Proc. London Math. Soc. (3) 51 (1985), no. 3, 385–414. MR 805714, DOI 10.1112/plms/s3-51.3.385
Joel Specter, Polynomials with surjective arboreal Galois representation exist in every degree, arXiv:1803.00434v2 (2018).