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A question for iterated Galois groups in arithmetic dynamics

Part of: Arithmetic problems. Diophantine geometry Arithmetic algebraic geometry Arithmetic and non-Archimedean dynamical systems Algebraic number theory: global fields

Published online by Cambridge University Press:  10 July 2020

Andrew Bridy ,
John R. Doyle ,
Dragos Ghioca ,
Liang-Chung Hsia  and
Thomas J. Tucker
Andrew Bridy
Affiliation:
Departments of Political Science and Computer Science, Yale University, New Haven, CT06511, USA e-mail: andrew.bridy@yale.edu
John R. Doyle
Affiliation:
Department of Mathematics and Statistics, Louisiana Tech University, Ruston, LA71272, USA e-mail: jdoyle@latech.edu
Dragos Ghioca*
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BCV6T 1Z2, Canada
Liang-Chung Hsia
Affiliation:
Department of Mathematics, National Taiwan Normal University, Taipei, Taiwan, ROC e-mail: hsia@math.ntnu.edu.tw
Thomas J. Tucker
Affiliation:
Department of Mathematics, University of Rochester, Rochester, NY14620, USA e-mail: thomas.tucker@rochester.edu
*
e-mail: dghioca@math.ubc.ca
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Abstract

We formulate a general question regarding the size of the iterated Galois groups associated with an algebraic dynamical system and then we discuss some special cases of our question. Our main result answers this question for certain split polynomial maps whose coordinates are unicritical polynomials.

Keywords

Arithmetic dynamicsarboreal Galois representationsiterated Galois groups

MSC classification

Primary: 37P15: Global ground fields
Secondary: 11G50: Heights 11R32: Galois theory 14G25: Global ground fields 37P05: Polynomial and rational maps 37P30: Height functions; Green functions; invariant measures
Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 2 , June 2021 , pp. 401 - 417
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Copyright
© Canadian Mathematical Society 2020

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Footnotes

D.G. was partially supported by an NSERC Discovery grant. L.-C. H. was partially supported by MOST Grant 108-2115-M-003-005-MY2.

References

Bachmakov, M., Un théorème de finitude sur la cohomologie des courbes elliptiques . C. R. Acad. Sci. Paris Sér. A-B 270(1970), A999–A1001.Google Scholar
Benedetto, R. L., Heights and preperiodic points of polynomials over function fields . Int. Math. Res. Not. (IMRN) 62(2005), 3855–3866. https://doi.org/10.1155/IMRN.2005.3855 CrossRefGoogle Scholar
Benedetto, R. L., Faber, X., Hutz, B., Juul, J., and Yasufuku, Y., A large arboreal Galois representation for a cubic postcritically finite polynomial . Res. Number Theory 3(2017), Paper No. 29. https://doi.org/10.1007/s40993-017-0092-8 CrossRefGoogle Scholar
Benedetto, R. L., Ghioca, D., Juul, J., and Tucker, T. J., Iterated Galois groups associated to postcritically finite quadratic polynomials . In preparation.Google Scholar
Bombieri, E. and Gubler, W., Heights in Diophantine geometry . New Mathematical Monographs, 4, Cambridge University Press, Cambridge, 2006. https://doi.org/10.1017/CB9780511542879 Google Scholar
Boston, N. and Jones, R., Arboreal Galois representations . Geom. Dedicata 124(2007), 27–35. https://doi.org/10.1007/s10711-006-9113-9 CrossRefGoogle Scholar
Boston, N. and Jones, R., The image of an arboreal Galois representation . Pure Appl. Math. Q. 5 (2009), no. 1, 213–225. https://doi.org/10.4310/PAMQ.2009.v5.n1.a6 CrossRefGoogle Scholar
Bridy, A., Doyle, J., Ghioca, D., Hsia, L.-C., and Tucker, T. J., Finite index theorems for iterated Galois groups of unicritical polynomials . Trans. Amer. Math. Soc., 2020, to appear.CrossRefGoogle Scholar
Bridy, A., Ingram, P., Jones, R., Juul, J., Levy, A., Manes, M., Rubinstein-Salzedo, S., and Silverman, J. H., Finite ramification for preimage fields of post-critically finite morphisms . Math. Res. Lett. 24(2017), no. 6, 1633–1647. https://doi.org/10.4310/MRL.2017.v24.n6.a3 CrossRefGoogle Scholar
Bridy, A. and Tucker, T. J., Finite index theorems for iterated Galois groups of cubic polynomials. Math. Ann. 373(2019), no. 1–2, 37–72. https://doi.org/10.1007/s00208-018-1670-3 CrossRefGoogle Scholar
Call, G. S. and Silverman, J. H., Canonical heights on varieties with morphisms . Compos. Math. 89(1993), no. 2, 163–205.Google Scholar
DeMarco, L., Ghioca, D., Krieger, H., Nguyen, K. D., Tucker, T. J., and Ye, H., Bounded height in families of dynamical systems . Int. Math. Res. Not. (IMRN) 2019(2019), no. 8, 2453–2482. https://doi.org/10.1093/imrn/rnx174 Google Scholar
Ghioca, D. and Nguyen, K. D., Dynamical anomalous subvarieties: structure and bounded height theorems . Adv. Math. 288(2016), 1433–1462. https://doi.org/10.1016/j.aim.2015.11.015 CrossRefGoogle Scholar
Ghioca, D. and Nguyen, K. D., Dynamics of split polynomial maps: uniform bounds for periods and applications . Int. Math. Res. Not. (IMRN) 2017(2017), 213–231. https://doi.org/10.1093/imrn/rnw041 Google Scholar
Ghioca, D., Nguyen, K. D., and Ye, H., The dynamical Manin-Mumford conjecture and the dynamical Bogomolov conjecture for endomorphisms of ${\left({\mathbb{P}}^1\right)}^n$ . Compos. Math. 154(2018), no. 7, 1441–1472. https://doi.org/10.1112/s0010437x18007157 CrossRefGoogle Scholar
Ghioca, D., Nguyen, K. D., and Ye, H., The Dynamical Manin-Mumford Conjecture and the Dynamical Bogomolov Conjecture for split rational maps . J. Eur. Math. Soc. (JEMS) 21(2019), no. 5, 1571–1594. https://doi.org/10.4171/JEMS/869 CrossRefGoogle Scholar
Ghioca, D., Tucker, T. J., and Zhang, S., Towards a dynamical Manin-Mumford conjecture . Int. Math. Res. Not. (IMRN) 2011(2011), no. 22, 5109–5122. https://doi.org/10.1093/imrn/rnq283 Google Scholar
Jones, R., Galois representations from pre-image trees: an arboreal survey . Actes de la Conférence “Théorie des Nombres et Applications,” Publ. Math. Besançon Algèbre Théorie Nr., 2013, Presses Univ. Franche-Comté, Besançon, 2013, pp. 107–136.Google Scholar
Jones, R., Iterated Galois towers, their associated martingales, and the $\;p$ -adic Mandelbrot set. Compos. Math. 143(2007), no. 5, 1108–1126. https://doi.org/10.1112/S0010437X07002667 CrossRefGoogle Scholar
Jones, R. and Levy, A., Eventually stable rational functions. Int. J. Number Theory 13(2017), no. 9, 2299–2318.CrossRefGoogle Scholar
Jones, R. and Manes, M., Galois theory of quadratic rational functions. Comment. Math. Helv. 89(2014), no. 1, 173–213.CrossRefGoogle Scholar
Juul, J., Iterates of generic polynomials and generic rational functions. Trans. Amer. Math. Soc. 371(2019), no. 2, 809–831.CrossRefGoogle Scholar
Juul, J., Kurlberg, P., Madhu, K., and Tucker, T. J., Wreath products and proportions of periodic points. Int. Math. Res. Not. (IMRN) 2016(2016), no. 13, 3944–3969.Google Scholar
Medvedev, A. and Scanlon, T., Invariant varieties for polynomial dynamical systems. Ann. of Math. (2) 179(2014), no. 1, 81–177.CrossRefGoogle Scholar
Odoni, R. W. K., The Galois theory of iterates and composites of polynomials. Proc. Lond. Math. Soc. 51(1985), no. 3, 385–414.CrossRefGoogle Scholar
Odoni, R. W. K., Realising wreath products of cyclic groups as Galois groups. Mathematika 35(1988), no. 1, 101–113.CrossRefGoogle Scholar
Pink, R., Profinite iterated monodromy groups arising from quadratic polynomials . Preprint, 2013. arXiv:1307.5678.Google Scholar
Pink, R., Profinite iterated monodromy groups arising from quadratic morphisms with infinite postcritical orbits . Preprint, 2013. arXiv:1309.5804.Google Scholar
Pink, R., Kummer theory for Drinfeld modules. Algebr. Number Theory 10(2016), no. 2, 215–234.CrossRefGoogle Scholar
Ribet, K., Kummer theory on extensions of abelian varieties by tori. Duke Math. J. 46(1979), no. 4, 745–761.CrossRefGoogle Scholar
Silverman, J. H., The arithmetic of dynamical systems. Graduate Texts in Mathematics, 241, Springer, New York, 2007.Google Scholar
Yuan, X. and Zhang, S., The arithmetic Hodge index theorem for adelic line bundles. Math. Ann. 367(2017), no. 3–4, 1123–1171.CrossRefGoogle Scholar
Zhang, S., Distributions in algebraic dynamics . Surv. Differ. Geom. 10(2006), 381–430.Google Scholar