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p-adic Uniformization and the Action of Galois on Certain Affine Correspondences

  • Patrick Ingram (a1)
Abstract

Given two monic polynomials ƒ and g with coefficients in a number field K, and some K, we examine the action of the absolute Galois group Gal(/K) on the directed graph of iterated preimages of under the correspondence g(y) = ƒ(x), assuming that deg(ƒ) > deg(g) and that gcd (deg(ƒ), deg(g)) = 1. If a prime of K exists at which ƒ and g have integral coefficients and at which is not integral, we show that this directed graph of preimages consists of finitely many Gal(/K)-orbits. We obtain this result by establishing a p-adic uniformization of such correspondences, tenuously related to Böttcher’s uniformization of polynomial dynamical systems over , although the construction of a Böttcher coordinate for complex holomorphic correspondences remains unresolved.

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[1] Boston, N. and Jones, R., Arboreal Galois representations. Geom. Dedicata. 124(2007), 2735. http://dx.doi.org/10.1007/s10711-006-9113-9
[2] Ingram, P., Arboreal Galois representations and uniformization of polynomial dynamics. Bull. Lond. Math. Soc. 45(2013), no. 2, 301308. http://dx.doi.Org/10.1112/blms/bdsO88
[3] Ingram, P., Canonical heights for correspondences. Trans. Amer. Math. Soc, to appear. http://dx.doi.Org/10.1090/tran/7288
[4] Ingram, P., Critical dynamics of variable-separated affine correspondences. J. Lond. Math. Soc. (2) 95(2017), no. 3, 10111034. http://dx.doi.Org/10.1142/S1793042117501263
[5] Jones, R., Galois representations from pre-image trees: an arboreal survey. In: Actes de la Conference “Theorie des Nombres et Applications”, Publ. Math. Besancon Theorie Nr., 2013, Presses Univ. Franche-Comte, Besancon, 2013, pp. 107-136.
[6] Jones, R. and Levy, A., Eventually stable rational funetions. Int. J. Number Theory 13(2017), 22992318.
[7] Milnor, J., Dynamics in one complex variable. Third ed., Annais of Mathematics Studies, Princeton University Press, Princeton, NJ, 2006.
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Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
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