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

Published online by Cambridge University Press:  20 November 2018

Patrick Ingram*
Mathematics Department, Colorado State University, Fort Collins, Colorado, USA and Department of Mathematics and Statistics, York University, Toronto, ON, e-mail :
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Given two monic polynomials $f$ and $g$ with coefficients in a number field $K$, and some $\alpha \,\in \,K$, we examine the action of the absolute Galois group $Gal\left( \bar{K}/K \right)$ on the directed graph of iterated preimages of $\alpha $ under the correspondence $g\left( y \right)\,=\,f\left( x \right)$, assuming that $\deg \left( f \right)\,>\,\deg \left( g \right)$ and that $\gcd \left( \deg \left( f \right),\deg \left( g \right) \right)\,=1$. If a prime of $K$ exists at which $f$ and $g$ have integral coefficients and at which $\alpha $ is not integral, we show that this directed graph of preimages consists of finitely many $Gal\left( \bar{K}/K \right)$-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 $\mathbb{C}$, although the construction of a Böttcher coordinate for complex holomorphic correspondences remains unresolved.

Research Article
Copyright © Canadian Mathematical Society 2018


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