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$p$-adic uniformization and the action of Galois on certain affine correspondences

  • Patrick Ingram,
    Mathematics Department, Colorado State University, Fort Collins, Colorado, USA
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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 $\operatorname{Gal}(\overline{K}/K)$ on the directed graph of iterated preimages of $\alpha$ under the correspondence $g(y)=f(x)$, assuming that $\deg(f)\gt \deg(g)$ and that $\gcd(\deg(f), \deg(g))=1$. 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 $\operatorname{Gal}(\overline{K}/K)$-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 $\mathbb{CC}$, although the construction of a Böttcher coordinate for complex holomorphic correspondences remains unresolved.
Keywords: arithmetic dynamics arithmetic dynamics
MSC Classifications: 37P20, 11S20 show english descriptions Non-Archimedean local ground fields
Galois theory
37P20 - Non-Archimedean local ground fields
11S20 - Galois theory

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