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

Ingram, Patrick

Canadian mathematical bulletin, Tome 61 (2018) no. 3, pp. 531-542

Résumé

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.

Détail

DOI : 10.4153/CMB-2017-082-7

Mots-clés : 37P20, 11S20, arithmetic dynamics