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

Abstract

AbstractGiven 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.