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

Source abstract

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.