Relating two genus 0 problems of John Thompson

Abstract

The "relating" entwines three problems: 1. Davenport's Problem, describing pairs of polynomials over Q whose ranges on Z/p are the same for almost all p. 2. Showing that the monodromy groups of rational function maps over the complexes are limited to a finite set of groups, outside of groups close to alternating groups (example, symmetric groups) with special representations, and dihedral and cyclic groups. 3. Relating the genus 0 modular curves to the character group of the Monster simple group, so-called Monstrous Moonshine. http://www.math.uci.edu/~mfried/pathlist-cov/thomp-genus0.html has a more detailed exposition on the paper; http://www.math.uci.edu/~mfried/deflist-cov/Genus0-Prob.html gives a separate description of genus 0 problem #2.