Relating two genus 0 problems of John Thompson

TL;DR

This paper explores three interconnected genus 0 problems in mathematics, including polynomial range coincidences, monodromy group classifications, and the Monstrous Moonshine conjecture, revealing deep links between algebra, geometry, and group theory.

Contribution

It establishes connections between three classical genus 0 problems, advancing understanding of polynomial mappings, monodromy groups, and the Monster group's character theory.

Findings

Classification of monodromy groups for rational functions

Relations between genus 0 modular curves and Monster group characters

Insights into polynomial range coincidences over finite fields

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.