On a conjecture of Parker about Dessins d'Enfants

TL;DR

This paper disproves the broadest form of Parker's conjecture, which links group theory to the field of moduli in Dessins d'Enfants, clarifying limitations of the conjecture.

Contribution

It provides the first disproof of the most general case of Parker's conjecture, advancing understanding of the relationship between group theory and algebraic geometry.

Findings

The conjecture does not hold in its most general form.

Counterexamples are constructed demonstrating the disproof.

The result refines the scope of Parker's conjecture.

Abstract

This note is concerned with the disproof of the most general case of Parker's conjecture. The conjecture relates a certain group theoretic objects to the field of moduli of a Dessin d'enfant.