On the fields of definition of genus-one covers of $\mathbb{P}^1$

Abstract

It is known that sometimes a Belyi pair is not defined over its field of moduli. Instead, it is defined over a finite degree extension of its field of moduli, called a field of definition. We show that given a number $m$ there exists a Belyi pair such that the degree of a field of definition over the field of moduli is greater than $m$. As a byproduct, we obtain a counterexample to the local-global principle for Belyi pairs.