On the fields of definition of genus‐one covers of P1$\mathbb {P}^1$

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 , there exists a Belyi pair such that the degree of a field of definition over the field of moduli is greater than . As a by‐product, we obtain a counterexample to the local‐global principle for Belyi pairs.