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

TL;DR

This paper constructs examples of genus-one covers of the projective line with fields of definition significantly larger than their fields of moduli, providing counterexamples to the local-global principle for Belyi pairs.

Contribution

It demonstrates the existence of Belyi pairs with arbitrarily large degrees of field of definition over their moduli, challenging previous assumptions about their definability.

Findings

Existence of Belyi pairs with large degree of field of definition

Counterexamples to the local-global principle for Belyi pairs

Field of definition can be arbitrarily larger than the field of moduli

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.