Fields of moduli of classical Humbert curves

TL;DR

This paper computes the fields of moduli for a specific family of genus five Riemann surfaces with automorphism group ${\

Contribution

It determines the field of moduli for genus five Humbert curves and proves these fields are also fields of definition, contrasting with other cases.

Findings

Fields of moduli are explicitly computed for the family.

These fields are proven to be fields of definition.

Contrasts with cases where the property fails.

Abstract

The computation of the field of moduli of a closed Riemann surface seems to be a very difficult problem and even more difficult is to determine if the field of moduli is a field of definition. In this paper we consider the family of closed Riemann surfaces of genus five admitting a group of conformal automorphisms isomorphic to ${\mathbb Z}_{2}^{4}$. These surfaces are non-hyperelliptic ones and turn out to be the highest branched abelian covers of the orbifolds of genus zero and five cone points of order two. We compute the field of moduli of these surfaces and we prove that they are fields of definition. This result is in contrast with the case of the highest branched abelian covers of the orbifolds of genus zero and six cone points of order two as there are cases for which the above property fails.