Field of moduli and field of definition for curves of genus 2

TL;DR

This paper investigates the relationship between the field of moduli and the field of definition for genus 2 curves, proving that for most automorphism groups, the moduli point over a field k corresponds to a curve defined over k, and providing explicit constructions.

Contribution

It extends Mestre's results by showing that for all automorphism groups other than C_2, the field of moduli always equals the field of definition for genus 2 curves, with explicit construction methods.

Findings

For automorphism groups other than C_2, the field of moduli is a field of definition.

Explicit constructions of genus 2 curves from moduli points are provided.

The obstruction in Br_2(k) vanishes for these cases.

Abstract

Let M_2 be the moduli space that classifies genus 2 curves. If a curve C is defined over a field k, the corresponding moduli point P=[C] is defined over k. Mestre solved the converse problem for curves with Aut(C) isomorphic to C_2. Given a moduli point defined over k, Mestre finds an obstruction to the existence of a corresponding curve defined over k, that is an element in Br_2(k) not always trivial. In this paper we prove that for all the other possibilities of Aut(C), every moduli point defined over k is represented by a curve defined over k. We also give an explicit construction of such a curve in terms of the coordinates of the moduli point.