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

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

2002-07-02

arxiv.org/abs/math/0207015v1

Mathematics

Number Theory

14 pages

Scientific paper

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.

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