  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.
