On fields of moduli of curves

The field of moduli K of a curve X a priori defined over the separable closure K5 of K need not be a field of definition. This paper shows that the obstruction is essentially the same as the obstruction to K being a field of definition of the cover X → X/Aut(X). Using previous results of Dèbes-Douai, we then obtain a cohomological measure of the obstruction. This yields concrete criteria for the field of moduli to be a field of definition. An interesting application is the following local-global principle. If a curve X, together with all of its automorphisms, is defined over p for all primes p, then it is defined over .