Fields of moduli and fields of definition of odd signature curves

Abstract

Let X be a smooth projective curve of genus \({g \geq 2}\) defined over a field K. We show that X can be defined over its field of moduli KX if the signature of the covering \({X \rightarrow X/ Aut(X)}\) is of type \({(0;c_1,\dots,c_k)}\) , where some ci appears an odd number of times. This result is applied to cyclic q-gonal curves and to plane quartics.