Fields of moduli and fields of definition of odd signature curves

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 K _X if the signature of the covering ${X \rightarrow X/ Aut(X)}$ is of type ${(0;c_1,\dots,c_k)}$ , where some c _i appears an odd number of times. This result is applied to cyclic q-gonal curves and to plane quartics.