Group actions on Jacobian varieties

Consider a finite group $G$ acting on a Riemann surface $S$, and the associated branched Galois cover ${\pi }_G:S\to Y=S\mathrm{/}G$. We introduce the concept of geometric signature for the action of $G$, and we show that it captures the information of the geometric structure of the lattice of intermediate covers, the information about the isotypical decomposition of the rational representation of the group $G$ acting on the Jacobian variety $JS$ of $S$, and the dimension of the subvarieties of the isogeny decomposition of $JS$. We also give a version of Riemann’s existence theorem, adjusted to the present setting.