Group actions on Jacobian varieties

Abstract

Consider a finite group G acting on a Riemann surface S, and the associated branched Galois cover πG : S → Y = S/G. We introduce the concept of geometric signature for the action of G, and we show that it captures much information: the geometric structure of the lattice of intermediate covers, 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.