Jacobians with group actions and rational idempotents

The object of this paper is to prove some general results about rational idempotents for a finite group G and deduce from them geometric information about the components that appear in the decomposition of the Jacobian variety of a curve with G-action.We give an algorithm to find explicit primitive rational idempotents for any G, as well as for rational projectors invariant under any given subgroup. These explicit constructions allow geometric descriptions of the factors appearing in the decomposition of a Jacobian with group action: from them we deduce the decomposition of any Prym or Jacobian variety of an intermediate cover, in the case of a Jacobian with G-action. In particular, we give a necessary and sufficient condition for a Prym variety of an intermediate cover to be such a factor.