Hecke algebras acting on Abelian varieties

We also show that the notion of Prym variety for covers of curves may be extended to abelian varieties, and describe its isotypical decomposition with respect to the action of a natural induced subalgebra of its endomorphism ring. We apply the results to the decomposition of the Jacobian and Prym varieties of the intermediate cover given by H, in the case of smooth projective curves with G-action. We work out several examples that give rise to families of principally polarized abelian varieties, of Jacobian and Prym varieties, with large endomorphism rings.