Prym varieties and fourfold covers

We describe the isotypical decomposition of the Jacobian variety JW of the Galois extension W–>T of any fourfold cover of smooth connected irreducible projective complex curves X–>T, in terms of Prym’s of intermediate covers. We also compute the degree of the isogenies involved and as a result we obtain new proofs of the bigonal and trigonal constructions. Furthermore, we give examples of families of Jacobians isogenous to a product of Jacobians and of families of Prym varieties isogenous to the product of elliptic curves.