Prym differentials and Teichmüller spaces

In this article we study multiplicative meromorphic functions and differentials on Riemann surfaces of finite type. We proved an analogue of P. Appell's formula on decomposition of multiplicative functions with poles of arbitrary multiplicity into a sum of elementary Prym integrals. We construct explicit bases for some important factor spaces and prove a theorem on a fiber isomorphism of vector bundles and $n!$-sheeted mappings over Teichmüller spaces. This theorem gives an important relation between spaces of Prym differentials on a compact Riemann surfaces and on a Riemann surfaces of finite type.