Prym differentials as solutions to boundary value problems on Riemann surfaces

Construction of multiplicative functions and Prym differentials, including the case of characters with branch points, reduces to solving a homogeneous boundary value problem on the Riemann surface. The use of the well-established theory of boundary value problems creates additional possibilities for studying Prym differentials and related bundles. Basing on the theory of boundary value problems, we fully describe the class of divisors of Prym differentials and obtain new integral expressions for Prym differentials, which enable us to study them directly and, in particular, to study their dependence on the point of the Teichmüller space and characters. Relying on this, we obtain and generalize certain available results on Prym differentials by a new method.