The spaces of meromorphic Prym differentials on a finite Riemann surface

In the previous articles the second author started constructing a general theory of multiplicative functions and Prym differentials on a compact Riemann surface for arbitrary characters. Function theory on compact Riemann surfaces differs substantially from that on finite Riemann surfaces. In this article we start constructing a general function theory on variable finite Riemann surfaces for multiplicative meromorphic functions and differentials. We construct the forms of all elementary Prym differentials for arbitrary characters and find the dimensions of, and also construct explicit bases for, two important quotient spaces of Prym differentials. This yields the dimension of and a basis for the first holomorphic de Rham cohomology group of Prym differentials for arbitrary characters.