Harmonic and holomorphic Prym differentials on a compact Riemann surface (Q1972178)

The author considers Prym differentials on compact Riemann surfaces. These are one-forms defined on the universal covering \(U\) of the surface \(F=U/\Gamma\) such that \(\Gamma\) acts on the form by multipliers which are given by a one-dimensional complex representation of \(\Gamma\). If the representation is trivial and a differential is holomorphic, this is exactly an Abelian differential which is uniquely determined by its periods over some finite family of one-cycles on \(F\). In the article under review, the opposite case in which the representation is essential, i.e. is rather different from trivial, is considered and the sets of periods which uniquely determine holomorphic and harmonic Prym differentials are described.