Inversion of the Abel--Prym map for real curves with involutions

Abstract: Riemann vanishing theorem is a main ingredient of the conventional technique related to the Jacobi inversion problem. In the case of curves with a holomorphic involution, it has been exhaustively expounded in wellknown Fay's Lectures on theta functions. The case of real algebraic curves with involution is presented with less completeness in the literature. We give a detailed presentation of that case, including real curves of non-separating type (with involution) not considered before with this relation. We obtain the Novikov--Veselov realness conditions in a different set-up.