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

TL;DR

This paper extends the inversion of the Abel--Prym map to real algebraic curves with involutions, including non-separating types, and discusses the symmetry of the Prym theta function.

Contribution

It provides a detailed treatment of real curves with involutions, especially non-separating types, filling gaps in the existing literature.

Findings

Formulation of the symmetry of the Prym theta function for real curves with involution.

Extension of Abel--Prym map inversion to non-separating real curves.

Enhanced understanding of real algebraic 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 presented quite fully 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 provide a detailed presentation of that case, including the case of real curves of the non-separating type with a holomorphic involution, not considered before with this relation. In particular, we formulate the symmetry of the Prym theta function in this case.