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

2025/11/06 by О. К. Шейнман, Sheinman, Oleg K.
Mathematics · #Algebraic Geometry and Number Theory #Holomorphic and Operator Theory #Advanced Algebra and Geometry

paper · pdf · doi:10.48550/arxiv.2511.04229

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.

Citations

• Inversion of the Abel--Prym map in presence of an additional involution
• Separation of variables for Hitchin systems with the structure group SO(4), on genus two curves
• Hitchin systems: some recent advances
• Hitchin systems on hyperelliptic curves
• Multi-Dimensional Sigma-Functions

Related

• Inversion of the Abel--Prym map in presence of an additional involution
• A new approach to separation of variables for the Clebsch integrable system. Part II: Inversion of the Abel--Prym map
• A generalization of Riemann's theta functions for singular curves
• Recovering of curves with involution by extended Prym data
• Jacobi inversion on strata of the Jacobian of the C_rs curve y^r = f(x). II