Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Oleg K. Sheinman

TL;DR

The work extends the classical Jacobi inversion via the Riemann vanishing theorem to real algebraic curves equipped with an involution, by developing the Abel–Prym framework and Prym theta-function formalism on the isoPrym covering. It derives symmetry properties of Prym data under real structures, and provides an inversion theorem that characterizes the preimage of real isoPrym subvarieties in terms of tau- or sigma tau-invariant divisors satisfying a Prym-related relation. A key methodological advance is a theta-function based procedure to compute symmetric functions of the zeros of the relevant auxiliary function, enabling effective recovery of divisor data without solving the original transcendental equations. The results unify and extend prior real-curve theory (including separating and non-separating cases) and yield realness conditions aligned with the Novikov–Veselov framework, broadening the applicability to integrable systems with real spectral data.

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.

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

TL;DR

The work extends the classical Jacobi inversion via the Riemann vanishing theorem to real algebraic curves equipped with an involution, by developing the Abel–Prym framework and Prym theta-function formalism on the isoPrym covering. It derives symmetry properties of Prym data under real structures, and provides an inversion theorem that characterizes the preimage of real isoPrym subvarieties in terms of tau- or sigma tau-invariant divisors satisfying a Prym-related relation. A key methodological advance is a theta-function based procedure to compute symmetric functions of the zeros of the relevant auxiliary function, enabling effective recovery of divisor data without solving the original transcendental equations. The results unify and extend prior real-curve theory (including separating and non-separating cases) and yield realness conditions aligned with the Novikov–Veselov framework, broadening the applicability to integrable systems with real spectral data.

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.

Paper Structure

This paper contains 13 sections, 14 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Real curves, their Jacobians and $\theta$-functions
  3. Separating and non-separating curves. Topological type. Real bases
  4. Realness properties of Riemann matrices of real curves
  5. Riemann $\theta$-function of a real curve
  6. Real curves with involutions, their Prym varieties, and $\theta$-functions
  7. Prym matrix for a general curve with an involution
  8. Symmetries of the Prym matrix for a real curve with an involution
  9. Prym theta function, isoPrym variety and the Abel--Prym map
  10. The Abel--Prym map, and inversion theorem
  11. Riemann vanishing theorem
  12. $\theta$-function formula for symmetric functions of zeroes of $F_{\varepsilon^{-1}(z-\Delta)}(P)$
  13. Inversion theorem for real curves

Key Result

Theorem 2.1

Two real curves are topologically equivalent iff they have the same topological type.

Theorems & Definitions (28)

  • Theorem 2.1: Nat1
  • Theorem 2.2: Nat1
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Remark 2.5
  • proof : Proof of Lemma \ref{['L:8.2']}
  • Lemma 2.6
  • proof
  • Remark 2.7
  • ...and 18 more