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Computation of $\wp$-functions on plane algebraic curves

Julia Bernatska

TL;DR

This work develops a rigorous, analytically grounded framework for computing Kleinian $\wp$-functions on plane algebraic curves of low gonality by constructing explicit Riemann-surface models. It combines radical solutions for quadratic, cubic, and quartic equations with carefully designed Abel maps, theta/sigma functions, and Bolza-type relations to obtain direct Abel images and periods, which are then used to evaluate $\wp$-functions and their higher derivatives. The authors provide detailed hyperelliptic and trigonal case studies, including explicit period matrices, $\varkappa$-matrices, and Jacobi-inversion verifications, and they illustrate how to determine the vector of Riemann constants and build paths to points on the curves. The approach enables direct computation of $\wp$-functions without solely relying on theta-function representations, with robust numerical verification against Jacobi inversion and Bolza-type formulas. This has potential impact for applications in integrable systems (e.g., KdV, NLS) where explicit, numerically stable evaluations of multiply periodic functions are required.

Abstract

Numerical tools for computation of $\wp$-functions, also known as Kleinian, or multiply periodic, are proposed. In this connection, computation of periods of the both first and second kinds is reconsidered. An analytical approach to constructing Riemann surfaces of plane algebraic curves of low gonalities is used. The approach is based on explicit radical solutions to quadratic, cubic, and quartic equations, which serve for hyperelliptic, trigonal, and tetragonal curves, respectively. The proposed analytical models of Riemann surfaces give full control over computation of the Abel image of any point or divisor. Therefore, computation of $\wp$-functions at Abel images of given divisors can be done directly. An alternative computation with the help of the Jacobi inversion problem is used for verification. Hyperelliptic and trigonal curves are considered in detail, and illustrated by examples. A method of finding the unique characteristic corresponding to the vector of Riemann constants is suggested for non-hyperelliptic and hyperelliptic curves.

Computation of $\wp$-functions on plane algebraic curves

TL;DR

This work develops a rigorous, analytically grounded framework for computing Kleinian -functions on plane algebraic curves of low gonality by constructing explicit Riemann-surface models. It combines radical solutions for quadratic, cubic, and quartic equations with carefully designed Abel maps, theta/sigma functions, and Bolza-type relations to obtain direct Abel images and periods, which are then used to evaluate -functions and their higher derivatives. The authors provide detailed hyperelliptic and trigonal case studies, including explicit period matrices, -matrices, and Jacobi-inversion verifications, and they illustrate how to determine the vector of Riemann constants and build paths to points on the curves. The approach enables direct computation of -functions without solely relying on theta-function representations, with robust numerical verification against Jacobi inversion and Bolza-type formulas. This has potential impact for applications in integrable systems (e.g., KdV, NLS) where explicit, numerically stable evaluations of multiply periodic functions are required.

Abstract

Numerical tools for computation of -functions, also known as Kleinian, or multiply periodic, are proposed. In this connection, computation of periods of the both first and second kinds is reconsidered. An analytical approach to constructing Riemann surfaces of plane algebraic curves of low gonalities is used. The approach is based on explicit radical solutions to quadratic, cubic, and quartic equations, which serve for hyperelliptic, trigonal, and tetragonal curves, respectively. The proposed analytical models of Riemann surfaces give full control over computation of the Abel image of any point or divisor. Therefore, computation of -functions at Abel images of given divisors can be done directly. An alternative computation with the help of the Jacobi inversion problem is used for verification. Hyperelliptic and trigonal curves are considered in detail, and illustrated by examples. A method of finding the unique characteristic corresponding to the vector of Riemann constants is suggested for non-hyperelliptic and hyperelliptic curves.
Paper Structure (33 sections, 5 theorems, 132 equations, 11 figures)

This paper contains 33 sections, 5 theorems, 132 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Sato weight
  4. Abel map
  5. Theta and sigma functions
  6. Bolza formulas and generalizations
  7. Jacobi inversion problem
  8. Periods on a hyperelliptic curve
  9. Hyperelliptic curves
  10. Riemann surface
  11. Monodromy path
  12. Homology
  13. Cohomology
  14. Computation of periods
  15. Example 1: Arbitrary complex branch points
  16. ...and 18 more sections

Key Result

Theorem 1

Let $\sqrt{\Delta}$ be defined by SqrtDef. Then $y_{\textsf{s}}$ defined by yDefH have discontinuity over the contour $\Gamma = \{x \mid \arg \Delta(x) = 0\}$, and $y_{+}$ serves as the analytic continuation of $y_{-}$ on the other side of the contour, and vice versa.

Figures (11)

  • Figure 1: Canonical homology cycles.$\space$
  • Figure 2: The contour $\Gamma$ (blue), and a continuous path (orange).
  • Figure 3: Density plots of $\arg y_{\textsf{s}}$, and the contour $\Gamma$ (blue).
  • Figure 4: Connection of solutions $y_+$ and $y_-$ on Sheet a.
  • Figure 5: The contour $\Gamma$ (blue), and a continuous path $\gamma$ (orange).
  • ...and 6 more figures

Theorems & Definitions (15)

  • Theorem 1
  • proof
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Theorem 2
  • proof
  • Theorem 3
  • ...and 5 more