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Abelian function fields on Jacobian varieties

Julia Bernatska

TL;DR

The paper advances abelian function theory on Jacobians of plane curves by leveraging Kleinian $\wp$-functions and the multi-variable $\sigma$-function to uniformly model $\mathrm{Jac}(\mathcal{C})$ and $\mathrm{Kum}(\mathcal{C})$, derive addition laws, and solve dynamical equations. It delivers concrete Jacobi inversion solutions, builds basis $\wp$-functions, and constructs algebraic models for hyperelliptic and trigonal cases via Klein’s formula and residue methods, with explicit examples. It further develops algebraic models of Jacobians and Kummer varieties, and formulates addition laws through groupoid structure and multi-linear Baker–Hirota operators, linking to integrable hierarchies like KdV. The work yields practical tools for explicit computation of Abelian functions, enabling uniformization, algebraic descriptions, and identities that govern the higher-genus generalizations of classical elliptic function theory, while outlining open problems for higher gonality and degenerate families.

Abstract

In this paper the fields of multiply periodic, or Kleinian $\wp$-functions are exposed. Such a field arises on the Jacobian variety of an algebraic curve, and provides natural algebraic models of the Jacobian and Kummer varieties, possesses the addition law, and accommodates dynamical equations with solutions. All this will be explained in detail for plane algebraic curves in their canonical forms. Example of hyperelliptic and non-hyperelliptic curves are presented.

Abelian function fields on Jacobian varieties

TL;DR

The paper advances abelian function theory on Jacobians of plane curves by leveraging Kleinian -functions and the multi-variable -function to uniformly model and , derive addition laws, and solve dynamical equations. It delivers concrete Jacobi inversion solutions, builds basis -functions, and constructs algebraic models for hyperelliptic and trigonal cases via Klein’s formula and residue methods, with explicit examples. It further develops algebraic models of Jacobians and Kummer varieties, and formulates addition laws through groupoid structure and multi-linear Baker–Hirota operators, linking to integrable hierarchies like KdV. The work yields practical tools for explicit computation of Abelian functions, enabling uniformization, algebraic descriptions, and identities that govern the higher-genus generalizations of classical elliptic function theory, while outlining open problems for higher gonality and degenerate families.

Abstract

In this paper the fields of multiply periodic, or Kleinian -functions are exposed. Such a field arises on the Jacobian variety of an algebraic curve, and provides natural algebraic models of the Jacobian and Kummer varieties, possesses the addition law, and accommodates dynamical equations with solutions. All this will be explained in detail for plane algebraic curves in their canonical forms. Example of hyperelliptic and non-hyperelliptic curves are presented.

Paper Structure

This paper contains 34 sections, 19 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Canonical form of plane algebraic curves
  4. Sato weight
  5. Cohomology basis
  6. Examples
  7. Abel map
  8. Theta function
  9. Sigma function
  10. Vector of Riemann constants
  11. Multiply periodic $\wp$-functions
  12. Divisor classes
  13. Polynomial functions on a curve
  14. Abelian function fields associated with curves
  15. Jacobi inversion problem
  16. ...and 19 more sections

Key Result

Theorem 1

$\theta[K]$ as a function of not normalized coordinates has the maximal weighted order of vanishing equal to $\mathfrak{d}={-}\mathop{\mathrm{wgt}}\nolimits \sigma$, and defined by WgtSigma, at $u=0$, that is

Theorems & Definitions (46)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Theorem 1
  • Remark 6
  • Remark 7
  • Theorem 2
  • proof
  • ...and 36 more