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Solution of the Jacobi inversion problem on non-hyperelliptic curves

Julia Bernatska, Dmitry Leykin

Abstract

In this paper we propose a method of solving the Jacobi inversion problem in terms of multiply periodic $\wp$ functions, also called Kleinian $\wp$ functions. This result is based on the recently developed theory of multivariable sigma functions for $(n,s)$-curves. Considering $(n,s)$-curves as canonical representatives in the corresponding classes of bi-rationally equivalent plane algebraic curves, we claim that the Jacobi inversion problem on plane algebraic curves is solved completely. Explicit solutions on trigonal, tetragonal and pentagonal curves are given as an illustration.

Solution of the Jacobi inversion problem on non-hyperelliptic curves

Abstract

In this paper we propose a method of solving the Jacobi inversion problem in terms of multiply periodic functions, also called Kleinian functions. This result is based on the recently developed theory of multivariable sigma functions for -curves. Considering -curves as canonical representatives in the corresponding classes of bi-rationally equivalent plane algebraic curves, we claim that the Jacobi inversion problem on plane algebraic curves is solved completely. Explicit solutions on trigonal, tetragonal and pentagonal curves are given as an illustration.
Paper Structure (23 sections, 10 theorems, 157 equations)

This paper contains 23 sections, 10 theorems, 157 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $(n,s)$-Curves
  4. Abel's map
  5. Sigma function
  6. The Jacobi inversion problem
  7. Entire rational functions on a curve
  8. Solving the Jacobi inversion problem
  9. Jacobi inversion problem on trigonal curves
  10. Proof of Theorem \ref{['T:C33m1']} ($(3,3\mathfrak{m}+1)$-Curves)
  11. Example: $(3,4)$-Curve
  12. Example: $(3,7)$-Curve
  13. Proof of Theorem \ref{['T:C33m2']} ($(3,3\mathfrak{m}+2)$-Curves)
  14. Example: $(3,5)$-Curve
  15. Jacobi inversion problem on tetragonal curves
  16. ...and 8 more sections

Key Result

Theorem 1

Let $D$ be a positive non-special divisor of degree $g$ on a non-degenerate $(n,s)$-curve of genus $g$. Then there exist $n-1$ entire rational functions $\mathcal{R}_{2g+l}$ of the weights $2g+l$, $l=0$, …, $n-2$, which vanish on $D$, and the following system of equations defines $D$ uniquely

Theorems & Definitions (19)

  • Definition
  • Example 1
  • Example 2
  • Example 3: Hyperelliptic curves
  • Theorem 1
  • Theorem 2
  • proof : Proof of Theorem \ref{['T1']}
  • proof : Proof of Theorem \ref{['T2']}
  • Example 4: Hyperelliptic curves
  • Theorem 3: $(3,3\mathfrak{m}+1)$-Curves
  • ...and 9 more