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Difference of solutions for the inversion problem of ultra-elliptic integrals

Takanori Ayano

Abstract

Let $V$ be a hyperelliptic curve of genus 2 defined by $Y^2=f(X)$, where $f(X)$ is a polynomial of degree 5. The sigma function associated with $V$ is a holomorphic function on $\mathbb{C}^2$. For a point $P$ on $V$, we consider the problem to express the $X$-coordinate of $P$ in terms of the image of $P$ under the Abel-Jacobi map. Two meromorphic functions $f_2$ and $g_2$ on $\mathbb{C}^2$ which give solutions of this problem are known. Since $f_2$ and $g_2$ coincide on the zero set of the sigma function, it is expected that $f_2-g_2$ can be divided by the sigma function. In this paper, we decompose $f_2-g_2$ into a product of the sigma function and a meromorphic function explicitly.

Difference of solutions for the inversion problem of ultra-elliptic integrals

Abstract

Let be a hyperelliptic curve of genus 2 defined by , where is a polynomial of degree 5. The sigma function associated with is a holomorphic function on . For a point on , we consider the problem to express the -coordinate of in terms of the image of under the Abel-Jacobi map. Two meromorphic functions and on which give solutions of this problem are known. Since and coincide on the zero set of the sigma function, it is expected that can be divided by the sigma function. In this paper, we decompose into a product of the sigma function and a meromorphic function explicitly.
Paper Structure (3 sections, 18 theorems, 60 equations)

This paper contains 3 sections, 18 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. The two-dimensional sigma function
  3. Inversion problem of ultra-elliptic integrals

Key Result

Proposition 2.1

For $m_1,m_2\in\mathbb{Z}^2$, let $\Omega=2\omega'm_1+2\omega"m_2$. Then, for $u\in\mathbb{C}^2$, we have

Theorems & Definitions (29)

  • Proposition 2.1: BEL-97-1
  • Theorem 2.2: BEL-99-R, N1
  • Remark 2.3
  • Lemma 3.1: Riemann vanishing theorem, e.g., Mumford-I
  • Lemma 3.2: G2, J
  • Lemma 3.3: M
  • Lemma 3.4: G, O-98
  • Lemma 3.5: BEL-2012, AB2019
  • Lemma 3.6: Riemann singularity theorem, e.g., Mumford-I
  • Lemma 3.7
  • ...and 19 more