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Reduction of bielliptic hyperelliptic functions of genus 3

Takanori Ayano

TL;DR

The paper tackles the reduction of bielliptic hyperelliptic functions for genus-$3$ curves by exploiting explicit degree-$2$ morphisms to an elliptic curve $E$ and a genus-$2$ curve $C$. Using sigma-function formalism and carefully constructed maps $\\varphi_1: V\\to E$ and $\\varphi_2: V\\to C$, it derives concrete pushforward/pullback relations on Jacobians and proves a key decomposition identity $\\varphi^*\\circ\\varphi_* = 2\\mathrm{Id}$. This framework yields explicit formulas expressing the genus-$3$ hyperelliptic functions $\\wp_{1,i}^V$ in terms of Weierstrass functions on $E$ and genus-$2$ hyperelliptic functions on $C$ (via linear maps $L_1$ and $L_2$), and, under certain parameter relations, further reduces genus-$2$ data to elliptic functions. The results advance both theoretical understanding and practical computation of higher-genus hyperelliptic functions and their reductions.

Abstract

The present paper is devoted to the problem about the reduction of hyperelliptic functions of genus 3. Our research was motivated by applications to the theory of equations and dynamical systems integrable in hyperelliptic functions. In this paper, we consider a hyperelliptic curve of genus 3 which admits a morphism of degree 2 to an elliptic curve. We express the hyperelliptic functions associated with the curve of genus 3 in terms of the Weierstrass elliptic functions and hyperelliptic functions of genus 2.

Reduction of bielliptic hyperelliptic functions of genus 3

TL;DR

The paper tackles the reduction of bielliptic hyperelliptic functions for genus- curves by exploiting explicit degree- morphisms to an elliptic curve and a genus- curve . Using sigma-function formalism and carefully constructed maps and , it derives concrete pushforward/pullback relations on Jacobians and proves a key decomposition identity . This framework yields explicit formulas expressing the genus- hyperelliptic functions in terms of Weierstrass functions on and genus- hyperelliptic functions on (via linear maps and ), and, under certain parameter relations, further reduces genus- data to elliptic functions. The results advance both theoretical understanding and practical computation of higher-genus hyperelliptic functions and their reductions.

Abstract

The present paper is devoted to the problem about the reduction of hyperelliptic functions of genus 3. Our research was motivated by applications to the theory of equations and dynamical systems integrable in hyperelliptic functions. In this paper, we consider a hyperelliptic curve of genus 3 which admits a morphism of degree 2 to an elliptic curve. We express the hyperelliptic functions associated with the curve of genus 3 in terms of the Weierstrass elliptic functions and hyperelliptic functions of genus 2.
Paper Structure (5 sections, 10 theorems, 76 equations)

This paper contains 5 sections, 10 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. The hyperelliptic sigma functions
  3. Bielliptic hyperelliptic curves of genus 3
  4. The maps between the Jacobian varieties
  5. Reduction of the hyperelliptic functions associated with $V$

Key Result

Proposition 2.1

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

Theorems & Definitions (21)

  • Proposition 2.1: BEL-97-1
  • Remark 2.2
  • Remark 2.3
  • Proposition 3.1
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3: KT
  • ...and 11 more