Reduction of bielliptic hyperelliptic functions of genus 3

TL;DR

This paper explores reducing genus 3 hyperelliptic functions to simpler forms involving elliptic and genus 2 hyperelliptic functions, aiding in solving equations and dynamical systems.

Contribution

It provides a method to express genus 3 hyperelliptic functions via elliptic and genus 2 hyperelliptic functions, expanding tools for mathematical analysis.

Findings

Reduction formulas for genus 3 hyperelliptic functions

Explicit expressions in terms of elliptic functions

Applications to integrable dynamical systems

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.