Relationships Between Hyperelliptic Functions of Genus 2 and Elliptic Functions

TL;DR

This paper explores the relationships between hyperelliptic functions of genus 2 and elliptic functions, providing new explicit constructions and descriptions of these functions in terms of Weierstrass elliptic functions, with applications to integrable systems.

Contribution

It introduces explicit methods to derive hyperelliptic functions of genus 2 from elliptic functions via morphisms between their Jacobian varieties, including new formulas and descriptions.

Findings

Hyperelliptic functions can be constructed from elliptic functions using degree 2 morphisms.

Restrictions of hyperelliptic functions to certain subspaces are elliptic functions.

Explicit relations between hyperelliptic and elliptic functions are derived through Jacobian homomorphisms.

Abstract

The article is devoted to the classical problems about the relationships between elliptic functions and hyperelliptic functions of genus 2. It contains new results, as well as a derivation from them of well-known results on these issues. Our research was motivated by applications to the theory of equations and dynamical systems integrable in hyperelliptic functions of genus 2. We consider a hyperelliptic curve $V$ of genus 2 which admits a morphism of degree 2 to an elliptic curve. Then there exist two elliptic curves $E_i$, $i=1,2$, and morphisms of degree 2 from $V$ to $E_i$. We construct hyperelliptic functions associated with $V$ from the Weierstrass elliptic functions associated with $E_i$ and describe them in terms of the fundamental hyperelliptic functions defined by the logarithmic derivatives of the two-dimensional sigma functions. We show that the restrictions of hyperelliptic functions associated with $V$ to the appropriate subspaces in $\mathbb{C}^2$ are elliptic functions and describe them in terms of the Weierstrass elliptic functions associated with $E_i$. Further, we express the hyperelliptic functions associated with $V$ on $\mathbb{C}^2$ in terms of the Weierstrass elliptic functions associated with $E_i$. We derive these results by describing the homomorphisms between the Jacobian varieties of the curves $V$ and $E_i$ induced by the morphisms from $V$ to $E_i$ explicitly.