Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 010, 30 pages      arXiv:2106.06764

Relationships Between Hyperelliptic Functions of Genus 2 and Elliptic Functions

Takanori Ayano a and Victor M. Buchstaber b
a) Osaka City University, Advanced Mathematical Institute, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan
b) Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina Street, Moscow, 119991, Russia

Received June 15, 2021, in final form January 20, 2022; Published online February 01, 2022

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.

Key words: hyperelliptic function; elliptic function; sigma function; reduction of hyperelliptic functions; Jacobian variety of an algebraic curve.

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