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Relationships Between Hyperelliptic Functions of Genus $2$ and Elliptic Functions
Takanori Ayanoa,
Victor M. Buchstaberb 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
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.
Keywords:
hyperelliptic function, elliptic function, sigma function, reduction of hyperelliptic functions, Jacobian variety of an algebraic curve.
MSC: 14H40,
14H42,
14K25,
32A20,
33E05 Received: June 15, 2021; in final form
January 20, 2022; Published online
February 1, 2022
Language: English
DOI:
10.3842/SIGMA.2022.010
Fulltext:
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