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Relationships Between Hyperelliptic Functions of Genus 2 and Elliptic Functionsстатья
- Авторы: Ayano Takanori, Buchstaber Victor M.
- Журнал: Symmetry, Integrability and Geometry - Methods and Applications
- Том: 18
- Год издания: 2022
- Издательство: Department of Applied Research, Institute of Mathematics of National Academy of Science of Ukraine
- Местоположение издательства: Ukraine
- Первая страница: 1
- Последняя страница: 30
- Номер статьи: 10
- DOI: 10.3842/sigma.2022.010
- Аннотация: 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 Ei 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 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 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.
- Добавил в систему: Бухштабер Виктор Матвеевич