Relationships Between Hyperelliptic Functions of Genus 2 and Elliptic Functions

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2022
• ISSN: 1815-0659
• Main Authors: Ayano, Takanori, Buchstaber, Victor M.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2022
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/211535
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Relationships Between Hyperelliptic Functions of Genus 2 and Elliptic Functions. Takanori Ayano and Victor M. Buchstaber. SIGMA 18 (2022), 010, 30 pages

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 of genus 2 that admits a morphism of degree 2 to an elliptic curve. Then there exist two elliptic curves ᵢ, i = 1, 2, and morphisms of degree 2 from to ᵢ. We construct hyperelliptic functions associated with from the Weierstrass elliptic functions associated with 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 to the appropriate subspaces in ℂ² are elliptic functions and describe them in terms of the Weierstrass elliptic functions associated with ᵢ. Further, we express the hyperelliptic functions associated with on ℂ² in terms of the Weierstrass elliptic functions associated with ᵢ. We derive these results by explicitly describing the homomorphisms between the Jacobian varieties of the curves and ᵢ induced by the morphisms from to ᵢ.