Relationships Between Hyperelliptic Functions of Genus 2 and Elliptic Functions

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