Relationships Between Hyperelliptic Functions of Genus $2$ and Elliptic Functions

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.