Relationships Between Hyperelliptic Functions of Genus 2 and Elliptic
Functions

Relationships Between Hyperelliptic Functions
of Genus 2 and Elliptic Functions

Takanori AYANO a and Victor M. BUCHSTABER b

T. Ayano and V.M. Buchstaber

a) Osaka City University, Advanced Mathematical Institute,
a) 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan<EMAIL_ADDRESS>cu.ac.jp https://researchmap.jp/ayano75?lang=en

b) Steklov Mathematical Institute of Russian Academy of Sciences,
b) 8 Gubkina Street, Moscow, 119991, Russia<EMAIL_ADDRESS>
Received June 15, 2021, in final form January 20, 2022; Published online
February 01, 2022

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.

hyperelliptic function; elliptic function; sigma function; reduction of
hyperelliptic functions; Jacobian variety of an algebraic curve

14H40; 14H42; 14K25; 32A20; 33E05

Dedicated to the memory of the remarkable
mathematician Victor Enolski $($1945–2019$)$

##  1 Introduction

Victor Enolski’s contributions to mathematics, mathematical and theoretical
physics are reflected in the memorial survey [44]. Our article is devoted to
the issues that were in the center of his attention; it contains our new
results on these issues and a description of their connection with classical
and relatively recent results, including the results obtained by Victor
Enolski together with his co-authors.

The problem whether the Jacobian variety of a hyperelliptic curve of genus 2
is isogenous to the direct product of elliptic curves is considered in many
papers of number theory (e.g., [12, 13, 14, 21, 27, 28, 39, 43, 46, 47, 53]).
The problems discussed in our article are related to the well-known open
problem about rational points on the Jacobian variety of a curve of genus 2
[31, 43]. The problem on the isogeny between the Jacobian variety of a curve
of genus 2 and the direct product of elliptic curves is naturally connected
with the following well-known problem (to which our research was devoted): Let
solutions of differential equations and dynamical systems in terms of
hyperelliptic functions of genus 2 are given. Under conditions when a
reduction of these functions to elliptic functions is possible, find an
explicit form of these solutions in terms of elliptic functions. In this
paper, we consider the algebraic curves over $\mathbb{C}$. The reductions of
the Riemann theta functions and the hyperelliptic functions turned out to be
important in real physical problems (cf. [22, 50]). In [22, 50], a
hyperelliptic curve of genus 2 defined by $y^{2}=f(x)$ with a square-free
polynomial $f(x)$ of degree 6 is considered. In the case when the curve is
defined by $y^{2}=\widetilde{f}(x)$ with a square-free polynomial
$\widetilde{f}(x)$ of degree 5, its sigma function is determined by the
coefficients of the polynomial $\widetilde{f}(x)$. In the case when the curve
is defined by $y^{2}=f(x)$ with a square-free polynomial $f(x)$ of degree 6,
one of the branch points can be chosen and an isomorphism can be explicitly
set with a curve defined by $y^{2}=\widetilde{f}(x)$ with a square-free
polynomial $\widetilde{f}(x)$ of degree 5. Thus, in the case under discussion,
the sigma function and the corresponding hyperelliptic functions are
determined by the coefficients of the polynomial of degree 6 and the choice of
one of its zeros. In some cases, an explicit transformation uses several
branch points (cf. Proposition 3.3).

For a positive integer $\Delta$, let $\mathcal{H}_{\Delta}$ be the set of
complex $2\times 2$ matrix
$\tau=\left(\begin{smallmatrix}\tau_{11}&\tau_{12}\\\
\tau_{12}&\tau_{22}\end{smallmatrix}\right)$ such that ${}^{t}\tau=\tau$,
$\operatorname{Im}\tau>0$, and

$\displaystyle
h_{1}\tau_{11}+h_{2}\tau_{12}+h_{3}\tau_{22}+h_{4}\big{(}\tau_{12}^{2}-\tau_{11}\tau_{22}\big{)}+h_{5}=0$

for some integers $h_{1}$, $h_{2}$, $h_{3}$, $h_{4}$, $h_{5}$ satisfying
$\Delta=h_{2}^{2}-4(h_{1}h_{3}+h_{4}h_{5})$. Then $\mathcal{H}_{\Delta}$ is
called Humbert surface with invariant $\Delta$ [33, 34, 35].

######  Theorem 1.1 ([38], [7, Theorem 5.10], [17, Theorem 7.4]).

Let $H$ be a hyperelliptic curve of genus $2$ and $N$ be an integer such that
$N\geq 2$. There exist an elliptic curve $W$ and a morphism $H\to W$ of degree
$N$ if and only if a normalized period matrix
$\tau=\left(\begin{smallmatrix}\tau_{11}&\tau_{12}\\\
\tau_{12}&\tau_{22}\end{smallmatrix}\right)$ of $H$ belongs to the Humbert
surface $\mathcal{H}_{\Delta}$ with $\Delta=N^{2}$.

In [9], the geometric characterizations of the Humbert surfaces in terms of
the presence of certain curves on the associated Kummer plane are given. In
[6, Section 3.1], [7, Section 5.3.3], [17, Section 7.3.3], [19, Section 6.3,
Section 11.5], [25], the Humbert surfaces are considered and relationships
between the hyperelliptic functions associated with a curve of genus 2 and the
Jacobi elliptic functions are derived. In [6, p. 3448], [7, p. 366], [17, pp.
79, 80], [19, pp. 99, 175], [25], it is mentioned that the Jacobi elliptic
functions and the hyperelliptic functions associated with a curve of genus 2
give coordinates on the Kummer surfaces. In [12], algebraic correspondences
between Kummer surfaces associated with the Jacobian variety of a
hyperelliptic curve of genus 2 admitting a morphism of degree 2 to an elliptic
curve and Kummer surfaces associated with the product of two non-isogenous
elliptic curves are considered. In the equations (2.56) and (2.57) of [12],
the formulae in [6, p. 3448], [19, pp. 99, 100], [25] are described in terms
of coordinates on the Kummer surfaces. In this paper, by an approach different
from [6, 7, 17, 19, 25], we derive relationships between the hyperelliptic
functions associated with the curve of genus 2 and the Weierstrass elliptic
functions.

For $\alpha,\beta\in\mathbb{C}$ satisfying $\alpha^{2},\beta^{2}\neq 0,1$,
$\alpha^{2}\neq\beta^{2}$, and $\alpha^{2}\beta^{2}\neq 1$, we consider the
nonsingular hyperelliptic curve of genus $2$

$\displaystyle V=\big{\\{}(x,y)\in\mathbb{C}^{2}\mid
y^{2}=x(x-1)\big{(}x-\alpha^{2}\big{)}\big{(}x-\beta^{2}\big{)}\big{(}x-\alpha^{2}\beta^{2}\big{)}\big{\\}}.$

For a hyperelliptic curve $H$ of genus 2, it is known that there exist an
elliptic curve $W$ and a morphism $H\to W$ of degree $2$ if and only if $H$ is
isomorphic to the curve $V$ for some $\alpha$, $\beta$ (see Section 3). Let
$E_{1}$ and $E_{2}$ be the elliptic curves defined by

$\displaystyle
E_{1}=\bigg{\\{}(X,Y)\in\mathbb{C}^{2}\,\bigg{|}\,Y^{2}=X(X-1)\bigg{(}X-\frac{(\alpha-\beta)^{2}}{(\alpha\beta-1)^{2}}\bigg{)}\bigg{\\}},$
$\displaystyle
E_{2}=\bigg{\\{}(X,Y)\in\mathbb{C}^{2}\,\bigg{|}\,Y^{2}=X(X-1)\bigg{(}X-\frac{(\alpha+\beta)^{2}}{(\alpha\beta+1)^{2}}\bigg{)}\bigg{\\}}.$

Then we can define the morphisms of degree 2 (see Section 3)

$\displaystyle\varphi_{1}\colon\quad V\to
E_{1},\qquad(x,y)\mapsto(X,Y)=\bigg{(}\frac{(\alpha-\beta)^{2}x}{(x-\alpha\beta)^{2}},\frac{(\alpha-\beta)^{2}y}{(\alpha\beta-1)(x-\alpha\beta)^{3}}\bigg{)},$
$\displaystyle\varphi_{2}\colon\quad V\to
E_{2},\qquad(x,y)\mapsto(X,Y)=\bigg{(}\frac{(\alpha+\beta)^{2}x}{(x+\alpha\beta)^{2}},\frac{-(\alpha+\beta)^{2}y}{(\alpha\beta+1)(x+\alpha\beta)^{3}}\bigg{)}.$

Let $\operatorname{Jac}(V)$, $\operatorname{Jac}(E_{1})$, and
$\operatorname{Jac}(E_{2})$ be the Jacobian varieties of $V$, $E_{1}$, and
$E_{2}$, respectively. The maps $\varphi_{i}$, $i=1,2$, induce the
homomorphisms of the Jacobian varieties

$\displaystyle\psi_{i,*}\colon\
\operatorname{Jac}(V)\to\operatorname{Jac}(E_{i}),\qquad\psi_{i}^{*}\colon\
\operatorname{Jac}(E_{i})\to\operatorname{Jac}(V),\qquad i=1,2.$

In this paper, we describe the maps $\psi_{i,*}$ and $\psi_{i}^{*}$ explicitly
in Propositions 4.1 and 5.6. Let $\sigma({\bf u})$, ${\bf
u}={}^{t}(u_{1},u_{3})\in\mathbb{C}^{2}$, be the sigma function associated
with $V$, which is the holomorphic function on $\mathbb{C}^{2}$,
$\wp_{j,k}=-\partial_{u_{j}}\partial_{u_{k}}\log\sigma$, and
$\wp_{j,k,\ell}=\partial_{u_{\ell}}\wp_{j,k}$, where
$\partial_{u_{j}}=\frac{\partial}{\partial u_{j}}$. Then the functions
$\wp_{j,k}$ and $\wp_{j,k,\ell}$ are the fundamental meromorphic functions on
$\operatorname{Jac}(V)$. For $i=1,2$, let $\wp_{E_{i}}$ be the Weierstrass
elliptic function associated with the elliptic curve $E_{i}$. Let $f_{i}({\bf
u})=\wp_{E_{i}}(\psi_{i,*}({\bf u}))$ for ${\bf u}\in\operatorname{Jac}(V)$
and $i=1,2$. Then the functions $f_{1}({\bf u})$ and $f_{2}({\bf u})$ are the
meromorphic functions on $\operatorname{Jac}(V)$. In this paper, we express
$f_{1}({\bf u})$ and $f_{2}({\bf u})$ in terms of $\wp_{1,1}({\bf
u}),\wp_{1,3}({\bf u})$, and $\wp_{3,3}({\bf u})$ explicitly in Theorem 4.3
and Corollary 4.4. We consider the homomorphisms of the Jacobian varieties

$\displaystyle\psi_{i}^{*}\colon\quad\operatorname{Jac}(E_{i})\to\operatorname{Jac}(V),\qquad
i=1,2.$

For $v\in\operatorname{Jac}(E_{i})$, the functions
$\wp_{j,k}(\psi_{i}^{*}(v))$ and $\wp_{j,k,\ell}(\psi_{i}^{*}(v))$ are the
meromorphic functions on $\operatorname{Jac}(E_{i})$. In this paper, we
describe the functions $\wp_{j,k}(\psi_{i}^{*}(v))$ and
$\wp_{j,k,\ell}(\psi_{i}^{*}(v))$ in terms of $\wp_{E_{i}}(v)$ explicitly in
Theorem 5.7 and Corollary 5.8, i.e., we show that the restrictions of the
hyperelliptic functions $\wp_{j,k}$ and $\wp_{j,k,\ell}$ to the appropriate
subspaces in $\mathbb{C}^{2}$ are elliptic functions and describe them in
terms of the Weierstrass elliptic functions $\wp_{E_{i}}$. Further, by using
the addition formulae for $\wp_{j,k}$, which are given in [30, Theorem 3.3],
we express $\wp_{j,k}$ on $\mathbb{C}^{2}$ in terms of $\wp_{E_{1}}$ and
$\wp_{E_{2}}$ explicitly in Theorem 5.9. From Theorem 5.9, we obtain the
decompositions of the functions $\wp_{1,1}$, $\wp_{1,3}-\alpha^{2}\beta^{2}$,
and $\wp_{3,3}$ into the products of meromorphic functions (Remark 5.10), a
solution of the KdV-hierarchy in terms of functions constructed by the
elliptic functions $\wp_{E_{1}}$ and $\wp_{E_{2}}$ (Remark 5.11), and the
expressions of the coordinates of the Kummer surface in terms of the
Weierstrass elliptic functions $\wp_{E_{1}}$ and $\wp_{E_{2}}$ (Remark 5.12).
In Section 6, we compare our results with the results of [6, 19, 25].

The elliptic sigma functions, which are defined and studied by Weierstrass,
are generalized to the sigma functions associated with the hyperelliptic
curves and many applications in integrable systems and mathematical physics
are derived (cf. [15, 16, 17, 19]). In [2], we consider the solutions of the
inversion problem of the ultra-elliptic integrals of the hyperelliptic curves
of genus 2 in terms of the sigma functions associated with the hyperelliptic
curves of genus 2. In [3], we consider the series expansion of the sigma
functions associated with the hyperelliptic curves of genus 2 by using the
heat equations in a nonholonomic frame derived in [20]. The problem of the
reduction of the hyperelliptic integrals to the elliptic integrals has been
studied in many papers (e.g., [5, 6, 7, 10, 11]). In particular, O. Bolza
derived many examples of the reduction of hyperelliptic integrals to elliptic
integrals (cf. [10, 11]). The problem of the reduction of a hyperelliptic
integral to an elliptic integral is closely related to that of the morphism
from a hyperelliptic curve to an elliptic curve. In this paper, when a
hyperelliptic curve of genus 2 admits a morphism of degree 2 to an elliptic
curve, we derive the relationships between the hyperelliptic functions of
genus 2, which are defined by the logarithmic derivatives of the sigma
functions associated with the hyperelliptic curves of genus 2, and the
Weierstrass elliptic functions, which are defined by the logarithmic
derivatives of the elliptic sigma functions.

##  2 Preliminaries

In this section we recall the definitions of the Jacobi elliptic functions and
the sigma functions for the curves of genus 1 and 2 and give facts about them
which will be used later on. For the details of the Jacobi elliptic functions,
see [36, 54]. For the details of the sigma functions, see [19].

###  2.1 The Jacobi elliptic functions

We denote the imaginary unit by ${\bf i}$. Let
$\mathbb{H}=\\{\tau\in\mathbb{C}\mid\operatorname{Im}\tau>0\\}$. For
$(z,\tau)\in\mathbb{C}\times\mathbb{H}$ and
$\varepsilon_{1},\varepsilon_{2}\in\mathbb{R}$, the theta function with
characteristics ${}^{t}(\varepsilon_{1},\varepsilon_{2})$ is defined by

$\displaystyle\theta\begin{bmatrix}\varepsilon_{1}\\\
\varepsilon_{2}\end{bmatrix}(z,\tau)=\sum_{n\in\mathbb{Z}}\exp\left(\pi{\bf
i}(n+\varepsilon_{1})^{2}\tau+2\pi{\bf
i}(n+\varepsilon_{1})(z+\varepsilon_{2})\right).$

Let

$\displaystyle\theta_{00}(z,\tau)=\theta\begin{bmatrix}0\\\
0\end{bmatrix}(z,\tau),\qquad$
$\displaystyle\theta_{01}(z,\tau)=\theta\begin{bmatrix}0\\\
1/2\end{bmatrix}(z,\tau),$
$\displaystyle\theta_{10}(z,\tau)=\theta\begin{bmatrix}1/2\\\
0\end{bmatrix}(z,\tau),\qquad$
$\displaystyle\theta_{11}(z,\tau)=\theta\begin{bmatrix}1/2\\\
1/2\end{bmatrix}(z,\tau).$

The Jacobi elliptic functions associated with $\tau\in\mathbb{H}$ are defined
by

$\displaystyle\mbox{sn}(u,\tau)=-\frac{\theta_{00}(0,\tau)\theta_{11}(z,\tau)}{\theta_{10}(0,\tau)\theta_{01}(z,\tau)},\qquad$
$\displaystyle\mbox{cn}(u,\tau)=\frac{\theta_{01}(0,\tau)\theta_{10}(z,\tau)}{\theta_{10}(0,\tau)\theta_{01}(z,\tau)},$
$\displaystyle\mbox{dn}(u,\tau)=\frac{\theta_{01}(0,\tau)\theta_{00}(z,\tau)}{\theta_{00}(0,\tau)\theta_{01}(z,\tau)},\qquad$
$\displaystyle z=\frac{u}{\pi\theta_{00}(0,\tau)^{2}}.$

The modulus $m$ of the Jacobi elliptic functions associated with $\tau$ is
defined by

$\displaystyle m=\frac{\theta_{10}(0,\tau)^{2}}{\theta_{00}(0,\tau)^{2}}.$

Then we have $m^{2}\neq 0,1$. Let
$\omega^{\prime}_{\tau}=\pi\theta_{00}(0,\tau)^{2}/2$,
$\omega^{\prime\prime}_{\tau}=\omega^{\prime}_{\tau}\tau$, and
$\Lambda_{\tau}=\\{4\omega^{\prime}_{\tau}m_{1}+2\omega^{\prime\prime}_{\tau}m_{2}\mid
m_{1},m_{2}\in\mathbb{Z}\\}$. Let us consider the elliptic curve defined by

$\displaystyle E_{\tau}=\big{\\{}(x,y)\in\mathbb{C}^{2}\mid
y^{2}=\big{(}1-x^{2}\big{)}\big{(}1-m^{2}x^{2}\big{)}\big{\\}}.$

It is well known that the map

$\displaystyle\mathbb{C}/\Lambda_{\tau}\to E_{\tau},\qquad
u\mapsto(\mbox{sn}(u,\tau),\mbox{sn}^{\prime}(u,\tau))$

is the isomorphism, where $\mbox{sn}^{\prime}(u,\tau)=\frac{\rm d}{{\rm
d}u}\mbox{sn}(u,\tau)$.

###  2.2 The elliptic sigma function

We set

$\displaystyle
M_{1}(x)=x^{3}+\lambda_{2}x^{2}+\lambda_{4}x+\lambda_{6},\qquad\lambda_{i}\in\mathbb{C}.$

We assume that $M_{1}(x)$ has no multiple root. We consider the elliptic curve

$\displaystyle E=\big{\\{}(x,y)\in\mathbb{C}^{2}\mid y^{2}=M_{1}(x)\big{\\}}.$

For $(x,y)\in E$, let

$\displaystyle\omega=-\frac{{\rm d}x}{2y},$

which is the basis of the vector space of holomorphic one forms on $E$.
Further, let

$\displaystyle\eta=-\frac{x}{2y}{\rm d}x,$

which is the meromorphic one form on $E$ with a pole only at $\infty$. Let
$\\{\mathfrak{a},\mathfrak{b}\\}$ be a canonical basis in the one-dimensional
homology group of the curve $E$. We define the periods by

$\displaystyle 2\omega^{\prime}=\int_{\mathfrak{a}}\omega,\qquad
2\omega^{\prime\prime}=\int_{\mathfrak{b}}\omega,\qquad-2\eta^{\prime}=\int_{\mathfrak{a}}\eta,\qquad-2\eta^{\prime\prime}=\int_{\mathfrak{b}}\eta.$

The normalized period has the form
$\tau=(\omega^{\prime})^{-1}\omega^{\prime\prime}$. The sigma function
associated with $E$ is defined by

$\displaystyle\sigma(u)=\frac{2\omega^{\prime}}{\theta_{11}^{\prime}(0,\tau)}\exp\bigg{(}\frac{\eta^{\prime}}{2\omega^{\prime}}u^{2}\bigg{)}\theta_{11}\bigg{(}\frac{u}{2\omega^{\prime}},\tau\bigg{)},$

where $\theta_{11}^{\prime}(0,\tau)=\frac{\partial}{\partial
z}\theta_{11}(z,\tau)|_{z=0}$. For $m_{1},m_{2}\in\mathbb{Z}$, let
$\Omega=2\omega^{\prime}m_{1}+2\omega^{\prime\prime}m_{2}$. Then, for
$u\in\mathbb{C}$, we have

$\displaystyle\sigma(u+\Omega)/\sigma(u)=(-1)^{m_{1}-m_{2}+m_{1}m_{2}}\exp\big{\\{}(2\eta^{\prime}m_{1}+2\eta^{\prime\prime}m_{2})(u+\omega^{\prime}m_{1}+\omega^{\prime\prime}m_{2})\big{\\}}.$

We set $\deg\lambda_{2i}=2i$ for $1\leq i\leq 3$. The sigma function
$\sigma(u)$ is an entire function on $\mathbb{C}$ and the series expansion of
$\sigma(u)$ around $u=0$ has the following form:

$\displaystyle\sigma(u)=u+\sum_{n\geq 3}\mu_{n}u^{n},$

where the coefficient $\mu_{n}$ is a homogeneous polynomial in
$\mathbb{Q}[\lambda_{2},\lambda_{4},\lambda_{6}]$ of degree $n-1$ if
$\mu_{n}\neq 0$. In particular, the sigma function $\sigma(u)$ does not depend
on the choice of a canonical basis $\\{\mathfrak{a},\mathfrak{b}\\}$ in the
one-dimensional homology group of the elliptic curve $E$ and is determined by
the coefficients $\lambda_{2}$, $\lambda_{4}$, $\lambda_{6}$ of the defining
equation of the elliptic curve $E$.

Let $\wp(u)=-\frac{{\rm d}^{2}}{{\rm d}u^{2}}\log\sigma(u)$, which is the
Weierstrass elliptic function. Then the following relations hold (cf. [36])

$\displaystyle\wp(u)-\wp(\omega^{\prime})=\mbox{al}_{1}(u)^{2},\qquad\wp(u)-\wp(\omega^{\prime}+\omega^{\prime\prime})=\mbox{al}_{2}(u)^{2},\qquad\wp(u)-\wp(\omega^{\prime\prime})=\mbox{al}_{3}(u)^{2},$

where

$\displaystyle\mbox{al}_{1}(u)=\frac{\exp(\eta^{\prime}u)\sigma(\omega^{\prime}-u)}{\sigma(u)\sigma(\omega^{\prime})},\qquad\mbox{al}_{2}(u)=\frac{\exp(\eta^{\prime}u+\eta^{\prime\prime}u)\sigma(\omega^{\prime}+\omega^{\prime\prime}-u)}{\sigma(u)\sigma(\omega^{\prime}+\omega^{\prime\prime})},$
$\displaystyle\mbox{al}_{3}(u)=\frac{\exp(\eta^{\prime\prime}u)\sigma(\omega^{\prime\prime}-u)}{\sigma(u)\sigma(\omega^{\prime\prime})}.$

The functions $\mbox{al}_{i}(u)$, $i=1,2,3$, are called Weierstrass elliptic
al functions (cf. [52]).

###  2.3 The two-dimensional sigma function

We set

$\displaystyle
M_{2}(x)=x^{5}+\lambda_{2}x^{4}+\lambda_{4}x^{3}+\lambda_{6}x^{2}+\lambda_{8}x+\lambda_{10},\qquad\lambda_{i}\in\mathbb{C}.$

We assume that $M_{2}(x)$ has no multiple root. We consider the nonsingular
hyperelliptic curve of genus $2$

$\displaystyle V=\big{\\{}(x,y)\in\mathbb{C}^{2}\mid y^{2}=M_{2}(x)\big{\\}}.$

For $(x,y)\in V$, let

$\displaystyle\omega_{1}=-\frac{x}{2y}{\rm
d}x,\qquad\omega_{3}=-\frac{1}{2y}{\rm d}x,$

which are the basis of the vector space of holomorphic one forms on $V$.
Further, let

$\displaystyle\eta_{1}=-\frac{x^{2}}{2y}{\rm
d}x,\qquad\eta_{3}=\frac{-\lambda_{4}x-2\lambda_{2}x^{2}-3x^{3}}{2y}{\rm d}x,$

which are the meromorphic one forms on $V$ with a pole only at $\infty$. Let
$\\{\mathfrak{a}_{i},\mathfrak{b}_{i}\\}_{i=1}^{2}$ be a canonical basis in
the one-dimensional homology group of the curve $V$. We define the matrices of
periods by

$\displaystyle 2\omega^{\prime}=\int_{\mathfrak{a}_{j}}\omega_{i},\qquad
2\omega^{\prime\prime}=\int_{\mathfrak{b}_{j}}\omega_{i},\qquad-2\eta^{\prime}=\int_{\mathfrak{a}_{j}}\eta_{i},\qquad-2\eta^{\prime\prime}=\int_{\mathfrak{b}_{j}}\eta_{i}.$

The matrix of normalized periods has the form
$\tau=(\omega^{\prime})^{-1}\omega^{\prime\prime}$. Let
$\delta=\tau\delta^{\prime}+\delta^{\prime\prime}$,
$\delta^{\prime},\delta^{\prime\prime}\in\mathbb{R}^{2},$ be the vectors of
Riemann’s constants with respect to
$(\\{\mathfrak{a}_{i},\mathfrak{b}_{i}\\}_{i=1}^{2},\infty)$ and
$\delta:={}^{t}({}^{t}\delta^{\prime},{}^{t}\delta^{\prime\prime})$. Then we
have $\delta^{\prime}={}^{t}\big{(}\frac{1}{2},\frac{1}{2}\big{)}$ and
$\delta^{\prime\prime}={}^{t}\big{(}1,\frac{1}{2}\big{)}$ (cf. [26, 41]). The
sigma function $\sigma({\bf u})$ associated with $V$, ${\bf
u}={}^{t}(u_{1},u_{3})\in\mathbb{C}^{2}$, is defined by

$\displaystyle\sigma({\bf u})=\varepsilon\exp\bigg{(}\frac{1}{2}{}^{t}{\bf
u}\eta^{\prime}(\omega^{\prime})^{-1}{\bf
u}\bigg{)}\theta[\delta]\big{(}(2\omega^{\prime})^{-1}{\bf u},\tau\big{)},$

where $\theta[\delta]({\bf u})$ is the Riemann’s theta function with
characteristics $\delta$, which is defined by

$\displaystyle\theta[\delta]({\bf
u})=\sum_{n\in\mathbb{Z}^{2}}\exp\big{\\{}\pi{\bf
i}\,{}^{t}(n+\delta^{\prime})\tau(n+\delta^{\prime})+2\pi{\bf
i}\,{}^{t}(n+\delta^{\prime})({\bf u}+\delta^{\prime\prime})\big{\\}},$

and $\varepsilon$ is a constant.

######  Proposition 2.1 ([17, Theorem 1.1], [42, p. 193]).

For $m_{1},m_{2}\in\mathbb{Z}^{2}$, let
$\Omega=2\omega^{\prime}m_{1}+2\omega^{\prime\prime}m_{2}$. Then, for ${\bf
u}\in\mathbb{C}^{2}$, we have

$\displaystyle\sigma({\bf u}+\Omega)/\sigma({\bf
u})=(-1)^{2({}^{t}\delta^{\prime}m_{1}-{}^{t}\delta^{\prime\prime}m_{2})+{}^{t}m_{1}m_{2}}\exp\big{\\{}{}^{t}(2\eta^{\prime}m_{1}+2\eta^{\prime\prime}m_{2})({\bf
u}+\omega^{\prime}m_{1}+\omega^{\prime\prime}m_{2})\big{\\}}.$

We set $\deg\lambda_{2i}=2i$ for $1\leq i\leq 5$.

######  Theorem 2.2 ([18, Theorem 6.3], [19, Theorem 7.7], [42, Theorem 3]).

The sigma function $\sigma({\bf u})$ is an entire function on $\mathbb{C}^{2}$
and we have the unique constant $\varepsilon$ such that the series expansion
of $\sigma({\bf u})$ around the origin has the following form:

$\displaystyle\sigma({\bf
u})=\frac{1}{3}u_{1}^{3}-u_{3}+\sum_{n_{1}+3n_{3}\geq
5}\mu_{n_{1},n_{3}}u_{1}^{n_{1}}u_{3}^{n_{3}},$ (2.1)

where the coefficient $\mu_{n_{1},n_{3}}$ is a homogeneous polynomial in
$\mathbb{Q}[\lambda_{2},\lambda_{4},\lambda_{6},\lambda_{8},\lambda_{10}]$ of
degree $n_{1}+3n_{3}-3$ if $\mu_{n_{1},n_{3}}\neq 0$.

We take the constant $\varepsilon$ such that the expansion (2.1) holds, see
the expression for the sigma function above, which involves the constant
$\varepsilon$. Then the sigma function $\sigma({\bf u})$ does not depend on
the choice of a canonical basis
$\\{\mathfrak{a}_{i},\mathfrak{b}_{i}\\}_{i=1}^{2}$ in the one-dimensional
homology group of the curve $V$ and is determined by the coefficients
$\lambda_{2}$, $\lambda_{4}$, $\lambda_{6}$, $\lambda_{8}$, $\lambda_{10}$ of
the defining equation of the curve $V$.

######  Remark 2.3.

The constant $\varepsilon$ is given explicitly in [23, p. 906] and [24, p. 9].

###  2.4 Inversion problem of the elliptic integrals for the Jacobi
and Legendre forms

Let

$\displaystyle f_{J}(s)=s^{4}+a_{J,2}s^{2}+a_{J,4},$

where $a_{J,2},a_{J,4}\in\mathbb{C}$ satisfying
$a_{J,4}\big{(}a_{J,2}^{2}-4a_{J,4}\big{)}\neq 0$. We consider the elliptic
curve in the Jacobi form

$\displaystyle E_{J}=\big{\\{}(s,t)\in\mathbb{C}^{2}\mid
t^{2}=f_{J}(s)\big{\\}}$

and the elliptic curve in the Weierstrass normal form

$\displaystyle\widetilde{E}_{J}=\bigg{\\{}(x,y)\in\mathbb{C}^{2}\,\bigg{|}\,y^{2}=x^{3}-\bigg{(}4a_{J,4}+\frac{1}{3}a_{J,2}^{2}\bigg{)}x-\frac{8}{3}a_{J,2}a_{J,4}+\frac{2}{27}a_{J,2}^{3}\bigg{\\}}.$

We take $a_{J}\in\mathbb{C}$ such that $f_{J}(a_{J})=0$. We have the
isomorphism

$\displaystyle\xi_{J}\colon\quad
E_{J}\to\widetilde{E}_{J},\qquad(s,t)\mapsto(x,y),$

with

$\displaystyle
x=\frac{f_{J}^{\prime}(a_{J})}{s-a_{J}}+\frac{f_{J}^{\prime\prime}(a_{J})}{6},\qquad
y=\frac{f_{J}^{\prime}(a_{J})t}{(s-a_{J})^{2}}.$ (2.2)

Let $\\{\mathfrak{a}_{J},\mathfrak{b}_{J}\\}$ be a canonical basis in the one-
dimensional homology group of the curve $E_{J}$. We define the periods by

$\displaystyle 2\Omega_{J}^{\prime}=\int_{\mathfrak{a}_{J}}\frac{{\rm
d}s}{2t},\qquad 2\Omega_{J}^{\prime\prime}=\int_{\mathfrak{b}_{J}}\frac{{\rm
d}s}{2t}.$

We define the period lattice
$\Lambda_{J}=\\{2\Omega_{J}^{\prime}m_{1}+2\Omega_{J}^{\prime\prime}m_{2}\mid
m_{1},m_{2}\in\mathbb{Z}\\}$ and consider the Jacobian variety
$\operatorname{Jac}(E_{J})=\mathbb{C}/\Lambda_{J}$. Let
$\sigma_{\widetilde{E}_{J}}(u)$ be the sigma function associated with the
elliptic curve $\widetilde{E}_{J}$ and $\wp_{\widetilde{E}_{J}}=-\frac{{\rm
d}^{2}}{{\rm d}u^{2}}\log\sigma_{\widetilde{E}_{J}}$, which is the Weierstrass
elliptic function. We consider the map

$\displaystyle I_{J}\colon\quad E_{J}\to\operatorname{Jac}(E_{J}),\qquad
S\mapsto\int_{(a_{J},0)}^{S}\frac{{\rm d}s}{2t}.$

The map $I_{J}$ is the isomorphism. Let $u=I_{J}(S)$ for $S=(s,t)\in E_{J}$.

######  Proposition 2.4 ([54, pp. 453, 454]).

We have

$\displaystyle
s=a_{J}+\frac{f_{J}^{\prime}(a_{J})}{\wp_{\widetilde{E}_{J}}(u)-f_{J}^{\prime\prime}(a_{J})/6},\qquad
t=-\frac{f_{J}^{\prime}(a_{J})\wp_{\widetilde{E}_{J}}^{\prime}(u)}{2(\wp_{\widetilde{E}_{J}}(u)-f_{J}^{\prime\prime}(a_{J})/6)^{2}}.$

###### Proof.

Since the pullback of the holomorphic one form $-{\rm d}x/(2y)$ on
$\widetilde{E}_{J}$ with respect to $\xi_{J}$ is ${\rm d}s/(2t)$, we have

$\displaystyle u=\int_{(a_{J},0)}^{(s,t)}\frac{{\rm
d}s}{2t}=\int_{\infty}^{(x,y)}\bigg{(}{-}\frac{{\rm d}x}{2y}\bigg{)}.$

We have

$\displaystyle x=\wp_{\widetilde{E}_{J}}(u),\qquad
y=-\frac{\wp_{\widetilde{E}_{J}}^{\prime}(u)}{2}.$ (2.3)

From (2.2) and (2.3), we obtain the statement of the proposition. ∎

We consider the elliptic curve in the Legendre form

$\displaystyle E_{L}=\big{\\{}(s,t)\in\mathbb{C}^{2}\mid
t^{2}=s(s-b_{L})(s-c_{L})\big{\\}},$

where $b_{L},c_{L}\in\mathbb{C}$ satisfying $b_{L}c_{L}(b_{L}-c_{L})\neq 0$,
and the elliptic curve in the Weierstrass normal form

$\displaystyle\widetilde{E}_{L}=\bigg{\\{}(x,y)\in\mathbb{C}^{2}\,\bigg{|}\,y^{2}=x^{3}-\bigg{(}\frac{b_{L}^{2}+c_{L}^{2}-b_{L}c_{L}}{3}\bigg{)}x+\frac{(2b_{L}-c_{L})(b_{L}+c_{L})(2c_{L}-b_{L})}{27}\bigg{\\}}.$

We have the isomorphism

$\displaystyle\xi_{L}\colon\quad
E_{L}\to\widetilde{E}_{L},\qquad(s,t)\mapsto(x,y),$

with

$\displaystyle x=s-\frac{b_{L}+c_{L}}{3},\qquad y=t.$ (2.4)

Let $\\{\mathfrak{a}_{L},\mathfrak{b}_{L}\\}$ be a canonical basis in the one-
dimensional homology group of the curve $E_{L}$. We define the periods by

$\displaystyle
2\Omega_{L}^{\prime}=\int_{\mathfrak{a}_{L}}\bigg{(}{-}\frac{{\rm
d}s}{2t}\bigg{)},\qquad
2\Omega_{L}^{\prime\prime}=\int_{\mathfrak{b}_{L}}\bigg{(}{-}\frac{{\rm
d}s}{2t}\bigg{)}.$

We define the period lattice
$\Lambda_{L}=\\{2\Omega_{L}^{\prime}m_{1}+2\Omega_{L}^{\prime\prime}m_{2}\mid
m_{1},m_{2}\in\mathbb{Z}\\}$ and consider the Jacobian variety
$\operatorname{Jac}(E_{L})=\mathbb{C}/\Lambda_{L}$. Let
$\sigma_{\widetilde{E}_{L}}(u)$ be the sigma function associated with the
elliptic curve $\widetilde{E}_{L}$ and $\wp_{\widetilde{E}_{L}}=-\frac{{\rm
d}^{2}}{{\rm d}u^{2}}\log\sigma_{\widetilde{E}_{L}}$, which is the Weierstrass
elliptic function. We consider the map

$\displaystyle I_{L}\colon\quad E_{L}\to\operatorname{Jac}(E_{L}),\qquad
S\mapsto\int_{\infty}^{S}\bigg{(}{-}\frac{{\rm d}s}{2t}\bigg{)}.$

The map $I_{L}$ is the isomorphism. Let $u=I_{L}(S)$ for $S=(s,t)\in E_{L}$.

######  Proposition 2.5.

We have

$\displaystyle s=\wp_{\widetilde{E}_{L}}(u)+\frac{b_{L}+c_{L}}{3},\qquad
t=-\frac{\wp_{\widetilde{E}_{L}}^{\prime}(u)}{2}.$

###### Proof.

Since the pullback of the holomorphic one form $-{\rm d}x/(2y)$ on
$\widetilde{E}_{L}$ with respect to $\xi_{L}$ is $-{\rm d}s/(2t)$, we have

$\displaystyle u=\int_{\infty}^{(s,t)}\bigg{(}{-}\frac{{\rm
d}s}{2t}\bigg{)}=\int_{\infty}^{(x,y)}\bigg{(}{-}\frac{{\rm d}x}{2y}\bigg{)}.$

We have

$\displaystyle x=\wp_{\widetilde{E}_{L}}(u),\qquad
y=-\frac{\wp_{\widetilde{E}_{L}}^{\prime}(u)}{2}.$ (2.5)

From (2.4) and (2.5), we obtain the statement of the proposition. ∎

##  3 The curve of genus 2 and the elliptic curves

######  Theorem 3.1 ([37, 48]).

Let $H$ be a hyperelliptic curve of genus $2$. There exist an elliptic curve
$W$ and a morphism $H\to W$ of degree $2$ if and only if $H$ is isomorphic to
the hyperelliptic curve $H^{\prime}$ of genus $2$ defined by

$\displaystyle H^{\prime}=\big{\\{}(s,t)\in\mathbb{C}^{2}\mid
t^{2}=\big{(}s^{2}-1\big{)}\big{(}s^{2}-e_{1}^{2}\big{)}\big{(}s^{2}-e_{2}^{2}\big{)}\big{\\}},$
(3.1)

where $e_{1},e_{2}\in\mathbb{C}$ satisfying

$\displaystyle e_{1}^{2}\neq 0,1,\qquad e_{2}^{2}\neq 0,1,\qquad e_{1}^{2}\neq
e_{2}^{2}.$ (3.2)

In appendix, we give a proof of this theorem. In [12, Section 1.3.1], [21,
Chapter 14], [27, Section 4], and [53, Theorem 4.2] , the elliptic curves

$\displaystyle W_{1}=\big{\\{}(s,t)\in\mathbb{C}^{2}\mid
t^{2}=(s-1)\big{(}s-e_{1}^{2}\big{)}\big{(}s-e_{2}^{2}\big{)}\big{\\}},$
$\displaystyle W_{2}=\big{\\{}(s,t)\in\mathbb{C}^{2}\mid
t^{2}=(1-s)\big{(}1-e_{1}^{2}s\big{)}\big{(}1-e_{2}^{2}s\big{)}\big{\\}}$

and the morphisms

$\displaystyle\phi_{1}\colon\quad H^{\prime}\to
W_{1},\qquad(s,t)\mapsto\big{(}s^{2},t\big{)},\qquad\phi_{2}\colon\quad
H^{\prime}\to W_{2},\qquad(s,t)\mapsto\big{(}1/s^{2},t/s^{3}\big{)}$

are considered. It is well known that the Jacobian variety of $H^{\prime}$ is
isogenous to $W_{1}\times W_{2}$ (cf. [21, Chapter 14], [27, Section 4], [28,
Section 1], [39, Section 2], and [53, Theorem 4.2]). In [43, Section 7], many
examples of the curves of genus 2 whose Jacobian varieties are isogenous to
the direct products of elliptic curves are given. These examples are important
from the point of view of problems in number theory.

Throughout this paper, for a complex number $z$, we denote a complex number
whose square is $z$ by $\sqrt{z}$. For $\alpha,\beta\in\mathbb{C}$ satisfying

$\displaystyle\alpha^{2},\beta^{2}\neq
0,1,\qquad\alpha^{2}\neq\beta^{2},\qquad\alpha^{2}\beta^{2}\neq 1,$ (3.3)

we consider the nonsingular hyperelliptic curve of genus $2$111The
hyperelliptic curve of genus 2 defined by  $\displaystyle
y^{2}=x(1-x)\big{(}1-\nu_{1}^{2}x\big{)}\big{(}1-\nu_{2}^{2}x\big{)}\big{(}1-\nu_{3}^{2}x\big{)},\qquad\nu_{1},\nu_{2},\nu_{3}\in\mathbb{C},$
is called Richelot normal form (cf. [10, p. 450], [19, pp. 97, 236]).

$\displaystyle V=\big{\\{}(x,y)\in\mathbb{C}^{2}\mid
y^{2}=x(x-1)\big{(}x-\alpha^{2}\big{)}\big{(}x-\beta^{2}\big{)}\big{(}x-\alpha^{2}\beta^{2}\big{)}\big{\\}}.$
(3.4)

This curve $V$ is considered in [5, 6, 7, 12, 17, 19, 25]. For $i=2,4,6,8$, we
define $\lambda_{i}$ such that the following relation holds:

$\displaystyle
x(x-1)\big{(}x-\alpha^{2}\big{)}\big{(}x-\beta^{2}\big{)}\big{(}x-\alpha^{2}\beta^{2}\big{)}=x^{5}+\lambda_{2}x^{4}+\lambda_{4}x^{3}+\lambda_{6}x^{2}+\lambda_{8}x,$

i.e., we set

$\displaystyle\lambda_{2}=-1-\alpha^{2}-\beta^{2}-\alpha^{2}\beta^{2},\qquad\lambda_{4}=\alpha^{2}+\beta^{2}+2\alpha^{2}\beta^{2}+\alpha^{4}\beta^{2}+\alpha^{2}\beta^{4},$
$\displaystyle\lambda_{6}=-\alpha^{4}\beta^{4}-\alpha^{2}\beta^{4}-\alpha^{4}\beta^{2}-\alpha^{2}\beta^{2},\qquad\lambda_{8}=\alpha^{4}\beta^{4}.$

######  Proposition 3.2 ([5, p. 235], [6, p. 3442], [12, Proposition 1.10],
[21, Theorem 14.1.1]).

Given $e_{1},e_{2}\in\mathbb{C}$ satisfying (3.2), the curve $H^{\prime}$ is
isomorphic to the curve $V$ with

$\displaystyle\alpha=\sqrt{\frac{(e_{1}+1)(e_{2}+1)}{(e_{1}-1)(e_{2}-1)}},\qquad\beta=\sqrt{\frac{(e_{1}-1)(e_{2}+1)}{(e_{1}+1)(e_{2}-1)}}.$
(3.5)

Conversely, given $\alpha,\beta\in\mathbb{C}$ satisfying (3.3), the curve $V$
is isomorphic to the curve $H^{\prime}$ with

$\displaystyle e_{1}=\frac{\alpha+\beta}{\alpha-\beta},\qquad
e_{2}=\frac{\alpha\beta+1}{\alpha\beta-1}.$ (3.6)

######  Proposition 3.3 ([5, p. 235], [6, p. 3442]).

Given $e_{1},e_{2}\in\mathbb{C}$ satisfying (3.2), an isomorphism from
$H^{\prime}$ to $V$ with (3.5) is given by

$\displaystyle\zeta\colon\quad H^{\prime}\to V,\qquad(s,t)\mapsto(x,y)$

with

$\displaystyle(x,y)=\bigg{(}\frac{(e_{2}+1)(s+1)}{(e_{2}-1)(s-1)},\frac{4(e_{2}+1)^{2}t}{(e_{2}-1)^{3}\sqrt{e_{1}^{2}-1}(s-1)^{3}}\bigg{)}.$

Furthermore, given $\alpha,\beta\in\mathbb{C}$ satisfying (3.3), an
isomorphism from $V$ to $H^{\prime}$ with (3.6) is given by

$\displaystyle\widetilde{\zeta}\colon\quad V\to
H^{\prime},\qquad(x,y)\mapsto(s,t)$

with

$\displaystyle(s,t)=\bigg{(}\frac{x+\alpha\beta}{x-\alpha\beta},\frac{8\alpha\beta\sqrt{\alpha\beta}y}{(\alpha\beta-1)(\alpha-\beta)(x-\alpha\beta)^{3}}\bigg{)}.$

The elliptic curve $W_{1}$ with (3.6) is isomorphic to the elliptic curve in
Legendre form

$\displaystyle
E_{1}=\bigg{\\{}(X,Y)\in\mathbb{C}^{2}\,\bigg{|}\,Y^{2}=X(X-1)\bigg{(}X-\frac{(\alpha-\beta)^{2}}{(\alpha\beta-1)^{2}}\bigg{)}\bigg{\\}}$
(3.7)

by the morphism

$\displaystyle\xi_{1}\colon\quad W_{1}\to E_{1},\qquad(s,t)\mapsto(X,Y)$

with

$\displaystyle(X,Y)=\bigg{(}\frac{(\alpha-\beta)^{2}(s-1)}{4\alpha\beta},\frac{(\alpha-\beta)^{3}t}{8\alpha\beta\sqrt{\alpha\beta}}\bigg{)}.$

The elliptic curve $W_{2}$ with (3.6) is isomorphic to the elliptic curve in
Legendre form

$\displaystyle
E_{2}=\bigg{\\{}(X,Y)\in\mathbb{C}^{2}\,\bigg{|}\,Y^{2}=X(X-1)\bigg{(}X-\frac{(\alpha+\beta)^{2}}{(\alpha\beta+1)^{2}}\bigg{)}\bigg{\\}}$
(3.8)

by the morphism

$\displaystyle\xi_{2}\colon\quad W_{2}\to E_{2},\qquad(s,t)\mapsto(X,Y)$

with

$\displaystyle(X,Y)=\bigg{(}{-}\frac{(\alpha+\beta)^{2}(s-1)}{4\alpha\beta},-\frac{(\alpha+\beta)^{2}(\alpha-\beta)(\alpha\beta-1)t}{8\alpha\beta\sqrt{\alpha\beta}(\alpha\beta+1)}\bigg{)}.$

We consider the maps

$\displaystyle\varphi_{i}=\xi_{i}\circ\phi_{i}\circ\widetilde{\zeta}\colon\quad
V\to E_{i},\qquad i=1,2.$

Then the maps $\varphi_{i}$, $i=1,2$, are described by

$\displaystyle\varphi_{1}\colon\quad V\to
E_{1},\qquad(x,y)\mapsto(X,Y)=\bigg{(}\frac{(\alpha-\beta)^{2}x}{(x-\alpha\beta)^{2}},\frac{(\alpha-\beta)^{2}y}{(\alpha\beta-1)(x-\alpha\beta)^{3}}\bigg{)},$
(3.9) $\displaystyle\varphi_{2}\colon\quad V\to
E_{2},\qquad(x,y)\mapsto(X,Y)=\bigg{(}\frac{(\alpha+\beta)^{2}x}{(x+\alpha\beta)^{2}},\frac{-(\alpha+\beta)^{2}y}{(\alpha\beta+1)(x+\alpha\beta)^{3}}\bigg{)}.$
(3.10)

######  Remark 3.4.

For $P=(x,y)\in V\backslash\\{\infty\\}$ such that $x=\alpha\beta$, we have
$\varphi_{1}(P)=\infty$. For $P=(x,y)\in V\backslash\\{\infty\\}$ such that
$x=-\alpha\beta$, we have $\varphi_{2}(P)=\infty$. Furthermore, we have
$\varphi_{i}(\infty)=(0,0)$ for $i=1,2$.

Let

$\displaystyle\kappa_{1}=\frac{\sqrt{-1}(\alpha-\beta)}{\sqrt{\big{(}1-\alpha^{2}\big{)}\big{(}1-\beta^{2}\big{)}}},\qquad\kappa_{2}=\frac{\sqrt{-1}(\alpha+\beta)}{\sqrt{\big{(}1-\alpha^{2}\big{)}\big{(}1-\beta^{2}\big{)}}}.$

######  Remark 3.5.

In [5, Section 7.1], [6, Section 3.1.1], [19, Chapter 6], [25], the elliptic
curves

$\displaystyle E_{+}=\bigg{\\{}(X_{+},Y_{+})\in\mathbb{C}^{2}\mid
Y_{+}^{2}=X_{+}(1-X_{+})\big{(}1-\kappa_{1}^{2}X_{+}\big{)}\bigg{\\}},$
$\displaystyle E_{-}=\bigg{\\{}(X_{-},Y_{-})\in\mathbb{C}^{2}\mid
Y_{-}^{2}=X_{-}(1-X_{-})\big{(}1-\kappa_{2}^{2}X_{-}\big{)}\bigg{\\}}$

and the morphisms

$\displaystyle\pi_{+}\colon\quad V\to E_{+},\qquad(x,y)\mapsto(X_{+},Y_{+})$

with

$\displaystyle(X_{+},Y_{+})=\bigg{(}\frac{\big{(}1-\alpha^{2}\big{)}\big{(}1-\beta^{2}\big{)}x}{\big{(}x-\alpha^{2}\big{)}\big{(}x-\beta^{2}\big{)}},\frac{-\sqrt{\big{(}1-\alpha^{2}\big{)}\big{(}1-\beta^{2}\big{)}}(x-\alpha\beta)y}{\big{(}x-\alpha^{2}\big{)}^{2}\big{(}x-\beta^{2}\big{)}^{2}}\bigg{)},$
$\displaystyle\pi_{-}\colon\quad V\to E_{-},\qquad(x,y)\mapsto(X_{-},Y_{-})$

with

$\displaystyle(X_{-},Y_{-})=\bigg{(}\frac{\big{(}1-\alpha^{2}\big{)}\big{(}1-\beta^{2}\big{)}x}{\big{(}x-\alpha^{2}\big{)}\big{(}x-\beta^{2}\big{)}},\frac{-\sqrt{\big{(}1-\alpha^{2}\big{)}\big{(}1-\beta^{2}\big{)}}(x+\alpha\beta)y}{\big{(}x-\alpha^{2}\big{)}^{2}\big{(}x-\beta^{2}\big{)}^{2}}\bigg{)}$

are considered. We have the isomorphisms

$\displaystyle\xi_{+}\colon\quad E_{1}\to
E_{+},\qquad(X,Y)\mapsto(X_{+},Y_{+})$

with

$\displaystyle(X_{+},Y_{+})=\bigg{(}\frac{X}{\kappa_{1}^{2}(X-1)},\frac{(1-\alpha\beta)\sqrt{\big{(}1-\alpha^{2}\big{)}\big{(}1-\beta^{2}\big{)}}Y}{(\alpha-\beta)^{2}(X-1)^{2}}\bigg{)},$

and

$\displaystyle\xi_{-}\colon\quad E_{2}\to
E_{-},\qquad(X,Y)\mapsto(X_{-},Y_{-})$

with

$\displaystyle(X_{-},Y_{-})=\bigg{(}\frac{X}{\kappa_{2}^{2}(X-1)},\frac{(1+\alpha\beta)\sqrt{\big{(}1-\alpha^{2}\big{)}\big{(}1-\beta^{2}\big{)}}Y}{(\alpha+\beta)^{2}(X-1)^{2}}\bigg{)}.$

Furthermore, we have $\pi_{+}=\xi_{+}\circ\varphi_{1}$ and
$\pi_{-}=\xi_{-}\circ\varphi_{2}$.

######  Remark 3.6.

In [12, Lemma 1.7 and Proposition 1.8], for $i=1,2$, the elliptic curves

$\displaystyle\mathcal{E}_{i}=\big{\\{}(\mathcal{X}_{i},\mathcal{Y}_{i})\in\mathbb{C}^{2}\mid\mathcal{Y}_{i}^{2}=\mathcal{X}_{i}(\mathcal{X}_{i}-1)\big{(}\mathcal{X}_{i}-\kappa_{i}^{2}\big{)}\big{\\}}$

and the morphisms

$\displaystyle\pi_{\mathcal{E}_{i}}\colon\quad
V\to\mathcal{E}_{i},\qquad(x,y)\mapsto(\mathcal{X}_{i},\mathcal{Y}_{i})$

with

$\displaystyle(\mathcal{X}_{i},\mathcal{Y}_{i})=\bigg{(}\frac{\big{(}x-\alpha^{2}\big{)}\big{(}x-\beta^{2}\big{)}}{\big{(}1-\alpha^{2}\big{)}\big{(}1-\beta^{2}\big{)}x},\frac{\big{(}x+(-1)^{i}\alpha\beta\big{)}y}{\big{(}1-\alpha^{2}\big{)}\big{(}1-\beta^{2}\big{)}\sqrt{\big{(}1-\alpha^{2}\big{)}\big{(}1-\beta^{2}\big{)}}x^{2}}\bigg{)}$

are considered. For $i=1,2$, we have the isomorphisms

$\displaystyle\overline{\xi}_{i}\colon\quad
E_{i}\to\mathcal{E}_{i},\qquad(X,Y)\mapsto(\mathcal{X}_{i},\mathcal{Y}_{i})$

with

$\displaystyle(\mathcal{X}_{i},\mathcal{Y}_{i})=\left(\frac{\kappa_{i}^{2}(X-1)}{X},\frac{\kappa_{i}^{2}(1+(-1)^{i}\alpha\beta)Y}{\sqrt{\big{(}1-\alpha^{2}\big{)}\big{(}1-\beta^{2}\big{)}}X^{2}}\right)$

and $\pi_{\mathcal{E}_{i}}=\overline{\xi}_{i}\circ\varphi_{i}$.

######  Remark 3.7.

In [12, Section 1.3.4], [13, Section 6.1], [46, Section 3.3], given two
elliptic curves in Legendre form satisfying a condition, a method to construct
a hyperelliptic curve of genus 2 whose Jacobian variety is $(2,2)$-isogenous
to the direct product of the two elliptic curves is introduced, which is
called Legendre’s glueing method.

We consider the hyperelliptic curve $V$ defined by (3.4) and the elliptic
curves $E_{1}$ and $E_{2}$ defined by (3.7) and (3.8), respectively. We define
the period lattice
$\Lambda=\big{\\{}2\omega^{\prime}m_{1}+2\omega^{\prime\prime}m_{2}\mid
m_{1},m_{2}\in\mathbb{Z}^{2}\big{\\}}$ and consider the Jacobian variety
$\operatorname{Jac}(V)=\mathbb{C}^{2}/\Lambda$. For $i=1,2$, let
$\big{\\{}\mathfrak{a}^{(i)},\mathfrak{b}^{(i)}\big{\\}}$ be a canonical basis
in the one-dimensional homology group of the curve $E_{i}$. For $i=1,2$, we
consider the holomorphic one form on $E_{i}$

$\displaystyle\Omega_{i}=-\frac{{\rm d}X}{2Y}.$

We define the periods by

$\displaystyle 2\Omega_{i}^{\prime}=\int_{\mathfrak{a}^{(i)}}\Omega_{i},\qquad
2\Omega_{i}^{\prime\prime}=\int_{\mathfrak{b}^{(i)}}\Omega_{i}.$

We define the period lattice
$\Lambda_{i}=\big{\\{}2\Omega_{i}^{\prime}m_{1}+2\Omega_{i}^{\prime\prime}m_{2}\mid
m_{1},m_{2}\in\mathbb{Z}\big{\\}}$ and consider the Jacobian variety
$\operatorname{Jac}(E_{i})=\mathbb{C}/\Lambda_{i}$. Let
$\pi\colon\mathbb{C}^{2}\to\operatorname{Jac}(V)$ and
$\pi_{i}\colon\mathbb{C}\to\operatorname{Jac}(E_{i})$ be the natural
projections for $i=1,2$.

In [43], the curves of genus 2 defined by

$\displaystyle C_{48,1}=\big{\\{}(s,t)\in\mathbb{C}^{2}\mid
t^{2}=(s-2)(s+2)\big{(}s^{4}-10s^{2}-3\big{)}\big{\\}},$ $\displaystyle
C_{48,2}=\big{\\{}(s,t)\in\mathbb{C}^{2}\mid
t^{2}=\big{(}s^{2}+s+2\big{)}\big{(}s^{4}+3s^{3}+21s^{2}-27s+18\big{)}\big{\\}}$

are considered. The groups of torsion points over $\mathbb{Q}$ in the Jacobian
varieties of $C_{48,1}$ and $C_{48,2}$ are isomorphic to the cyclic groups of
order 48 [43, Theorem 6.2]. The curve $C_{48,1}$ is isomorphic to the curve of
genus 2

$\displaystyle\widetilde{C}_{48,1}=\bigg{\\{}(s,t)\in\mathbb{C}^{2}\,\bigg{|}\,t^{2}=\big{(}s^{2}-1\big{)}\bigg{(}s^{2}-\frac{a_{48,1}^{2}}{4}\bigg{)}\bigg{(}s^{2}-\frac{b_{48,1}^{2}}{4}\bigg{)}\bigg{\\}}$

by the morphism

$\displaystyle C_{48,1}\to\widetilde{C}_{48,1},\qquad(s,t)\mapsto(s/2,t/8),$

where

$\displaystyle a_{48,1}=\sqrt{5+2\sqrt{7}},\qquad
b_{48,1}=\sqrt{5-2\sqrt{7}}.$

The curve $\widetilde{C}_{48,1}$ is isomorphic to the curve $V$ defined by
(3.4) with

$\displaystyle\alpha=\sqrt{\frac{(a_{48,1}+2)(b_{48,1}+2)}{(a_{48,1}-2)(b_{48,1}-2)}},\qquad\beta=\sqrt{\frac{(a_{48,1}-2)(b_{48,1}+2)}{(a_{48,1}+2)(b_{48,1}-2)}}.$
(3.11)

Then the Jacobian variety $\operatorname{Jac}(V)$ is isogenous to $E_{1}\times
E_{2}$ defined by (3.7) and (3.8) with (3.11). The curve $C_{48,2}$ is
isomorphic to the curve of genus 2

$\displaystyle\widetilde{C}_{48,2}=\bigg{\\{}(s,t)\in\mathbb{C}^{2}\,\bigg{|}\,t^{2}=\big{(}s^{2}-1\big{)}\bigg{(}s^{2}-\frac{a_{48,2}^{2}}{6}\bigg{)}\bigg{(}s^{2}-\frac{b_{48,2}^{2}}{6}\bigg{)}\bigg{\\}}$

by the morphism

$\displaystyle
C_{48,2}\to\widetilde{C}_{48,2},\qquad(s,t)\mapsto\bigg{(}\frac{\sqrt{-7}(s+1)}{s-3},\frac{56\sqrt{-1}t}{3(s-3)^{3}}\bigg{)},$

where

$\displaystyle a_{48,2}=\sqrt{-7-7\sqrt{-7}},\qquad
b_{48,2}=\sqrt{-7+7\sqrt{-7}}.$

The curve $\widetilde{C}_{48,2}$ is isomorphic to the curve $V$ defined by
(3.4) with

$\displaystyle\alpha=\sqrt{\frac{\big{(}a_{48,2}+\sqrt{6}\big{)}\big{(}b_{48,2}+\sqrt{6}\big{)}}{\big{(}a_{48,2}-\sqrt{6}\big{)}\big{(}b_{48,2}-\sqrt{6}\big{)}}},\qquad\beta=\sqrt{\frac{\big{(}a_{48,2}-\sqrt{6}\big{)}\big{(}b_{48,2}+\sqrt{6}\big{)}}{\big{(}a_{48,2}+\sqrt{6}\big{)}\big{(}b_{48,2}-\sqrt{6}\big{)}}}.$
(3.12)

Then the Jacobian variety $\operatorname{Jac}(V)$ is isogenous to $E_{1}\times
E_{2}$ defined by (3.7) and (3.8) with (3.12).

##  4 The map from $\boldsymbol{\operatorname{Jac}(V)}$ to
$\boldsymbol{\operatorname{Jac}(E_{i})}$

We denote the coordinates in $\mathbb{C}^{2}$ and $\mathbb{C}$ by
${}^{t}(u_{1},u_{3})$ and $u$, respectively. We take the following two points
on $V$:

$\displaystyle
O_{1}=\big{(}\alpha\beta,\alpha\beta\sqrt{\alpha\beta}(\alpha\beta-1)(\alpha-\beta)\big{)},\qquad
O_{2}=\big{(}{-}\alpha\beta,\alpha\beta\sqrt{-\alpha\beta}(\alpha\beta+1)(\alpha+\beta)\big{)}.$

We have $\varphi_{i}(O_{i})=\infty$ for $i=1,2$. Let
${\boldsymbol{\omega}}={}^{t}(\omega_{1},\omega_{3})$. We consider the
holomorphic maps

$\displaystyle I_{1}\colon\quad V\to\operatorname{Jac}(V),\qquad
P\mapsto\int_{O_{1}}^{P}{\boldsymbol{\omega}},$ $\displaystyle
I_{2}\colon\quad V\to\operatorname{Jac}(V),\qquad
P\mapsto\int_{O_{2}}^{P}{\boldsymbol{\omega}}.$

Note that the elliptic curves $E_{1}$ and $E_{2}$ have the group structures
(cf. [49, Chapter III, Section 2]). Since $\operatorname{Jac}(V)$ is the
Albanese variety of $V$, we have the unique holomorphic map

$\displaystyle\varphi_{1,*}\colon\ \operatorname{Jac}(V)\to E_{1}$

such that $\varphi_{1,*}$ is the group homomorphism and
$\varphi_{1}=\varphi_{1,*}\circ I_{1}$ and we have the unique holomorphic map

$\displaystyle\varphi_{2,*}\colon\ \operatorname{Jac}(V)\to E_{2}$

such that $\varphi_{2,*}$ is the group homomorphism and
$\varphi_{2}=\varphi_{2,*}\circ I_{2}$ (cf. [40, Proposition 6.1, p. 104]).

$\textstyle{\operatorname{Jac}(V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{1,*}}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{I_{1}}$$\scriptstyle{\varphi_{1}}$$\textstyle{E_{1},}$$\textstyle{\operatorname{Jac}(V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{2,*}}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{I_{2}}$$\scriptstyle{\varphi_{2}}$$\textstyle{E_{2}.}$

For $i=1,2$, we consider the maps

$\displaystyle J_{i}\colon\quad E_{i}\to\operatorname{Jac}(E_{i}),\qquad
S\mapsto\int_{\infty}^{S}\Omega_{i}.$

It is well known that the map $J_{i}$ is the isomorphism. We consider the maps

$\displaystyle\psi_{i,*}=J_{i}\circ\varphi_{i,*}\colon\quad\operatorname{Jac}(V)\to\operatorname{Jac}(E_{i}),\qquad
i=1,2.$

For $i=1,2$, the map $\psi_{i,*}$ is the holomorphic map between the complex
manifolds $\operatorname{Jac}(V)$ and $\operatorname{Jac}(E_{i})$, and the map
$\psi_{i,*}$ is the homomorphism between the groups $\operatorname{Jac}(V)$
and $\operatorname{Jac}(E_{i})$.

######  Proposition 4.1.

For $i=1,2$, we consider the maps

$\displaystyle\widetilde{\psi}_{i,*}\colon\quad\mathbb{C}^{2}\to\mathbb{C},\qquad{}^{t}(u_{1},u_{3})\mapsto
a_{i}u_{1}+b_{i}u_{3},$

where

$\displaystyle a_{1}=1-\alpha\beta,\qquad
b_{1}=\alpha\beta(1-\alpha\beta),\qquad a_{2}=\alpha\beta+1,\qquad
b_{2}=-\alpha\beta(\alpha\beta+1).$ (4.1)

Then we have $\psi_{i,*}\circ\pi=\pi_{i}\circ\widetilde{\psi}_{i,*}$, i.e., we
have the following commutative diagram:

$\displaystyle\begin{CD}\mathbb{C}^{2}@>{\widetilde{\psi}_{i,*}}>{}>\mathbb{C}\\\
@V{\pi}V{}V@V{\pi_{i}}V{}V\\\
\operatorname{Jac}(V)@>{{\psi_{i,*}}}>{}>\operatorname{Jac}(E_{i}).\end{CD}$

###### Proof.

Since $\psi_{1,*}$ is holomorphic and the group homomorphism, it is well known
that there exist complex numbers $a_{1}$ and $b_{1}$ such that
$\psi_{1,*}\circ\pi=\pi_{1}\circ\widetilde{\psi}_{1,*}$, where

$\displaystyle\widetilde{\psi}_{1,*}\colon\quad\mathbb{C}^{2}\to\mathbb{C},\qquad{}^{t}(u_{1},u_{3})\mapsto
a_{1}u_{1}+b_{1}u_{3},$

(e.g., [8, Proposition 1.2.1]). We consider the following commutative diagram:

$\textstyle{\operatorname{Jac}(V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{1,*}}$$\textstyle{\operatorname{Jac}(E_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{J_{1}^{-1}}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{I_{1}}$$\scriptstyle{\varphi_{1}}$$\textstyle{E_{1}.}$

We consider the holomorphic one forms ${\rm d}u$ on
$\operatorname{Jac}(E_{1})$ and ${\rm d}u_{1},{\rm d}u_{3}$ on
$\operatorname{Jac}(V)$. The pullback of the holomorphic one form $\Omega_{1}$
on $E_{1}$ with respect to $J_{1}^{-1}$ is ${\rm d}u$. The pullback of ${\rm
d}u$ with respect to $\psi_{1,*}$ is $a_{1}{\rm d}u_{1}+b_{1}{\rm d}u_{3}$.
The pullback of $a_{1}{\rm d}u_{1}+b_{1}{\rm d}u_{3}$ with respect to $I_{1}$
is $a_{1}\omega_{1}+b_{1}\omega_{3}$. Thus, the pullback of $\Omega_{1}$ with
respect to $J_{1}^{-1}\circ\psi_{1,*}\circ I_{1}$ is
$a_{1}\omega_{1}+b_{1}\omega_{3}$. On the other hand, from the definition of
$\varphi_{1}$ and the direct calculations, we find that the pullback of
$\Omega_{1}$ with respect to $\varphi_{1}$ is
$(1-\alpha\beta)\omega_{1}+\alpha\beta(1-\alpha\beta)\omega_{3}$. Since
$\omega_{1}$ and $\omega_{3}$ are linearly independent, we obtain
$a_{1}=1-\alpha\beta$ and $b_{1}=\alpha\beta(1-\alpha\beta)$. We can obtain
the defining equation of the elliptic curve $E_{2}$ and the map $\varphi_{2}$
by changing $\beta$ with $-\beta$ in the defining equation of the elliptic
curve $E_{1}$ and the map $\varphi_{1}$ (see (3.7), (3.8), (3.9), and (3.10)).
On the other hand, when we change $\beta$ with $-\beta$ in the defining
equation of the curve $V$, the defining equation of $V$ does not change (see
(3.4)). Therefore we can obtain $a_{2}$ and $b_{2}$ by changing $\beta$ with
$-\beta$ in the expressions of $a_{1}$ and $b_{1}$, respectively. Thus we have
$a_{2}=\alpha\beta+1$ and $b_{2}=-\alpha\beta(\alpha\beta+1)$. ∎

Let $\mbox{Sym}^{2}(V)$ be the symmetric square of $V$. Then
$\mbox{Sym}^{2}(V)$ is a complex manifold of dimension 2. We consider the
holomorphic map

$\displaystyle
I\colon\quad\mbox{Sym}^{2}(V)\to\operatorname{Jac}(V),\qquad(P,Q)\mapsto\int_{\infty}^{P}{\boldsymbol{\omega}}+\int_{\infty}^{Q}{\boldsymbol{\omega}}.$

Let $\sigma({\bf u})$, ${\bf u}={}^{t}(u_{1},u_{3})\in\mathbb{C}^{2}$, be the
sigma function associated with the curve $V$,
$\wp_{j,k}=-\partial_{u_{k}}\partial_{u_{j}}\log\sigma$, and
$\wp_{j,k,\ell}=\partial_{u_{\ell}}\wp_{j,k}$, where
$\partial_{u_{j}}=\frac{\partial}{\partial u_{j}}$. For $i=1,2$, let
$\sigma_{E_{i}}(u)$ be the sigma function associated with the elliptic curve
$E_{i}$, $\wp_{E_{i}}=-\frac{{\rm d}^{2}}{{\rm d}u^{2}}\log\sigma_{E_{i}}$,
which is the Weierstrass elliptic function, and
$\wp_{E_{i}}^{\prime}=\frac{\rm d}{{\rm d}u}\wp_{E_{i}}$. Let $f_{i}({\bf
u})=\wp_{E_{i}}(\psi_{i,*}({\bf u}))$ for ${\bf u}\in\operatorname{Jac}(V)$
and $i=1,2$. Then the functions $f_{1}({\bf u})$ and $f_{2}({\bf u})$ are the
meromorphic functions on $\operatorname{Jac}(V)$. When we regard $f_{1}({\bf
u})$ and $f_{2}({\bf u})$ as meromorphic functions on $\mathbb{C}^{2}$, from
Proposition 4.1, we have

$\displaystyle f_{1}({\bf
u})=\wp_{E_{1}}\big{(}(1-\alpha\beta)(u_{1}+\alpha\beta u_{3})\big{)},\qquad
f_{2}({\bf u})=\wp_{E_{2}}\big{(}(\alpha\beta+1)(u_{1}-\alpha\beta
u_{3})\big{)},$

where ${\bf u}={}^{t}(u_{1},u_{3})$.

######  Lemma 4.2 ([1], [4, p. 39], [19, pp. 228, 229]).

The following relations hold:

$\displaystyle\wp_{1,1,1}^{2}=4\wp_{3,3}+4\lambda_{4}\wp_{1,1}+4\wp_{1,1}^{3}+4\wp_{1,3}\wp_{1,1}+4\lambda_{2}\wp_{1,1}^{2}+4\lambda_{6},$
(4.2)
$\displaystyle\wp_{1,1,1}\wp_{1,1,3}=2\lambda_{8}+2\wp_{1,3}^{2}-2\wp_{3,3}\wp_{1,1}+2\lambda_{4}\wp_{1,3}+4\wp_{1,3}\wp_{1,1}^{2}+4\lambda_{2}\wp_{1,3}\wp_{1,1},$
(4.3)
$\displaystyle\wp_{1,1,3}^{2}=-4\wp_{3,3}\wp_{1,3}+4\lambda_{2}\wp_{1,3}^{2}+4\wp_{1,1}\wp_{1,3}^{2},$
(4.4)
$\displaystyle\wp_{3,3}=-\wp_{1,1}\wp_{1,3}+\frac{1}{4}\wp_{1,1,1}^{2}-\wp_{1,1}^{3}-\lambda_{2}\wp_{1,1}^{2}-\lambda_{4}\wp_{1,1}-\lambda_{6},$
(4.5) $\displaystyle\wp_{1,3,3}=\wp_{1,1,1}\wp_{1,3}-\wp_{1,1}\wp_{1,1,3},$
(4.6)
$\displaystyle\wp_{3,3,3}=2\wp_{1,1}\wp_{1,3,3}-\wp_{3,3}\wp_{1,1,1}-\wp_{1,3}\wp_{1,1,3}+2\lambda_{2}\wp_{1,3,3}-\lambda_{4}\wp_{1,1,3}.$
(4.7)

######  Theorem 4.3.

The hyperelliptic function $f_{1}$ is expressed in terms of $\wp_{1,1}$,
$\wp_{1,3}$, and $\wp_{3,3}$ as follows:

$\displaystyle f_{1}({\bf
u})=\frac{-\alpha\beta\wp_{1,3}^{2}-A_{1}\wp_{1,3}-\alpha\beta(\wp_{1,1}-\alpha\beta)\big{(}\wp_{3,3}-\alpha^{3}\beta^{3}\big{)}}{(\alpha\beta-1)^{2}\big{(}\wp_{1,3}+\alpha^{2}\beta^{2}\big{)}^{2}},$
(4.8)

where ${\bf u}\in\mathbb{C}^{2}$ and

$\displaystyle
A_{1}=\wp_{3,3}+\alpha^{2}\beta^{2}\wp_{1,1}+\alpha\beta(\alpha\beta-1)^{2}(\alpha-\beta)^{2}-2\alpha^{3}\beta^{3}.$

Here for simplicity we denote $\wp_{j,k}({\bf u})$ by $\wp_{j,k}$.

###### Proof.

Let $U$ be the subset of $\mbox{Sym}^{2}(V)$ consisting of elements $(P,Q)$
such that $P,Q\in V\backslash\\{\infty\\}$, $x_{1},x_{2}\neq\alpha\beta$,
$x_{1}\neq x_{2}$, and $x_{1}x_{2}\neq\alpha^{2}\beta^{2}$, where
$P=(x_{1},y_{1})$ and $Q=(x_{2},y_{2})$. Then $U$ is an open set in
$\mbox{Sym}^{2}(V)$. Let $U^{\prime}=I(U)$. Since the restriction of $I$ to
$U$ is injective, $U^{\prime}$ is an open set in $\operatorname{Jac}(V)$. We
take a point ${\bf u}\in U^{\prime}$. Then there exists $(P,Q)\in U$ such that
${\bf u}=I\left((P,Q)\right)$. We have

$\displaystyle{\bf
u}=\int_{\infty}^{P}{\boldsymbol{\omega}}+\int_{\infty}^{Q}{\boldsymbol{\omega}}=\int_{O_{1}}^{P}{\boldsymbol{\omega}}+\int_{O_{1}}^{Q}{\boldsymbol{\omega}}-2\int_{O_{1}}^{\infty}{\boldsymbol{\omega}}.$

Let $P=(x_{1},y_{1})$ and $Q=(x_{2},y_{2})$. We have
$\varphi_{1}(\infty)=(0,0)$, which is a branch point of $E_{1}$. Since the map
$\varphi_{1,*}$ is the group homomorphism, we have

$\displaystyle\varphi_{1,*}({\bf u})$
$\displaystyle=\varphi_{1,*}\bigg{(}\int_{O_{1}}^{P}{\boldsymbol{\omega}}\bigg{)}+\varphi_{1,*}\bigg{(}\int_{O_{1}}^{Q}{\boldsymbol{\omega}}\bigg{)}-2\varphi_{1,*}\bigg{(}\int_{O_{1}}^{\infty}{\boldsymbol{\omega}}\bigg{)}$
$\displaystyle=\varphi_{1}(P)+\varphi_{1}(Q)-2\varphi_{1}(\infty)=\varphi_{1}(P)+\varphi_{1}(Q).$

Let $\varphi_{1}(P)=(X_{1},Y_{1})$ and $\varphi_{1}(Q)=(X_{2},Y_{2})$. Then we
have

$\displaystyle
X_{i}=\frac{(\alpha-\beta)^{2}x_{i}}{(x_{i}-\alpha\beta)^{2}},\qquad
Y_{i}=\frac{(\alpha-\beta)^{2}y_{i}}{(\alpha\beta-1)(x_{i}-\alpha\beta)^{3}},\qquad
i=1,2.$ (4.9)

From $x_{1}\neq x_{2}$ and $x_{1}x_{2}\neq\alpha^{2}\beta^{2}$, we have
$X_{1}\neq X_{2}$. Let $\varphi_{1}(P)+\varphi_{1}(Q)=(X_{3},Y_{3})$. Then,
from the definition of the addition defined in $E_{1}$ (cf. [49, Chapter III,
Section 2]), we have

$\displaystyle
X_{3}=\left(\frac{Y_{2}-Y_{1}}{X_{2}-X_{1}}\right)^{2}-X_{1}-X_{2}+1+\frac{(\alpha-\beta)^{2}}{(\alpha\beta-1)^{2}}.$
(4.10)

From the well-known solution of the Jacobi inversion problem, we have

$\displaystyle x_{1}+x_{2}=\wp_{1,1}({\bf u}),\qquad
x_{1}x_{2}=-\wp_{1,3}({\bf u}),$ (4.11) $\displaystyle
2y_{i}=-\wp_{1,1,1}({\bf u})x_{i}-\wp_{1,1,3}({\bf u}),\qquad i=1,2.$ (4.12)

From (4.9), (4.11), and (4.12), we have

$\displaystyle\frac{Y_{2}-Y_{1}}{X_{2}-X_{1}}=B({\bf u}),\qquad
X_{1}+X_{2}=C({\bf u}),$ (4.13)

where

$\displaystyle B({\bf
u})=\frac{\big{(}\wp_{1,1}\wp_{1,3}-3\alpha\beta\wp_{1,3}-\alpha^{3}\beta^{3}\big{)}\wp_{1,1,1}-\big{(}\wp_{1,1}^{2}+\wp_{1,3}-3\alpha\beta\wp_{1,1}+3\alpha^{2}\beta^{2}\big{)}\wp_{1,1,3}}{2(\alpha\beta-1)\big{(}{-}\alpha^{2}\beta^{2}+\alpha\beta\wp_{1,1}+\wp_{1,3}\big{)}\big{(}\alpha^{2}\beta^{2}+\wp_{1,3}\big{)}},$
$\displaystyle C({\bf
u})=\frac{(\alpha-\beta)^{2}\big{(}\alpha^{2}\beta^{2}\wp_{1,1}+4\alpha\beta\wp_{1,3}-\wp_{1,1}\wp_{1,3}\big{)}}{\big{(}\alpha^{2}\beta^{2}-\alpha\beta\wp_{1,1}-\wp_{1,3}\big{)}^{2}},$

where for simplicity we denote $\wp_{j,k}({\bf u})$ and $\wp_{j,k,\ell}({\bf
u})$ by $\wp_{j,k}$ and $\wp_{j,k,\ell}$, respectively. From (4.10) and
(4.13), for ${\bf u}\in U^{\prime}$, we obtain

$\displaystyle f_{1}({\bf u})=\wp_{E_{1}}(\psi_{1,*}({\bf u}))=X_{3}=B({\bf
u})^{2}-C({\bf u})+1+\frac{(\alpha-\beta)^{2}}{(\alpha\beta-1)^{2}}.$

We replace $\wp_{1,1,1}^{2}$, $\wp_{1,1,1}\wp_{1,1,3}$, and $\wp_{1,1,3}^{2}$
in the above expression of $f_{1}({\bf u})$ with the right hand sides of
(4.2), (4.3), and (4.4), respectively. Then we find that the relation (4.8)
holds on $U^{\prime}$. When we regard $f_{1}$ as a meromorphic function on
$\mathbb{C}^{2}$, the relation (4.8) holds on $\pi^{-1}(U^{\prime})$. Since
$\pi^{-1}(U^{\prime})$ is an open set in $\mathbb{C}^{2}$, the relation (4.8)
holds on $\mathbb{C}^{2}$. ∎

######  Corollary 4.4.

The hyperelliptic function $f_{2}$ is expressed in terms of $\wp_{1,1}$,
$\wp_{1,3}$, and $\wp_{3,3}$ as follows:

$\displaystyle f_{2}({\bf
u})=\frac{\alpha\beta\wp_{1,3}^{2}-A_{2}\wp_{1,3}+\alpha\beta(\wp_{1,1}+\alpha\beta)\big{(}\wp_{3,3}+\alpha^{3}\beta^{3}\big{)}}{(\alpha\beta+1)^{2}\big{(}\wp_{1,3}+\alpha^{2}\beta^{2}\big{)}^{2}},$
(4.14)

where ${\bf u}\in\mathbb{C}^{2}$ and

$\displaystyle
A_{2}=\wp_{3,3}+\alpha^{2}\beta^{2}\wp_{1,1}-\alpha\beta(\alpha\beta+1)^{2}(\alpha+\beta)^{2}+2\alpha^{3}\beta^{3}.$

Here for simplicity we denote $\wp_{j,k}({\bf u})$ by $\wp_{j,k}$.

###### Proof.

In the same way as the proof of Proposition 4.1, we can obtain the expression
(4.14) of $f_{2}$ by changing $\beta$ with $-\beta$ in the expression of
$f_{1}$ given in (4.8). ∎

##  5 The map from $\boldsymbol{\operatorname{Jac}(E_{i})}$ to
$\boldsymbol{\operatorname{Jac}(V)}$

Since the degree of $\varphi_{i}$ is 2 for $i=1,2$, the preimage
$\varphi_{i}^{-1}(S)$ consists of two elements with multiplicity for any $S\in
E_{i}$. We consider the morphisms

$\displaystyle\chi\colon\quad V\to V,\qquad(x,y)\mapsto(x,-y),$
$\displaystyle\chi_{i}\colon\quad E_{i}\to
E_{i},\qquad(X,Y)\mapsto(X,-Y),\qquad i=1,2.$

We consider the maps

$\displaystyle\varphi_{1}^{*}\colon\quad E_{1}\to\operatorname{Jac}(V),\qquad
S\to\int_{O_{1}}^{S_{1}}{\boldsymbol{\omega}}+\int_{\chi(O_{1})}^{S_{2}}{\boldsymbol{\omega}},$

where $\varphi_{1}^{-1}(S)=\\{S_{1},S_{2}\\}$, and

$\displaystyle\varphi_{2}^{*}\colon\quad E_{2}\to\operatorname{Jac}(V),\qquad
T\to\int_{O_{2}}^{T_{1}}{\boldsymbol{\omega}}+\int_{\chi(O_{2})}^{T_{2}}{\boldsymbol{\omega}},$

where $\varphi_{2}^{-1}(T)=\\{T_{1},T_{2}\\}$.

######  Lemma 5.1.

We have $\varphi_{1}^{-1}(\infty)=\\{O_{1},\chi(O_{1})\\}$ and
$\varphi_{1}^{-1}((0,0))=\\{(0,0),\infty\\}$. For $S=(X,Y)\in
E_{1}\backslash\\{\infty,(0,0)\\}$, let
$\varphi_{1}^{-1}(S)=\\{S_{1},S_{2}\\}$ and $S_{i}=(x_{i},y_{i})$ for $i=1,2$.
Then we have

$\displaystyle
x_{1}=\alpha\beta+\frac{(\alpha-\beta)\big{(}\alpha-\beta+\sqrt{4\alpha\beta
X+(\alpha-\beta)^{2}}\big{)}}{2X},$ $\displaystyle
y_{1}=\frac{(\alpha\beta-1)(\alpha-\beta)Y\big{(}\alpha-\beta+\sqrt{4\alpha\beta
X+(\alpha-\beta)^{2}}\big{)}^{3}}{8X^{3}},$ $\displaystyle
x_{2}=\alpha\beta+\frac{(\alpha-\beta)\big{(}\alpha-\beta-\sqrt{4\alpha\beta
X+(\alpha-\beta)^{2}}\big{)}}{2X},$ $\displaystyle
y_{2}=\frac{(\alpha\beta-1)(\alpha-\beta)Y\big{(}\alpha-\beta-\sqrt{4\alpha\beta
X+(\alpha-\beta)^{2}}\big{)}^{3}}{8X^{3}}.$

###### Proof.

From the definition of $\varphi_{1}$ and the direct calculations, we obtain
the statement of the lemma. ∎

######  Lemma 5.2.

We have $\varphi_{2}^{-1}(\infty)=\\{O_{2},\chi(O_{2})\\}$ and
$\varphi_{2}^{-1}((0,0))=\\{(0,0),\infty\\}$. For $T=(X,Y)\in
E_{2}\backslash\\{\infty,(0,0)\\}$, let
$\varphi_{2}^{-1}(T)=\\{T_{1},T_{2}\\}$ and $T_{i}=(x_{i},y_{i})$ for $i=1,2$.
Then we have

$\displaystyle
x_{1}=-\alpha\beta+\frac{(\alpha+\beta)\big{(}\alpha+\beta+\sqrt{-4\alpha\beta
X+(\alpha+\beta)^{2}}\big{)}}{2X},$ $\displaystyle
y_{1}=\frac{-(\alpha\beta+1)(\alpha+\beta)Y\big{(}\alpha+\beta+\sqrt{-4\alpha\beta
X+(\alpha+\beta)^{2}}\big{)}^{3}}{8X^{3}},$ $\displaystyle
x_{2}=-\alpha\beta+\frac{(\alpha+\beta)\big{(}\alpha+\beta-\sqrt{-4\alpha\beta
X+(\alpha+\beta)^{2}}\big{)}}{2X},$ $\displaystyle
y_{2}=\frac{-(\alpha\beta+1)(\alpha+\beta)Y\big{(}\alpha+\beta-\sqrt{-4\alpha\beta
X+(\alpha+\beta)^{2}}\big{)}^{3}}{8X^{3}}.$

###### Proof.

From the definition of $\varphi_{2}$ and the direct calculations, we obtain
the statement of the lemma. ∎

######  Lemma 5.3.

For $i=1,2$, the map $\varphi_{i}^{*}$ is the holomorphic map between the
complex manifolds $E_{i}$ and $\operatorname{Jac}(V)$, and the map
$\varphi_{i}^{*}$ is the homomorphism between the groups $E_{i}$ and
$\operatorname{Jac}(V)$.

###### Proof.

We consider the following point on $E_{1}$

$\displaystyle
D=\bigg{(}{-}\frac{(\alpha-\beta)^{2}}{4\alpha\beta},\frac{(\alpha-\beta)^{2}(\alpha+\beta)(\alpha\beta+1)}{8\alpha\beta\sqrt{-\alpha\beta}(\alpha\beta-1)}\bigg{)}.$

Let $E_{1}^{\prime}=E_{1}\backslash\\{(0,0),D,\chi_{1}(D),\infty\\}$. From the
definition of $\varphi_{1}^{*}$ and Lemma 5.1, the map $\varphi_{1}^{*}$ is
holomorphic on $E_{1}^{\prime}$. Since the map $\varphi_{1}^{*}$ is bounded
around the points $(0,0),D,\chi_{1}(D)$, and $\infty$, the map
$\varphi_{1}^{*}$ is holomorphic on $E_{1}$. Since
$\varphi_{1}^{-1}(\infty)=\\{O_{1},\chi(O_{1})\\}$, we have
$\varphi_{1}^{*}(\infty)={\bf 0}$, where ${\bf 0}$ is the unit element of
$\operatorname{Jac}(V)$. Therefore the map $\varphi_{1}^{*}$ is the
homomorphism between the groups $E_{1}$ and $\operatorname{Jac}(V)$. By
changing $\beta$ with $-\beta$, we can obtain the statement of the lemma for
$\varphi_{2}^{*}$. ∎

We consider the maps

$\displaystyle\varphi^{*}\colon\quad E_{1}\times
E_{2}\to\operatorname{Jac}(V),\qquad(S,T)\mapsto\varphi_{1}^{*}(S)+\varphi_{2}^{*}(T),$
$\displaystyle\varphi_{*}\colon\quad\operatorname{Jac}(V)\to E_{1}\times
E_{2},\qquad{\bf u}\mapsto(\varphi_{1,*}({\bf u}),\varphi_{2,*}({\bf u})).$

######  Remark 5.4.

It is known that the map $\varphi_{*}$ is a $(2,2)$-isogeny and $\varphi^{*}$
is its dual isogeny (cf. [9, Proposition 4.8], [12, Section 1.3.5, Theorem
2.40], [14, Lemma 2.6], [27, Section 4], [28, Section 1], [39, Section 2],
[53, Theorem 4.2]).

######  Lemma 5.5 ([12, Theorem 2.40], [14, Lemma 2.6], [27, Section 4], [53,
Theorem 4.2]).

We have

$\displaystyle\varphi_{*}\circ\varphi^{*}(S,T)=(2S,2T).$

###### Proof.

For the sake to be complete and self-contained, we give a proof of this lemma.
For $(S,T)\in E_{1}\times E_{2}$, we have

$\displaystyle\varphi_{1,*}\circ\varphi^{*}(S,T)$
$\displaystyle=\varphi_{1,*}\bigg{(}\int_{O_{1}}^{S_{1}}{\boldsymbol{\omega}}+\int_{\chi(O_{1})}^{S_{2}}{\boldsymbol{\omega}}+\int_{O_{2}}^{T_{1}}{\boldsymbol{\omega}}+\int_{\chi(O_{2})}^{T_{2}}{\boldsymbol{\omega}}\bigg{)}$
$\displaystyle=\varphi_{1,*}\bigg{(}\int_{O_{1}}^{S_{1}}\\!{\boldsymbol{\omega}}+\\!\int_{O_{1}}^{S_{2}}\\!{\boldsymbol{\omega}}-\\!\int_{O_{1}}^{\chi(O_{1})}\\!{\boldsymbol{\omega}}+\\!\int_{O_{1}}^{T_{1}}\\!{\boldsymbol{\omega}}-\\!\int_{O_{1}}^{O_{2}}\\!{\boldsymbol{\omega}}+\\!\int_{O_{1}}^{T_{2}}\\!{\boldsymbol{\omega}}-\\!\int_{O_{1}}^{\chi(O_{2})}\\!{\boldsymbol{\omega}}\bigg{)}$
$\displaystyle=\varphi_{1}(S_{1})+\varphi_{1}(S_{2})-\varphi_{1}(\chi(O_{1}))+\varphi_{1}(T_{1})-\varphi_{1}(O_{2})+\varphi_{1}(T_{2})-\varphi_{1}(\chi(O_{2}))$
$\displaystyle=2S+\varphi_{1}(T_{1})+\varphi_{1}(T_{2})-\varphi_{1}(O_{2})-\varphi_{1}(\chi(O_{2})),$

where $\varphi_{1}^{-1}(S)=\\{S_{1},S_{2}\\}$ and
$\varphi_{2}^{-1}(T)=\\{T_{1},T_{2}\\}$. From (3.9), Lemma 5.2, and the direct
calculations, we can check $\varphi_{1}(T_{2})=\chi_{1}(\varphi_{1}(T_{1}))$
and $\varphi_{1}(\chi(O_{2}))=\chi_{1}(\varphi_{1}(O_{2}))$. Therefore we
obtain

$\displaystyle\varphi_{1,*}\circ\varphi^{*}(S,T)=2S.$

By changing $\beta$ with $-\beta$, we can find

$\displaystyle\varphi_{2,*}\circ\varphi^{*}(S,T)=2T.$

Therefore we obtain the statement of the lemma. ∎

We consider the maps

$\displaystyle\psi_{i}^{*}=\varphi_{i}^{*}\circ
J_{i}^{-1}\colon\quad\operatorname{Jac}(E_{i})\to\operatorname{Jac}(V),\qquad
i=1,2.$

For $i=1,2$, the map $\psi_{i}^{*}$ is the holomorphic map between the complex
manifolds $\operatorname{Jac}(E_{i})$ and $\operatorname{Jac}(V)$, and the map
$\psi_{i}^{*}$ is the homomorphism between the groups
$\operatorname{Jac}(E_{i})$ and $\operatorname{Jac}(V)$. We consider the maps

$\displaystyle\psi^{*}\colon\quad\operatorname{Jac}(E_{1})\times\operatorname{Jac}(E_{2})\to\operatorname{Jac}(V),\qquad{}^{t}(v_{1},v_{2})\mapsto\psi_{1}^{*}(v_{1})+\psi_{2}^{*}(v_{2}),$
$\displaystyle\psi_{*}\colon\quad\operatorname{Jac}(V)\to\operatorname{Jac}(E_{1})\times\operatorname{Jac}(E_{2}),\qquad{\bf
u}\mapsto{}^{t}(\psi_{1,*}({\bf u}),\psi_{2,*}({\bf u})).$

######  Proposition 5.6.

For $i=1,2$, we consider the maps

$\displaystyle\widetilde{\psi}_{i}^{*}\colon\quad\mathbb{C}\to\mathbb{C}^{2},\qquad
v\mapsto{}^{t}\left(c_{i}v,d_{i}v\right),$

where

$\displaystyle c_{1}=\frac{1}{1-\alpha\beta},\qquad
d_{1}=\frac{1}{\alpha\beta(1-\alpha\beta)},\qquad
c_{2}=\frac{1}{1+\alpha\beta},\qquad
d_{2}=-\frac{1}{\alpha\beta(1+\alpha\beta)}.$ (5.1)

Then we have $\psi_{i}^{*}\circ\pi_{i}=\pi\circ\widetilde{\psi}_{i}^{*}$,
i.e., we have the following commutative diagram:

$\displaystyle\begin{CD}\mathbb{C}@>{\widetilde{\psi}_{i}^{*}}>{}>\mathbb{C}^{2}\\\
@V{\pi_{i}}V{}V@V{\pi}V{}V\\\
\operatorname{Jac}(E_{i})@>{{\psi_{i}^{*}}}>{}>\operatorname{Jac}(V).\end{CD}$

###### Proof.

For $i=1,2$, since $\psi_{i}^{*}$ is holomorphic and the group homomorphism,
it is well known that there exist complex numbers $c_{i}$ and $d_{i}$ such
that $\psi_{i}^{*}\circ\pi_{i}=\pi\circ\widetilde{\psi}_{i}^{*}$, where

$\displaystyle\widetilde{\psi}_{i}^{*}\colon\quad\mathbb{C}\to\mathbb{C}^{2},\qquad
v\mapsto{}^{t}\left(c_{i}v,d_{i}v\right),$

(e.g., [8, Proposition 1.2.1]). We consider the maps

$\displaystyle\widetilde{\psi}^{*}\colon\quad\mathbb{C}^{2}\to\mathbb{C}^{2},\qquad\begin{pmatrix}v_{1}\\\
v_{2}\end{pmatrix}\mapsto\begin{pmatrix}c_{1}&c_{2}\\\
d_{1}&d_{2}\end{pmatrix}\begin{pmatrix}v_{1}\\\ v_{2}\end{pmatrix}\\!,$
$\displaystyle\widetilde{\psi}_{*}\colon\quad\mathbb{C}^{2}\to\mathbb{C}^{2},\qquad\begin{pmatrix}u_{1}\\\
u_{3}\end{pmatrix}\mapsto\begin{pmatrix}a_{1}&b_{1}\\\
a_{2}&b_{2}\end{pmatrix}\begin{pmatrix}u_{1}\\\ u_{3}\end{pmatrix}\\!,$

where $a_{1}$, $b_{1}$, $a_{2}$, and $b_{2}$ are defined in Proposition 4.1.
We consider the map

$\displaystyle\pi_{1,2}\colon\quad\mathbb{C}^{2}\to\operatorname{Jac}(E_{1})\times\operatorname{Jac}(E_{2}),\qquad{}^{t}(v_{1},v_{2})\mapsto{}^{t}(\pi_{1}(v_{1}),\pi_{2}(v_{2})).$

Then we have the following commutative diagrams:

$\displaystyle\begin{CD}\mathbb{C}^{2}@>{\widetilde{\psi}^{*}}>{}>\mathbb{C}^{2}\\\
@V{\pi_{1,2}}V{}V@V{\pi}V{}V\\\
\operatorname{Jac}(E_{1})\times\operatorname{Jac}(E_{2})@>{{\psi^{*}}}>{}>\operatorname{Jac}(V),\end{CD}\qquad\begin{CD}\mathbb{C}^{2}@>{\widetilde{\psi}_{*}}>{}>\mathbb{C}^{2}\\\
@V{\pi}V{}V@V{\pi_{1,2}}V{}V\\\
\operatorname{Jac}(V)@>{{\psi_{*}}}>{}>\operatorname{Jac}(E_{1})\times\operatorname{Jac}(E_{2}).\end{CD}$

Therefore we obtain the following commutative diagram:

$\displaystyle\begin{CD}\mathbb{C}^{2}@>{\widetilde{\psi}_{*}\circ\widetilde{\psi}^{*}}>{}>\mathbb{C}^{2}\\\
@V{\pi_{1,2}}V{}V@V{\pi_{1,2}}V{}V\\\
\operatorname{Jac}(E_{1})\times\operatorname{Jac}(E_{2})@>{{\psi_{*}\circ\psi^{*}}}>{}>\operatorname{Jac}(E_{1})\times\operatorname{Jac}(E_{2}).\end{CD}$

We have

$\displaystyle\widetilde{\psi}_{*}\circ\widetilde{\psi}^{*}\colon\quad\mathbb{C}^{2}\to\mathbb{C}^{2},\qquad\begin{pmatrix}v_{1}\\\
v_{2}\end{pmatrix}\mapsto\begin{pmatrix}a_{1}&b_{1}\\\
a_{2}&b_{2}\end{pmatrix}\begin{pmatrix}c_{1}&c_{2}\\\
d_{1}&d_{2}\end{pmatrix}\begin{pmatrix}v_{1}\\\ v_{2}\end{pmatrix}\\!.$

From Lemma 5.5, we obtain

$\displaystyle\begin{pmatrix}a_{1}&b_{1}\\\
a_{2}&b_{2}\end{pmatrix}\begin{pmatrix}c_{1}&c_{2}\\\
d_{1}&d_{2}\end{pmatrix}=\begin{pmatrix}2&0\\\ 0&2\end{pmatrix}\\!.$

From (4.1), we obtain (5.1). ∎

For $i=1,2$, let ${\bf k}_{i}={}^{t}(c_{i},d_{i})$ and $K=({\bf k}_{1},{\bf
k}_{2})=\left(\begin{smallmatrix}c_{1}&c_{2}\\\
d_{1}&d_{2}\end{smallmatrix}\right)$, where $c_{1}$, $d_{1}$, $c_{2}$, and
$d_{2}$ are given in (5.1).

######  Theorem 5.7.

For $v\in\mathbb{C}$, the following relations hold:

$\displaystyle\wp_{1,1}({\bf
k}_{1}v)=2\alpha\beta+\frac{(\alpha-\beta)^{2}}{\wp_{E_{1}}(v)},\qquad\wp_{1,3}({\bf
k}_{1}v)=-\alpha^{2}\beta^{2},$ (5.2) $\displaystyle\wp_{3,3}({\bf
k}_{1}v)=\alpha^{2}\beta^{2}\big{\\{}(\alpha\beta-1)^{2}\wp_{E_{1}}(v)+2\alpha\beta\big{\\}},$
(5.3) $\displaystyle\wp_{1,1,1}({\bf
k}_{1}v)=\frac{(\alpha\beta-1)\wp_{E_{1}}^{\prime}(v)\big{\\{}(\alpha-\beta)^{2}+\alpha\beta\wp_{E_{1}}(v)\big{\\}}}{\wp_{E_{1}}(v)^{2}},$
(5.4) $\displaystyle\wp_{1,1,3}({\bf
k}_{1}v)=-\frac{\alpha^{2}\beta^{2}(\alpha\beta-1)\wp_{E_{1}}^{\prime}(v)}{\wp_{E_{1}}(v)},$
(5.5) $\displaystyle\wp_{1,3,3}({\bf
k}_{1}v)=\frac{\alpha^{3}\beta^{3}(\alpha\beta-1)\wp_{E_{1}}^{\prime}(v)}{\wp_{E_{1}}(v)},$
(5.6) $\displaystyle\wp_{3,3,3}({\bf
k}_{1}v)=-\frac{\alpha^{3}\beta^{3}(\alpha\beta-1)\big{\\{}(\alpha\beta-1)^{2}\wp_{E_{1}}(v)+\alpha\beta\big{\\}}\wp_{E_{1}}^{\prime}(v)}{\wp_{E_{1}}(v)}.$
(5.7)

###### Proof.

We consider the set $E_{1}^{\prime}$ defined in the proof of Lemma 5.3 and
$U^{\prime\prime}=J_{1}(E_{1}^{\prime})$. Then $U^{\prime\prime}$ is an open
set in $\operatorname{Jac}(E_{1})$. We take a point $S=(X,Y)\in
E_{1}^{\prime}$ and set $\overline{v}=J_{1}(S)$. We have

$\displaystyle\varphi_{1}^{*}(S)=\int_{O_{1}}^{S_{1}}{\boldsymbol{\omega}}+\int_{\chi(O_{1})}^{S_{2}}{\boldsymbol{\omega}}=\int_{\infty}^{S_{1}}{\boldsymbol{\omega}}+\int_{\infty}^{S_{2}}{\boldsymbol{\omega}},$

where $\varphi_{1}^{-1}(S)=\\{S_{1},S_{2}\\}$. From Proposition 5.6, we have
$\varphi_{1}^{*}(S)=\pi({\bf k}_{1}v)$, where $v\in\mathbb{C}$ such that
$\pi_{1}(v)=\overline{v}$. From Lemma 5.1 and $S\neq(0,0)$, we have
$S_{1},S_{2}\neq\infty$. Let $S_{i}=(x_{i},y_{i})$ for $i=1,2$. From Lemma 5.1
and $S\in E_{1}^{\prime}$, we have $x_{1}\neq x_{2}$, i.e.,
$S_{2}\neq\chi(S_{1})$. Therefore, from the well-known solution of the Jacobi
inversion problem, we have

$\displaystyle\wp_{1,1}({\bf k}_{1}v)=x_{1}+x_{2},\qquad\wp_{1,3}({\bf
k}_{1}v)=-x_{1}x_{2},$ $\displaystyle 2y_{i}=-\wp_{1,1,1}({\bf
k}_{1}v)x_{i}-\wp_{1,1,3}({\bf k}_{1}v),\qquad i=1,2.$

Thus we have

$\displaystyle\wp_{1,1,1}({\bf
k}_{1}v)=-2\frac{y_{1}-y_{2}}{x_{1}-x_{2}},\qquad\wp_{1,1,3}({\bf
k}_{1}v)=2\frac{x_{2}y_{1}-x_{1}y_{2}}{x_{1}-x_{2}}.$

We have $X=\wp_{E_{1}}(v)$ and $Y=-\wp_{E_{1}}^{\prime}(v)/2$. Therefore, from
Lemma 5.1 and the direct calculations, we find that (5.2), (5.4), and (5.5)
hold for $v\in\pi_{1}^{-1}(U^{\prime\prime})$. Since
$\pi_{1}^{-1}(U^{\prime\prime})$ is an open set in $\mathbb{C}$, (5.2), (5.4),
and (5.5) hold on $\mathbb{C}$. From (4.5), (4.6), (4.7), and the direct
calculations, we obtain (5.3), (5.6), and (5.7). ∎

######  Corollary 5.8.

For $v\in\mathbb{C}$, the following relations hold:

$\displaystyle\wp_{1,1}({\bf
k}_{2}v)=-2\alpha\beta+\frac{(\alpha+\beta)^{2}}{\wp_{E_{2}}(v)},\qquad\wp_{1,3}({\bf
k}_{2}v)=-\alpha^{2}\beta^{2},$ (5.8) $\displaystyle\wp_{3,3}({\bf
k}_{2}v)=\alpha^{2}\beta^{2}\big{\\{}(\alpha\beta+1)^{2}\wp_{E_{2}}(v)-2\alpha\beta\big{\\}},$
(5.9) $\displaystyle\wp_{1,1,1}({\bf
k}_{2}v)=-\frac{(\alpha\beta+1)\wp_{E_{2}}^{\prime}(v)\big{\\{}(\alpha+\beta)^{2}-\alpha\beta\wp_{E_{2}}(v)\big{\\}}}{\wp_{E_{2}}(v)^{2}},$
(5.10) $\displaystyle\wp_{1,1,3}({\bf
k}_{2}v)=\frac{\alpha^{2}\beta^{2}(\alpha\beta+1)\wp_{E_{2}}^{\prime}(v)}{\wp_{E_{2}}(v)},$
(5.11) $\displaystyle\wp_{1,3,3}({\bf
k}_{2}v)=\frac{\alpha^{3}\beta^{3}(\alpha\beta+1)\wp_{E_{2}}^{\prime}(v)}{\wp_{E_{2}}(v)},$
(5.12) $\displaystyle\wp_{3,3,3}({\bf
k}_{2}v)=-\frac{\alpha^{3}\beta^{3}(\alpha\beta+1)\big{\\{}(\alpha\beta+1)^{2}\wp_{E_{2}}(v)-\alpha\beta\big{\\}}\wp_{E_{2}}^{\prime}(v)}{\wp_{E_{2}}(v)}.$
(5.13)

###### Proof.

In the same way as the proof of Proposition 4.1, we can obtain the expressions
(5.8)–(5.13) by changing $\beta$ with $-\beta$ in the expressions (5.2)–(5.7),
respectively. ∎

######  Theorem 5.9.

For ${\bf v}={}^{t}(v_{1},v_{2})\in\mathbb{C}^{2}$, we have

$\displaystyle\wp_{1,1}(K{\bf
v})=-\frac{p_{2}\big{\\{}(\alpha\beta-1)^{2}(\alpha+\beta)^{2}\wp_{E_{1}}^{2}+(\alpha\beta+1)^{2}(\alpha-\beta)^{2}\wp_{E_{2}}^{2}+p_{3}\big{\\}}}{2p_{1}^{2}},$
$\displaystyle\wp_{1,3}(K{\bf
v})=\alpha^{2}\beta^{2}+\frac{\alpha^{2}\beta^{2}p_{2}\big{\\{}\big{(}\alpha^{2}\beta^{2}-1\big{)}\wp_{E_{1}}^{\prime}\wp_{E_{2}}^{\prime}-p_{2}\big{\\}}}{2p_{1}^{2}},$
$\displaystyle\wp_{3,3}(K{\bf
v})=-\frac{\alpha^{2}\beta^{2}p_{2}\big{\\{}\big{(}\alpha^{2}\beta^{2}-1\big{)}^{2}\wp_{E_{1}}^{2}\wp_{E_{2}}^{2}+\big{(}\alpha^{2}-\beta^{2}\big{)}^{2}+p_{3}\big{\\}}}{2p_{1}^{2}},$

where

$\displaystyle
p_{1}=\big{\\{}(\alpha\beta-1)^{2}\wp_{E_{1}}-(\alpha\beta+1)^{2}\wp_{E_{2}}+8\alpha\beta\big{\\}}\wp_{E_{1}}\wp_{E_{2}}-(\alpha+\beta)^{2}\wp_{E_{1}}+(\alpha-\beta)^{2}\wp_{E_{2}},$
$\displaystyle
p_{2}=\big{(}\alpha^{2}\beta^{2}-1\big{)}\wp_{E_{1}}^{\prime}\wp_{E_{2}}^{\prime}-2\big{\\{}(\alpha\beta-1)^{2}\wp_{E_{1}}+(\alpha\beta+1)^{2}\wp_{E_{2}}-2\big{(}\alpha^{2}+1\big{)}\big{(}\beta^{2}+1\big{)}\big{\\}}\wp_{E_{1}}\wp_{E_{2}}$
$\displaystyle\hphantom{p_{2}=}{}-2\big{\\{}(\alpha+\beta)^{2}\wp_{E_{1}}+(\alpha-\beta)^{2}\wp_{E_{2}}\big{\\}},$
$\displaystyle p_{3}=-2\alpha\beta
p_{1}-2\big{(}\alpha^{2}\beta^{4}+\alpha^{4}\beta^{2}-4\alpha^{2}\beta^{2}+\alpha^{2}+\beta^{2}\big{)}\wp_{E_{1}}\wp_{E_{2}}.$

Here for simplicity we denote $\wp_{E_{1}}(v_{1})$ and $\wp_{E_{2}}(v_{2})$ by
$\wp_{E_{1}}$ and $\wp_{E_{2}}$.

###### Proof.

In [30, Theorem 3.3], for ${\bf u},{\bf v}\in\mathbb{C}^{2}$, the following
functions are introduced:

$\displaystyle q({\bf u},{\bf v})=\wp_{3,3}({\bf u})-\wp_{3,3}({\bf
v})+\wp_{1,3}({\bf u})\wp_{1,1}({\bf v})-\wp_{1,3}({\bf v})\wp_{1,1}({\bf
u}),$ $\displaystyle q_{1}({\bf u},{\bf v})=\wp_{1,3,3}({\bf
u})-\wp_{1,3,3}({\bf v})+\wp_{1,1,3}({\bf u})\wp_{1,1}({\bf
v})-\wp_{1,1,3}({\bf v})\wp_{1,1}({\bf u})+\wp_{1,1,1}({\bf v})\wp_{1,3}({\bf
u})$ $\displaystyle\hphantom{q_{1}({\bf u},{\bf v})=}{}-\wp_{1,1,1}({\bf
u})\wp_{1,3}({\bf v}),$ $\displaystyle q_{3}({\bf u},{\bf v})=\wp_{3,3,3}({\bf
u})-\wp_{3,3,3}({\bf v})+\wp_{1,3,3}({\bf u})\wp_{1,1}({\bf
v})-\wp_{1,3,3}({\bf v})\wp_{1,1}({\bf u})+\wp_{1,1,3}({\bf v})\wp_{1,3}({\bf
u})$ $\displaystyle\hphantom{q_{3}({\bf u},{\bf v})=}{}-\wp_{1,1,3}({\bf
u})\wp_{1,3}({\bf v}),$ $\displaystyle q_{1,1}({\bf u},{\bf
v})=8\lambda_{2}(\wp_{1,3}({\bf u})\wp_{1,1}({\bf v})-\wp_{1,3}({\bf
v})\wp_{1,1}({\bf u}))+4\lambda_{4}(\wp_{1,3}({\bf u})-\wp_{1,3}({\bf v}))$
$\displaystyle\hphantom{q_{1,1}({\bf u},{\bf v})=}{}-4(\wp({\bf u})-\wp({\bf
v}))-8(\wp_{3,3}({\bf u})\wp_{1,1}({\bf v})-\wp_{3,3}({\bf v})\wp_{1,1}({\bf
u}))+2(\wp_{1,1,3}({\bf u})\wp_{1,1,1}({\bf v})$
$\displaystyle\hphantom{q_{1,1}({\bf u},{\bf v})=}{}-\wp_{1,1,3}({\bf
v})\wp_{1,1,1}({\bf u})),$ $\displaystyle q_{1,3}({\bf u},{\bf
v})=4\lambda_{6}(\wp_{1,3}({\bf u})-\wp_{1,3}({\bf
v}))+2\lambda_{4}(\wp_{1,3}({\bf u})\wp_{1,1}({\bf v})-\wp_{1,3}({\bf
v})\wp_{1,1}({\bf u}))$ $\displaystyle\hphantom{q_{1,3}({\bf u},{\bf
v})=}{}-4(\wp_{3,3}({\bf u})\wp_{1,3}({\bf v})-\wp_{3,3}({\bf
v})\wp_{1,3}({\bf u}))+2(\wp({\bf u})\wp_{1,1}({\bf v})-\wp({\bf
v})\wp_{1,1}({\bf u}))$ $\displaystyle\hphantom{q_{1,3}({\bf u},{\bf
v})=}{}-2\lambda_{8}(\wp_{1,1}({\bf u})-\wp_{1,1}({\bf v}))+\wp_{1,1,1}({\bf
v})\wp_{1,3,3}({\bf u})-\wp_{1,1,1}({\bf u})\wp_{1,3,3}({\bf v}),$
$\displaystyle q_{3,3}({\bf u},{\bf v})=4\lambda_{6}q({\bf u},{\bf
v})+4\lambda_{8}(\wp_{1,3}({\bf u})-\wp_{1,3}({\bf v}))+4(\wp({\bf
u})\wp_{1,3}({\bf v})-\wp({\bf v})\wp_{1,3}({\bf u}))$
$\displaystyle\hphantom{q_{3,3}({\bf u},{\bf v})=}{}+2(\wp_{1,3,3}({\bf
u})\wp_{1,1,3}({\bf v})-\wp_{1,3,3}({\bf v})\wp_{1,1,3}({\bf u})),$

where we set $\wp=\wp_{1,1}\wp_{3,3}-\wp_{1,3}^{2}$. From $K{\bf v}={\bf
k}_{1}v_{1}+{\bf k}_{2}v_{2}$ and the addition formulae for $\wp_{j,k}$, which
are given in [30, Theorem 3.3], we obtain

$\displaystyle\wp_{1,1}(K{\bf v})=-\wp_{1,1}({\bf k}_{1}v_{1})-\wp_{1,1}({\bf
k}_{2}v_{2})+\frac{1}{4}\bigg{(}\frac{q_{1}({\bf k}_{1}v_{1},{\bf
k}_{2}v_{2})}{q({\bf k}_{1}v_{1},{\bf
k}_{2}v_{2})}\bigg{)}^{2}-\frac{q_{1,1}({\bf k}_{1}v_{1},{\bf
k}_{2}v_{2})}{4q({\bf k}_{1}v_{1},{\bf k}_{2}v_{2})},$
$\displaystyle\wp_{1,3}(K{\bf v})=-\wp_{1,3}({\bf k}_{1}v_{1})-\wp_{1,3}({\bf
k}_{2}v_{2})+\frac{q_{1}({\bf k}_{1}v_{1},{\bf k}_{2}v_{2})q_{3}({\bf
k}_{1}v_{1},{\bf k}_{2}v_{2})}{4q({\bf k}_{1}v_{1},{\bf
k}_{2}v_{2})^{2}}-\frac{q_{1,3}({\bf k}_{1}v_{1},{\bf k}_{2}v_{2})}{4q({\bf
k}_{1}v_{1},{\bf k}_{2}v_{2})},$ $\displaystyle\wp_{3,3}(K{\bf
v})=-\wp_{3,3}({\bf k}_{1}v_{1})-\wp_{3,3}({\bf
k}_{2}v_{2})+\frac{1}{4}\bigg{(}\frac{q_{3}({\bf k}_{1}v_{1},{\bf
k}_{2}v_{2})}{q({\bf k}_{1}v_{1},{\bf
k}_{2}v_{2})}\bigg{)}^{2}-\frac{q_{3,3}({\bf k}_{1}v_{1},{\bf
k}_{2}v_{2})}{4q({\bf k}_{1}v_{1},{\bf k}_{2}v_{2})}.$

From Theorem 5.7 and Corollary 5.8, we obtain the expressions of the theorem.
∎

######  Remark 5.10.

From Theorem 5.9, we obtain the decompositions of the functions $\wp_{1,1}$,
$\wp_{1,3}-\alpha^{2}\beta^{2}$, and $\wp_{3,3}$ into the products of the
meromorphic functions.

######  Remark 5.11.

For ${\bf u}={}^{t}(u_{1},u_{3})\in\mathbb{C}^{2}$, we set $F({\bf
u})=2\wp_{1,1}({\bf u})+\frac{2}{3}\lambda_{2}$ and $G({\bf
u})=2\wp_{1,3}({\bf u})$. It is known that the function $F$ is a solution of
the KdV-hierarchy (see [6, Theorem 2.25] and [19, Theorem 3.6])

$\displaystyle\frac{\partial F}{\partial
u_{3}}=\frac{1}{4}\frac{\partial^{3}F}{\partial
u_{1}^{3}}-\frac{3}{2}F\frac{\partial F}{\partial u_{1}},$ $\displaystyle
0=\frac{1}{4}\frac{\partial^{3}F}{\partial u_{1}^{2}\partial
u_{3}}-\bigg{(}F+\frac{\lambda_{2}}{3}\bigg{)}\frac{\partial F}{\partial
u_{3}}-\frac{1}{2}\frac{\partial F}{\partial u_{1}}G,\qquad\frac{\partial
F}{\partial u_{3}}=\frac{\partial G}{\partial u_{1}}.$

For ${\bf v}={}^{t}(v_{1},v_{2})\in\mathbb{C}^{2}$, we set $\widetilde{F}({\bf
v})=\wp_{1,1}(K{\bf v})$ and $\widetilde{G}({\bf v})=\wp_{1,3}(K{\bf v})$.
From Theorem 5.9, we can express $\widetilde{F}({\bf v})$ and
$\widetilde{G}({\bf v})$ in terms of $\wp_{E_{1}}(v_{1})$ and
$\wp_{E_{2}}(v_{2})$. We have $F({\bf u})=2\widetilde{F}\big{(}K^{-1}{\bf
u}\big{)}+\frac{2}{3}\lambda_{2}$ and $G({\bf
u})=2\widetilde{G}\big{(}K^{-1}{\bf u}\big{)}$. Therefore, we can obtain a
solution of the KdV-hierarchy in terms of the functions constructed by the
Weierstrass elliptic functions $\wp_{E_{1}}$ and $\wp_{E_{2}}$.

######  Remark 5.12.

It is well known that the functions $\wp_{1,1}$, $\wp_{1,3}$, and $\wp_{3,3}$
are coordinates of the Kummer surface (e.g., [17, 19, 32]). From Theorem 5.9,
we obtain the expressions of the coordinates of the Kummer surface in terms of
the Weierstrass elliptic functions $\wp_{E_{1}}$ and $\wp_{E_{2}}$.

##  6 Comparison of our results and the results of [6, 19, 25]

Let us consider the curve $V$ defined by (3.4). Let $\widetilde{E}_{1}$ and
$\widetilde{E}_{2}$ be the elliptic curves defined by

$\displaystyle\widetilde{E}_{1}=\big{\\{}\big{(}\widetilde{X},\widetilde{Y}\big{)}\in\mathbb{C}^{2}\mid\widetilde{Y}^{2}=\widetilde{X}\big{(}\widetilde{X}-1\big{)}\big{(}\widetilde{X}-1/\kappa_{1}^{2}\big{)}\big{\\}},$
$\displaystyle\widetilde{E}_{2}=\big{\\{}\big{(}\widetilde{X},\widetilde{Y}\big{)}\in\mathbb{C}^{2}\mid\widetilde{Y}^{2}=\widetilde{X}\big{(}\widetilde{X}-1\big{)}\big{(}\widetilde{X}-1/\kappa_{2}^{2}\big{)}\big{\\}}.$

For $i=1,2,$ let
$\\{\widetilde{\mathfrak{a}}^{(i)},\widetilde{\mathfrak{b}}^{(i)}\\}$ be a
canonical basis in the one-dimensional homology group of the curve
$\widetilde{E}_{i}$. For $i=1,2$, we define the periods

$\displaystyle
2\widetilde{\Omega}_{i}^{\prime}=\int_{\widetilde{\mathfrak{a}}^{(i)}}-\frac{{\rm
d}\widetilde{X}}{2\widetilde{Y}},\qquad
2\widetilde{\Omega}_{i}^{\prime\prime}=\int_{\widetilde{\mathfrak{b}}^{(i)}}-\frac{{\rm
d}\widetilde{X}}{2\widetilde{Y}}.$

We define the period lattice
$\widetilde{\Lambda}_{i}=\big{\\{}2\widetilde{\Omega}_{i}^{\prime}m_{1}+2\widetilde{\Omega}_{i}^{\prime\prime}m_{2}\mid
m_{1},m_{2}\in\mathbb{Z}\big{\\}}$ and consider the Jacobian variety
$\operatorname{Jac}\big{(}\widetilde{E}_{i}\big{)}=\mathbb{C}/\widetilde{\Lambda}_{i}$.
For $i=1,2$, we take the canonical basis
$\big{\\{}\widetilde{\mathfrak{a}}^{(i)},\widetilde{\mathfrak{b}}^{(i)}\big{\\}}$
in the one-dimensional homology group of the curve $\widetilde{E}_{i}$ such
that the following relations hold in
$\operatorname{Jac}\big{(}\widetilde{E}_{i}\big{)}$

$\displaystyle\int_{\infty}^{(1/\kappa_{i}^{2},0)}\bigg{(}{-}\frac{{\rm
d}\widetilde{X}}{2\widetilde{Y}}\bigg{)}=\widetilde{\Omega}_{i}^{\prime},\qquad\int_{\infty}^{(0,0)}\bigg{(}{-}\frac{{\rm
d}\widetilde{X}}{2\widetilde{Y}}\bigg{)}=\widetilde{\Omega}_{i}^{\prime\prime},\qquad\int_{\infty}^{(1,0)}\bigg{(}{-}\frac{{\rm
d}\widetilde{X}}{2\widetilde{Y}}\bigg{)}=\widetilde{\Omega}_{i}^{\prime}+\widetilde{\Omega}_{i}^{\prime\prime}.$

For $i=1,2$, let
$\widetilde{\tau}_{i}=\big{(}\widetilde{\Omega}_{i}^{\prime}\big{)}^{-1}\widetilde{\Omega}_{i}^{\prime\prime}$.
We consider the Jacobi elliptic functions $\mbox{sn}(u,\widetilde{\tau}_{i})$,
$\mbox{cn}(u,\widetilde{\tau}_{i})$, and $\mbox{dn}(u,\widetilde{\tau}_{i})$
associated with $\widetilde{\tau}_{i}$, where the definitions of the Jacobi
elliptic functions are written in Section 2.1. For $i=1,2$, let $m_{i}$ be the
modulus of the Jacobi elliptic functions associated with
$\widetilde{\tau}_{i}$. Then we have $m_{i}^{2}=\kappa_{i}^{2}$ (cf. [36]). In
[6, 19, 25], the Jacobi elliptic functions with modulus $\kappa_{i}$ are
considered. Since the Jacobi elliptic functions are determined by the second
power of the modulus, the Jacobi elliptic functions
$\mbox{sn}(u,\widetilde{\tau}_{i}),\mbox{cn}(u,\widetilde{\tau}_{i})$, and
$\mbox{dn}(u,\widetilde{\tau}_{i})$ are equal to the ones considered in [6,
19, 25]. For ${\bf u}={}^{t}(u_{1},u_{3})\in\mathbb{C}^{2}$, let

$\displaystyle
w_{1}=\sqrt{\big{(}1-\alpha^{2}\big{)}\big{(}1-\beta^{2}\big{)}}(u_{1}+\alpha\beta
u_{3}),\qquad
w_{2}=\sqrt{\big{(}1-\alpha^{2}\big{)}\big{(}1-\beta^{2}\big{)}}(u_{1}-\alpha\beta
u_{3})$

and

$\displaystyle
Z_{1}=\mbox{sn}(w_{1},\widetilde{\tau}_{1})\mbox{sn}(w_{2},\widetilde{\tau}_{2}),\qquad
Z_{2}=\mbox{cn}(w_{1},\widetilde{\tau}_{1})\mbox{cn}(w_{2},\widetilde{\tau}_{2}),\qquad
Z_{3}=\mbox{dn}(w_{1},\widetilde{\tau}_{1})\mbox{dn}(w_{2},\widetilde{\tau}_{2}).$

Let $\wp_{j,k}$ be the hyperelliptic functions associated with the curve $V$
defined by (3.4).

######  Theorem 6.1 ([6, p. 3448], [19, p. 99], [25]).

For ${\bf u}={}^{t}(u_{1},u_{3})\in\mathbb{C}^{2}$, the following relations
hold:

$\displaystyle
Z_{1}=-\frac{\big{(}1-\alpha^{2}\big{)}\big{(}1-\beta^{2}\big{)}\big{(}\alpha^{2}\beta^{2}+\wp_{1,3}({\bf
u})\big{)}}{\big{(}\alpha^{2}+\beta^{2}\big{)}\big{(}\wp_{1,3}({\bf
u})-\alpha^{2}\beta^{2}\big{)}+\alpha^{2}\beta^{2}\wp_{1,1}({\bf
u})+\wp_{3,3}({\bf u})},$ (6.1) $\displaystyle
Z_{2}=-\frac{\big{(}1+\alpha^{2}\beta^{2}\big{)}\big{(}\alpha^{2}\beta^{2}-\wp_{1,3}({\bf
u})\big{)}-\alpha^{2}\beta^{2}\wp_{1,1}({\bf u})-\wp_{3,3}({\bf
u})}{\big{(}\alpha^{2}+\beta^{2}\big{)}\big{(}\wp_{1,3}({\bf
u})-\alpha^{2}\beta^{2}\big{)}+\alpha^{2}\beta^{2}\wp_{1,1}({\bf
u})+\wp_{3,3}({\bf u})},$ (6.2) $\displaystyle
Z_{3}=-\frac{\alpha^{2}\beta^{2}\wp_{1,1}({\bf u})-\wp_{3,3}({\bf
u})}{\big{(}\alpha^{2}+\beta^{2}\big{)}\big{(}\wp_{1,3}({\bf
u})-\alpha^{2}\beta^{2}\big{)}+\alpha^{2}\beta^{2}\wp_{1,1}({\bf
u})+\wp_{3,3}({\bf u})}.$ (6.3)

######  Theorem 6.2 ([6, p. 3448], [19, p. 100], [25]).

For ${\bf u}={}^{t}(u_{1},u_{3})\in\mathbb{C}^{2}$, the following relations
hold:

$\displaystyle\wp_{1,1}({\bf
u})=\frac{\big{(}\alpha^{2}+\beta^{2}\big{)}(Z_{2}-Z_{3})+\big{(}1+\alpha^{2}\beta^{2}\big{)}(Z_{3}-1)}{Z_{1}+Z_{2}-1},$
$\displaystyle\wp_{1,3}({\bf
u})=\alpha^{2}\beta^{2}\frac{1+Z_{1}-Z_{2}}{Z_{1}+Z_{2}-1},$
$\displaystyle\wp_{3,3}({\bf
u})=\alpha^{2}\beta^{2}\frac{\big{(}\alpha^{2}+\beta^{2}\big{)}(Z_{2}+Z_{3})-\big{(}1+\alpha^{2}\beta^{2}\big{)}(Z_{3}+1)}{Z_{1}+Z_{2}-1}.$

######  Remark 6.3.

In [6, p. 3448], [7, p. 366], [17, pp. 79, 80], [19, pp. 99, 175], and [25],
it is mentioned that $(Z_{1},Z_{2},Z_{3})$ and
$(\wp_{1,1},\wp_{1,3},\wp_{3,3})$ are coordinates on the Kummer surfaces. In
[12], algebraic correspondences between Kummer surfaces associated with the
Jacobian variety of a hyperelliptic curve of genus 2 admitting a morphism of
degree 2 to an elliptic curve and Kummer surfaces associated with the product
of two non-isogenous elliptic curves are considered. In [12, equations (2.56)
and (2.57)], the formulae in Theorems 6.1 and 6.2 in the present paper are
described in terms of coordinates on the Kummer surfaces.

In this section, we give relationships between the above theorems (Theorems
6.1 and 6.2) and our results. For $i=1,2$, let $\wp_{\widetilde{E}_{i}}$ be
the Weierstrass elliptic function associated with $\widetilde{E}_{i}$. Then,
for $u\in\mathbb{C}$ and $i=1,2$, we have (cf. [36])

$\displaystyle\wp_{\widetilde{E}_{i}}(u)-\frac{1}{\kappa_{i}^{2}}=\frac{\mbox{cn}(u/\kappa_{i},\widetilde{\tau}_{i})^{2}}{\kappa_{i}^{2}\mbox{sn}(u/\kappa_{i},\widetilde{\tau}_{i})^{2}},$
$\displaystyle\wp_{\widetilde{E}_{i}}(u)-1=\frac{\mbox{dn}(u/\kappa_{i},\widetilde{\tau}_{i})^{2}}{\kappa_{i}^{2}\mbox{sn}(u/\kappa_{i},\widetilde{\tau}_{i})^{2}},$
$\displaystyle\wp_{\widetilde{E}_{i}}(u)=\frac{1}{\kappa_{i}^{2}\mbox{sn}(u/\kappa_{i},\widetilde{\tau}_{i})^{2}}.$

From the above equations, for $u\in\mathbb{C}$ and $i=1,2$, we have

$\displaystyle\mbox{sn}(u,\widetilde{\tau}_{i})^{2}=\frac{1}{\kappa_{i}^{2}\wp_{\widetilde{E}_{i}}(\kappa_{i}u)},$
(6.4)
$\displaystyle\mbox{cn}(u,\widetilde{\tau}_{i})^{2}=\frac{\wp_{\widetilde{E}_{i}}(\kappa_{i}u)-1/\kappa_{i}^{2}}{\wp_{\widetilde{E}_{i}}(\kappa_{i}u)},$
(6.5)
$\displaystyle\mbox{dn}(u,\widetilde{\tau}_{i})^{2}=\frac{\wp_{\widetilde{E}_{i}}(\kappa_{i}u)-1}{\wp_{\widetilde{E}_{i}}(\kappa_{i}u)}.$
(6.6)

For $i=1,2$, our elliptic curves $E_{i}$ are isomorphic to the elliptic curves
$\widetilde{E}_{i}$ by the following morphisms

$\displaystyle\widetilde{\xi}_{1}\colon\quad
E_{1}\to\widetilde{E}_{1},\qquad(X,Y)\mapsto\big{(}\widetilde{X},\widetilde{Y}\big{)}=\bigg{(}{-}\frac{(\alpha\beta-1)^{2}}{(\alpha-\beta)^{2}}X+1,\frac{\sqrt{-1}(\alpha\beta-1)^{3}Y}{(\alpha-\beta)^{3}}\bigg{)},$
(6.7) $\displaystyle\widetilde{\xi}_{2}\colon\quad
E_{2}\to\widetilde{E}_{2},\qquad(X,Y)\mapsto\big{(}\widetilde{X},\widetilde{Y}\big{)}=\bigg{(}{-}\frac{(\alpha\beta+1)^{2}}{(\alpha+\beta)^{2}}X+1,\frac{-\sqrt{-1}(\alpha\beta+1)^{3}Y}{(\alpha+\beta)^{3}}\bigg{)}.$

######  Proposition 6.4.

For $u\in\mathbb{C}$, the following relations hold:

$\displaystyle\wp_{E_{1}}\bigg{(}{-}\sqrt{-1}\frac{\alpha\beta-1}{\alpha-\beta}u\bigg{)}=\frac{(\alpha-\beta)^{2}}{(\alpha\beta-1)^{2}}\big{(}1-\wp_{\widetilde{E}_{1}}(u)\big{)},$
(6.8)
$\displaystyle\wp_{E_{2}}\bigg{(}\sqrt{-1}\frac{\alpha\beta+1}{\alpha+\beta}u\bigg{)}=\frac{(\alpha+\beta)^{2}}{(\alpha\beta+1)^{2}}\big{(}1-\wp_{\widetilde{E}_{2}}(u)\big{)}.$
(6.9)

###### Proof.

We take a point $u\in\mathbb{C}\backslash\widetilde{\Lambda}_{1}$. Then, there
exist a point
$\widetilde{S}=\big{(}\widetilde{X},\widetilde{Y}\big{)}\in\widetilde{E}_{1}\backslash\\{\infty\\}$
and a path $\widetilde{\gamma}$ from $\infty$ to $\widetilde{S}$ on
$\widetilde{E}_{1}$ such that

$\displaystyle u=\int_{\infty}^{\widetilde{S}}\bigg{(}{-}\frac{{\rm
d}\widetilde{X}}{2\widetilde{Y}}\bigg{)},$ (6.10)

where we take $\widetilde{\gamma}$ as the path of integration in the right
hand side of (6.10). Let $\gamma$ be the image of $\widetilde{\gamma}$ with
respect to the map $\widetilde{\xi}_{1}^{-1}\colon\widetilde{E}_{1}\to E_{1}$,
which is the inverse mapping of the map $\widetilde{\xi}_{1}$. Let
$S=\widetilde{\xi}_{1}^{-1}\big{(}\widetilde{S}\big{)}$ and $S=(X,Y)$. The
pullback of the holomorphic one form $-{\rm
d}\widetilde{X}/\big{(}2\widetilde{Y}\big{)}$ on $\widetilde{E}_{1}$ with
respect to the map $\widetilde{\xi}_{1}$ is

$\displaystyle\sqrt{-1}\frac{\alpha-\beta}{\alpha\beta-1}\bigg{(}{-}\frac{{\rm
d}X}{2Y}\bigg{)}.$

Then we have

$\displaystyle
u=\sqrt{-1}\frac{\alpha-\beta}{\alpha\beta-1}\int_{\infty}^{S}\bigg{(}{-}\frac{{\rm
d}X}{2Y}\bigg{)},$ (6.11)

where we take $\gamma$ as the path of integration in the right hand side of
(6.11). From (6.10), we have

$\displaystyle\widetilde{X}=\wp_{\widetilde{E}_{1}}(u).$ (6.12)

From (6.11), we have

$\displaystyle
X=\wp_{E_{1}}\bigg{(}{-}\sqrt{-1}\frac{\alpha\beta-1}{\alpha-\beta}u\bigg{)}.$
(6.13)

From (6.7), we have

$\displaystyle
X=\frac{(\alpha-\beta)^{2}}{(\alpha\beta-1)^{2}}\big{(}1-\widetilde{X}\big{)}.$
(6.14)

From (6.12), (6.13), and (6.14), we find that the relation (6.8) holds for
$u\in\mathbb{C}\backslash\widetilde{\Lambda}_{1}$. Since
$\mathbb{C}\backslash\widetilde{\Lambda}_{1}$ is an open set in $\mathbb{C}$,
the relation (6.8) holds for $u\in\mathbb{C}$. By changing $\beta$ with
$-\beta$ in (6.8), we obtain (6.9). ∎

By raising the both sides of (6.1), (6.2), and (6.3) to the second power and
substituting (6.4), (6.5), and (6.6), we can express
$\wp_{\widetilde{E}_{1}}(\kappa_{1}w_{1})$ and
$\wp_{\widetilde{E}_{2}}(\kappa_{2}w_{2})$ in terms of $\wp_{1,1}({\bf u})$,
$\wp_{1,3}({\bf u})$, and $\wp_{3,3}({\bf u})$. We have

$\displaystyle\kappa_{1}w_{1}=\sqrt{-1}(\alpha-\beta)(u_{1}+\alpha\beta
u_{3}),\qquad\kappa_{2}w_{2}=\sqrt{-1}(\alpha+\beta)(u_{1}-\alpha\beta
u_{3}).$

From Proposition 6.4 and the fact that the Weierstrass elliptic functions
$\wp_{E_{1}}$ and $\wp_{E_{2}}$ are the even functions, we have

$\displaystyle\wp_{E_{1}}\big{(}(1-\alpha\beta)(u_{1}+\alpha\beta
u_{3})\big{)}=\frac{(\alpha-\beta)^{2}}{(\alpha\beta-1)^{2}}\big{(}1-\wp_{\widetilde{E}_{1}}(\kappa_{1}w_{1})\big{)},$
$\displaystyle\wp_{E_{2}}\big{(}(\alpha\beta+1)(u_{1}-\alpha\beta
u_{3})\big{)}=\frac{(\alpha+\beta)^{2}}{(\alpha\beta+1)^{2}}\big{(}1-\wp_{\widetilde{E}_{2}}(\kappa_{2}w_{2})\big{)}.$

Therefore, we can express $\wp_{E_{1}}\big{(}(1-\alpha\beta)(u_{1}+\alpha\beta
u_{3})\big{)}$ and $\wp_{E_{2}}\big{(}(\alpha\beta+1)(u_{1}-\alpha\beta
u_{3})\big{)}$ in terms of $\wp_{1,1}({\bf u})$, $\wp_{1,3}({\bf u})$, and
$\wp_{3,3}({\bf u})$. These expressions are equal to (4.8) and (4.14). We set
${\bf u}={\bf k}_{1}v$. Then we have $w_{1}=2h$ and $w_{2}=0$, where

$\displaystyle
h=\frac{\sqrt{\big{(}1-\alpha^{2}\big{)}\big{(}1-\beta^{2}\big{)}}}{1-\alpha\beta}v.$

Since $\mbox{sn}(0,\widetilde{\tau}_{2})=0$ and
$\mbox{cn}(0,\widetilde{\tau}_{2})=\mbox{dn}(0,\widetilde{\tau}_{2})=1$, we
have

$\displaystyle Z_{1}=0,\qquad Z_{2}=\mbox{cn}(2h,\widetilde{\tau}_{1}),\qquad
Z_{3}=\mbox{dn}(2h,\widetilde{\tau}_{1}).$ (6.15)

By the addition formulae for the Jacobi elliptic functions, we have

$\displaystyle\mbox{cn}(2h,\widetilde{\tau}_{1})=\frac{\mbox{cn}(h,\widetilde{\tau}_{1})^{2}-\mbox{sn}(h,\widetilde{\tau}_{1})^{2}\mbox{dn}(h,\widetilde{\tau}_{1})^{2}}{1-\kappa_{1}^{2}\mbox{sn}(h,\widetilde{\tau}_{1})^{4}},$
(6.16)
$\displaystyle\mbox{dn}(2h,\widetilde{\tau}_{1})=\frac{\mbox{dn}(h,\widetilde{\tau}_{1})^{2}-\kappa_{1}^{2}\mbox{sn}(h,\widetilde{\tau}_{1})^{2}\mbox{cn}(h,\widetilde{\tau}_{1})^{2}}{1-\kappa_{1}^{2}\mbox{sn}(h,\widetilde{\tau}_{1})^{4}}.$
(6.17)

From Theorem 6.2, (6.15), (6.16), (6.17), (6.4), (6.5), and (6.6), we obtain

$\displaystyle\wp_{1,1}({\bf
k}_{1}v)=\frac{\alpha^{2}\beta^{2}-2\alpha\beta\wp_{\widetilde{E}_{1}}(\kappa_{1}h)+1}{1-\wp_{\widetilde{E}_{1}}(\kappa_{1}h)},$
$\displaystyle\wp_{1,3}({\bf k}_{1}v)=-\alpha^{2}\beta^{2},$
$\displaystyle\wp_{3,3}({\bf
k}_{1}v)=\alpha^{2}\beta^{2}\big{\\{}\alpha^{2}+\beta^{2}-(\alpha-\beta)^{2}\wp_{\widetilde{E}_{1}}(\kappa_{1}h)\big{\\}}.$

From (6.8), we obtain (5.2) and (5.3). In the same way as $\wp_{j,k}({\bf
k}_{1}v)$, we can obtain (5.8) and (5.9) from Theorem 6.2.

######  Remark 6.5.

For $i=1,2$ and $j=1,2,3$, let ${\rm al}^{(i)}_{j}(u)$ be the Weierstrass
elliptic al functions associated with $\widetilde{E}_{i}$, where the
definitions of the Weierstrass elliptic al functions are written in Section
2.2. From (6.4), (6.5), and (6.6), for $i=1,2$, we obtain

$\displaystyle{\rm sn}(w_{i},\widetilde{\tau}_{i})=\frac{1}{\kappa_{i}{\rm
al}^{(i)}_{3}(\kappa_{i}w_{i})},\qquad{\rm
cn}(w_{i},\widetilde{\tau}_{i})=\frac{{\rm
al}^{(i)}_{1}(\kappa_{i}w_{i})}{{\rm
al}^{(i)}_{3}(\kappa_{i}w_{i})},\qquad{\rm
dn}(w_{i},\widetilde{\tau}_{i})=\frac{{\rm
al}^{(i)}_{2}(\kappa_{i}w_{i})}{{\rm al}^{(i)}_{3}(\kappa_{i}w_{i})}.$

Let

$\displaystyle\mathcal{Z}_{j}={\rm al}^{(1)}_{j}(\kappa_{1}w_{1}){\rm
al}^{(2)}_{j}(\kappa_{2}w_{2}),\qquad j=1,2,3.$

From Theorem 6.1, for ${\bf u}={}^{t}(u_{1},u_{3})\in\mathbb{C}^{2}$, the
following relations hold:

$\displaystyle\mathcal{Z}_{1}=\frac{\alpha^{2}-\beta^{2}}{\big{(}1-\alpha^{2}\big{)}^{2}\big{(}1-\beta^{2}\big{)}^{2}}\frac{\big{(}1+\alpha^{2}\beta^{2}\big{)}\big{(}\wp_{1,3}({\bf
u})-\alpha^{2}\beta^{2}\big{)}+\alpha^{2}\beta^{2}\wp_{1,1}({\bf
u})+\wp_{3,3}({\bf u})}{\alpha^{2}\beta^{2}+\wp_{1,3}({\bf u})},$
$\displaystyle\mathcal{Z}_{2}=\frac{\alpha^{2}-\beta^{2}}{\big{(}1-\alpha^{2}\big{)}^{2}\big{(}1-\beta^{2}\big{)}^{2}}\frac{\wp_{3,3}({\bf
u})-\alpha^{2}\beta^{2}\wp_{1,1}({\bf u})}{\alpha^{2}\beta^{2}+\wp_{1,3}({\bf
u})},$
$\displaystyle\mathcal{Z}_{3}=\frac{\alpha^{2}-\beta^{2}}{\big{(}1-\alpha^{2}\big{)}^{2}\big{(}1-\beta^{2}\big{)}^{2}}\frac{\big{(}\alpha^{2}+\beta^{2}\big{)}\big{(}\wp_{1,3}({\bf
u})-\alpha^{2}\beta^{2}\big{)}+\alpha^{2}\beta^{2}\wp_{1,1}({\bf
u})+\wp_{3,3}({\bf u})}{\alpha^{2}\beta^{2}+\wp_{1,3}({\bf u})}.$

Furthermore, from Theorem 6.2, for ${\bf
u}={}^{t}(u_{1},u_{3})\in\mathbb{C}^{2}$, the following relations hold:

$\displaystyle\wp_{1,1}({\bf
u})=\big{(}1-\alpha^{2}\big{)}\big{(}1-\beta^{2}\big{)}\frac{\big{(}1+\alpha^{2}\beta^{2}\big{)}\mathcal{Z}_{3}-\big{(}1-\alpha^{2}\big{)}\big{(}1-\beta^{2}\big{)}\mathcal{Z}_{2}-\big{(}\alpha^{2}+\beta^{2}\big{)}\mathcal{Z}_{1}}{\big{(}1-\alpha^{2}\big{)}\big{(}1-\beta^{2}\big{)}(\mathcal{Z}_{3}-\mathcal{Z}_{1})+\alpha^{2}-\beta^{2}},$
$\displaystyle\wp_{1,3}({\bf
u})=\alpha^{2}\beta^{2}\frac{\big{(}1-\alpha^{2}\big{)}\big{(}1-\beta^{2}\big{)}(\mathcal{Z}_{1}-\mathcal{Z}_{3})+\alpha^{2}-\beta^{2}}{\big{(}1-\alpha^{2}\big{)}\big{(}1-\beta^{2}\big{)}(\mathcal{Z}_{3}-\mathcal{Z}_{1})+\alpha^{2}-\beta^{2}},$
$\displaystyle\wp_{3,3}({\bf
u})=\alpha^{2}\beta^{2}\big{(}1-\alpha^{2}\big{)}\big{(}1-\beta^{2}\big{)}\frac{\big{(}1+\alpha^{2}\beta^{2}\big{)}\mathcal{Z}_{3}+\big{(}1-\alpha^{2}\big{)}\big{(}1-\beta^{2}\big{)}\mathcal{Z}_{2}-\big{(}\alpha^{2}+\beta^{2}\big{)}\mathcal{Z}_{1}}{\big{(}1-\alpha^{2}\big{)}\big{(}1-\beta^{2}\big{)}(\mathcal{Z}_{3}-\mathcal{Z}_{1})+\alpha^{2}-\beta^{2}}.$

##  Appendix A Proof of Theorem 3.1

For the details of the theory of algebraic function fields, see [51]. Let $H$
be a hyperelliptic curve of genus 2, $W$ be an elliptic curve, and $\rho\colon
H\to W$ be a morphism of degree 2. Let $\mathcal{F}(H)$ and $\mathcal{F}(W)$
be the function fields of $H$ and $W$, respectively. The map $\rho$ induces
the injective homomorphism of fields
$\rho^{*}\colon\mathcal{F}(W)\to\mathcal{F}(H)$. Since the degree of the map
$\rho$ is 2, the degree of the extension $[\mathcal{F}(H):\mathcal{F}(W)]$ is
2. Therefore the extension $\mathcal{F}(W)\subset\mathcal{F}(H)$ is the Galois
extension. Let $g$ be the generator of the Galois group
$\mbox{Gal}(\mathcal{F}(H)/\mathcal{F}(W))$. The field $\mathcal{F}(H)$ has
the unique subfield $\mathbb{C}(x)$ of genus 0 such that
$[\mathcal{F}(H):\mathbb{C}(x)]=2$, where $x\in\mathcal{F}(H)$ is
transcendental over $\mathbb{C}$ (cf. [51, Proposition 6.2.4]). Since the
genus of $g(\mathbb{C}(x))$ is 0, we have $g(\mathbb{C}(x))=\mathbb{C}(x)$.
Therefore the map $g$ induces the automorphism of $\mathbb{C}(x)$ fixing
$\mathbb{C}$.

######  Lemma A.1.

We can choose $X\in\mathbb{C}(x)$ such that $\mathbb{C}(X)=\mathbb{C}(x)$ and
$g(X)=-X$.

###### Proof.

There exist $a,b,c,d\in\mathbb{C}$ such that $ad-bc=1$ and

$\displaystyle g(x)=\frac{ax+b}{cx+d}$

(cf. [45, Theorem 1.1.1]). Let $A=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)$. Since $g^{2}$ is the identity mapping, we have
$A^{2}=E$ or $A^{2}=-E$, where $E$ is the identity matrix. By the
Cayley–Hamilton theorem, we have

$\displaystyle A^{2}=(a+d)A-E.$ (A.1)

We assume $A^{2}=E$. From (A.1), we have $(a+d)A=2E$. Thus we have $a+d\neq 0$
and $A=\frac{2}{a+d}E$. Therefore we have $g(x)=x$. Then we have
$x\in\mathcal{F}(W)$, i.e., $\mathbb{C}(x)\subset\mathcal{F}(W)$. Since the
genus of $\mathbb{C}(x)$ is 0 and the genus of $\mathcal{F}(W)$ is 1, we have
$\mathbb{C}(x)\neq\mathcal{F}(W)$. Since $[\mathcal{F}(H):\mathcal{F}(W)]=2$,
we have $[\mathcal{F}(H):\mathbb{C}(x)]>2$. This contradicts
$[\mathcal{F}(H):\mathbb{C}(x)]=2$. Therefore we have $A^{2}=-E$. From (A.1),
we have $a+d=0$. From $a+d=0$ and $ad-bc=1$, we find that the eigenvalues of
$A$ are $\pm\sqrt{-1}$. Therefore there exists a regular matrix $R$ such that

$\displaystyle RA=\begin{pmatrix}-\sqrt{-1}&0\\\ 0&\sqrt{-1}\end{pmatrix}R.$
(A.2)

Let $R=\left(\begin{smallmatrix}r_{1}&r_{2}\\\
r_{3}&r_{4}\end{smallmatrix}\right)$ and

$\displaystyle X=\frac{r_{1}x+r_{2}}{r_{3}x+r_{4}}.$

Then we have $\mathbb{C}(X)=\mathbb{C}(x)$ (cf. [45, Example 1.1.3]). Let
$R^{\prime}=RA$ and
$R^{\prime}=\left(\begin{smallmatrix}r_{1}^{\prime}&r_{2}^{\prime}\\\
r_{3}^{\prime}&r_{4}^{\prime}\end{smallmatrix}\right)$. From (A.2), we have

$\displaystyle
g(X)=\frac{r_{1}^{\prime}x+r_{2}^{\prime}}{r_{3}^{\prime}x+r_{4}^{\prime}}=\frac{-\sqrt{-1}r_{1}x-\sqrt{-1}r_{2}}{\sqrt{-1}r_{3}x+\sqrt{-1}r_{4}}=-X.$
∎

Since $\mathcal{F}(H)$ is an algebraic function field of genus 2, there exists
$Y\in\mathcal{F}(H)$ such that $\mathcal{F}(H)=\mathbb{C}(X,Y)$ and
$Y^{2}=f(X)$ with a square-free polynomial $f(X)$ of degree 5 or 6 (cf. [51,
Example 3.7.6]). We assume that the degree of $f(X)$ is 5. Let
$f(X)=(X-\gamma_{1})\cdots(X-\gamma_{5})$, where $\gamma_{i}\in\mathbb{C}$ for
$1\leq i\leq 5$ and $\gamma_{i}\neq\gamma_{j}$ for $i\neq j$. A Weierstrass
point of $\mathcal{F}(H)$ is a place of $\mathcal{F}(H)$ which includes
$X-\gamma_{i}$ for some $1\leq i\leq 5$ or $1/X$ (cf. [29, Corollary I.49]).
Let $P_{i}$ be the place which includes $X-\gamma_{i}$. Then $g(P_{i})$
includes $X+\gamma_{i}$. Since $g(P_{i})$ is also a Weierstrass point of
$\mathcal{F}(H)$ (cf. [29, Proposition I.51]) and $g(P_{i})$ does not include
$1/X$, $g(P_{i})$ includes $X-\gamma_{j}$ for some $1\leq j\leq 5$. Therefore
we have $-\gamma_{i}=\gamma_{j}$. Thus the polynomial $f(X)$ has the following
form:

$\displaystyle
f(X)=X\big{(}X^{2}-\widetilde{\gamma}_{1}^{2}\big{)}\big{(}X^{2}-\widetilde{\gamma}_{2}^{2}\big{)},$

where $\widetilde{\gamma}_{1},\widetilde{\gamma}_{2}\in\mathbb{C}$,
$\widetilde{\gamma}_{1}\widetilde{\gamma}_{2}\neq 0$, and
$\widetilde{\gamma}_{1}^{2}\neq\widetilde{\gamma}_{2}^{2}$. From $Y^{2}=f(X)$,
we have $g\big{(}Y^{2}\big{)}=-Y^{2}$. Thus we have $g(Y)=\sqrt{-1}Y$ or
$g(Y)=-\sqrt{-1}Y$. Then we have $g^{2}(Y)=-Y$, which contradicts that $g^{2}$
is the identity mapping. Therefore the degree of $f(X)$ is 6. Let
$f(X)=(X-\gamma_{1})\cdots(X-\gamma_{6})$, where $\gamma_{i}\in\mathbb{C}$ for
$1\leq i\leq 6$ and $\gamma_{i}\neq\gamma_{j}$ for $i\neq j$. A Weierstrass
point of $\mathcal{F}(H)$ is a place of $\mathcal{F}(H)$ which includes
$X-\gamma_{i}$ for some $1\leq i\leq 6$ (cf. [29, Corollary I.49]). Let
$P_{i}$ be the place which includes $X-\gamma_{i}$. Then $g(P_{i})$ includes
$X+\gamma_{i}$. Since $g(P_{i})$ is also a Weierstrass point of
$\mathcal{F}(H)$ (cf. [29, Proposition I.51]), $g(P_{i})$ includes
$X-\gamma_{j}$ for some $1\leq j\leq 6$. Therefore we have
$-\gamma_{i}=\gamma_{j}$. Thus the polynomial $f(X)$ has the following form:

$\displaystyle
f(X)=\big{(}X^{2}-\widetilde{\gamma}_{1}^{2}\big{)}\big{(}X^{2}-\widetilde{\gamma}_{2}^{2}\big{)}\big{(}X^{2}-\widetilde{\gamma}_{3}^{2}\big{)},$

where
$\widetilde{\gamma}_{1},\widetilde{\gamma}_{2},\widetilde{\gamma}_{3}\in\mathbb{C}$,
$\widetilde{\gamma}_{1}\widetilde{\gamma}_{2}\widetilde{\gamma}_{3}\neq 0$,
and $\widetilde{\gamma}_{i}^{2}\neq\widetilde{\gamma}_{j}^{2}$ for $i\neq j$.
Let $\widetilde{H}$ be the hyperelliptic curve of genus 2 defined by

$\displaystyle\widetilde{H}=\big{\\{}(X,Y)\in\mathbb{C}^{2}\mid
Y^{2}=f(X)\big{\\}}.$

Since the function field of $\widetilde{H}$ is $\mathcal{F}(H)$, the curve $H$
is birationally equivalent to the curve $\widetilde{H}$. Since the curves $H$
and $\widetilde{H}$ are smooth, the curve $H$ is isomorphic to the curve
$\widetilde{H}$. The curve $\widetilde{H}$ is isomorphic to the curve

$\displaystyle H^{\prime}=\big{\\{}(s,t)\in\mathbb{C}^{2}\mid
t^{2}=\big{(}s^{2}-1\big{)}\big{(}s^{2}-e_{1}^{2}\big{)}\big{(}s^{2}-e_{2}^{2}\big{)}\big{\\}},$

where $e_{1}=\widetilde{\gamma}_{2}/\widetilde{\gamma}_{1}$ and
$e_{2}=\widetilde{\gamma}_{3}/\widetilde{\gamma}_{1}$, by the morphism

$\displaystyle\widetilde{\xi}\colon\quad\widetilde{H}\to
H^{\prime},\qquad(X,Y)\mapsto\big{(}X/\widetilde{\gamma}_{1},Y/\widetilde{\gamma}_{1}^{3}\big{)}.$

Thus the curve $H$ is isomorphic to the curve $H^{\prime}$. Next, we consider
the curve $H^{\prime}$ defined by (3.1). Let $W$ be the elliptic curve defined
by

$\displaystyle W=\big{\\{}(s,t)\in\mathbb{C}^{2}\mid
t^{2}=(s-1)\big{(}s-e_{1}^{2}\big{)}\big{(}s-e_{2}^{2}\big{)}\big{\\}}.$

Then we can define the morphism of degree 2

$\displaystyle H^{\prime}\to W\colon\quad(s,t)\mapsto\big{(}s^{2},t\big{)}.$

Therefore we obtain the statement of the theorem.

### Acknowledgements

The authors would like to thank the referees for reading our manuscript
carefully and giving the useful comments. The work of Takanori Ayano was
supported by JSPS KAKENHI Grant Number JP21K03296 and was partly supported by
Osaka City University Advanced Mathematical Institute (MEXT Joint
Usage/Research Center on Mathematics and Theoretical Physics JPMXP0619217849).

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