Relationships Between Hyperelliptic Functions of Genus 2 and Elliptic Functions

The article is devoted to the classical problems about the relationships between elliptic functions and hyperelliptic functions of genus 2. It contains new results, as well as a derivation from them of well-known results on these issues. Our research was motivated by applications to the theory of equations and dynamical systems integrable in hyperelliptic functions of genus 2. We consider a hyperelliptic curve $V$ of genus 2 which admits a morphism of degree 2 to an elliptic curve. Then there exist two elliptic curves $E_i$, $i=1,2$, and morphisms of degree 2 from $V$ to $E_i$. We construct hyperelliptic functions associated with $V$ from the Weierstrass elliptic functions associated with $E_i$ and describe them in terms of the fundamental hyperelliptic functions defined by the logarithmic derivatives of the two-dimensional sigma functions. We show that the restrictions of hyperelliptic functions associated with $V$ to the appropriate subspaces in $\mathbb{C}^2$ are elliptic functions and describe them in terms of the Weierstrass elliptic functions associated with $E_i$. Further, we express the hyperelliptic functions associated with $V$ on $\mathbb{C}^2$ in terms of the Weierstrass elliptic functions associated with $E_i$. We derive these results by describing the homomorphisms between the Jacobian varieties of the curves $V$ and $E_i$ induced by the morphisms from $V$ to $E_i$ explicitly.


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 arXiv:2106.06764v3 [math.AG] 1 Feb 2022 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 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 = f (x) with a square-free polynomial f (x) of degree 5, its sigma function is determined by the coefficients of the polynomial 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 = f (x) with a square-free polynomial 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).
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 α, β ∈ C satisfying α 2 , β 2 = 0, 1, α 2 = β 2 , and α 2 β 2 = 1, we consider the nonsingular hyperelliptic curve of genus 2 For a hyperelliptic curve H of genus 2, it is known that there exist an elliptic curve W and a morphism H → W of degree 2 if and only if H is isomorphic to the curve V for some α, β (see Section 3). Let E 1 and E 2 be the elliptic curves defined by Then we can define the morphisms of degree 2 (see Section 3) Let Jac(V ), Jac(E 1 ), and Jac(E 2 ) be the Jacobian varieties of V , E 1 , and E 2 , respectively. The maps ϕ i , i = 1, 2, induce the homomorphisms of the Jacobian varieties In this paper, we describe the maps ψ i, * and ψ * i explicitly in Propositions 4.1 and 5.6. Let σ(u), u = t (u 1 , u 3 ) ∈ C 2 , be the sigma function associated with V , which is the holomorphic function on Then the functions ℘ j,k and ℘ j,k, are the fundamental meromorphic functions on Jac(V ). For i = 1, 2, let ℘ E i be the Weierstrass elliptic function associated with the elliptic curve E i . Let f i (u) = ℘ E i (ψ i, * (u)) for u ∈ Jac(V ) and i = 1, 2. Then the functions f 1 (u) and f 2 (u) are the meromorphic functions on Jac(V ). In this paper, we express f 1 (u) and f 2 (u) in terms of ℘ 1,1 (u), ℘ 1,3 (u), and ℘ 3,3 (u) explicitly in Theorem 4.3 and Corollary 4.4. We consider the homomorphisms of the Jacobian varieties For v ∈ Jac(E i ), the functions ℘ j,k (ψ * i (v)) and ℘ j,k, (ψ * i (v)) are the meromorphic functions on Jac(E i ). In this paper, we describe the functions ℘ j,k (ψ * i (v)) and ℘ j,k, (ψ * i (v)) in terms of ℘ E i (v) explicitly in Theorem 5.7 and Corollary 5.8, i.e., we show that the restrictions of the hyperelliptic functions ℘ j,k and ℘ j,k, to the appropriate subspaces in C 2 are elliptic functions and describe them in terms of the Weierstrass elliptic functions ℘ E i . Further, by using the addition formulae for ℘ j,k , which are given in [30,Theorem 3.3], we express ℘ j,k on C 2 in terms of ℘ E 1 and ℘ E 2 explicitly in Theorem 5.9. From Theorem 5.9, we obtain the decompositions of the functions ℘ 1,1 , ℘ 1,3 − α 2 β 2 , and ℘ 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 ℘ E 1 and ℘ E 2 (Remark 5.11), and the expressions of the coordinates of the Kummer surface in terms of the Weierstrass elliptic functions ℘ E 1 and ℘ 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.

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].

The elliptic sigma function
We set We assume that M 1 (x) has no multiple root. We consider the elliptic curve For (x, y) ∈ E, let which is the basis of the vector space of holomorphic one forms on E. Further, let which is the meromorphic one form on E with a pole only at ∞. Let {a, b} be a canonical basis in the one-dimensional homology group of the curve E. We define the periods by The normalized period has the form τ = (ω ) −1 ω . The sigma function associated with E is defined by where θ 11 (0, τ ) = ∂ ∂z θ 11 (z, τ )| z=0 . For m 1 , m 2 ∈ Z, let Ω = 2ω m 1 + 2ω m 2 . Then, for u ∈ C, we have σ(u + Ω)/σ(u) = (−1) m 1 −m 2 +m 1 m 2 exp (2η m 1 + 2η m 2 )(u + ω m 1 + ω m 2 ) .
We set deg λ 2i = 2i for 1 ≤ i ≤ 3. The sigma function σ(u) is an entire function on C and the series expansion of σ(u) around u = 0 has the following form: where the coefficient µ n is a homogeneous polynomial in Q[λ 2 , λ 4 , λ 6 ] of degree n − 1 if µ n = 0. In particular, the sigma function σ(u) does not depend on the choice of a canonical basis {a, b} in the one-dimensional homology group of the elliptic curve E and is determined by the coefficients λ 2 , λ 4 , λ 6 of the defining equation of the elliptic curve E.

The two-dimensional sigma function
We set We assume that M 2 (x) has no multiple root. We consider the nonsingular hyperelliptic curve of genus 2 For (x, y) ∈ V , let which are the basis of the vector space of holomorphic one forms on V . Further, let which are the meromorphic one forms on V with a pole only at ∞.
be a canonical basis in the one-dimensional homology group of the curve V . We define the matrices of periods by The matrix of normalized periods has the form τ [26,41]). The sigma function σ(u) associated with V , where θ[δ](u) is the Riemann's theta function with characteristics δ, which is defined by and ε is a constant. [42, p. 193]). For m 1 , m 2 ∈ Z 2 , let Ω = 2ω m 1 + 2ω m 2 . Then, for u ∈ C 2 , we have We set deg λ 2i = 2i for 1 ≤ i ≤ 5. is an entire function on C 2 and we have the unique constant ε such that the series expansion of σ(u) around the origin has the following form: We take the constant ε such that the expansion (2.1) holds, see the expression for the sigma function above, which involves the constant ε. Then the sigma function σ(u) does not depend on the choice of a canonical basis in the one-dimensional homology group of the curve V and is determined by the coefficients λ 2 , λ 4 , λ 6 , λ 8 , λ 10 of the defining equation of the curve V .

Inversion problem of the elliptic integrals for the Jacobi and Legendre forms
Let where a J,2 , a J,4 ∈ C satisfying a J,4 a 2 J,2 − 4a J,4 = 0. We consider the elliptic curve in the Jacobi form and the elliptic curve in the Weierstrass normal form We take a J ∈ C such that f J (a J ) = 0. We have the isomorphism Let {a J , b J } be a canonical basis in the one-dimensional homology group of the curve E J . We define the periods by We define the period lattice Λ J = {2Ω J m 1 + 2Ω J m 2 | m 1 , m 2 ∈ Z} and consider the Jacobian variety Jac(E J ) = C/Λ J . Let σ E J (u) be the sigma function associated with the elliptic curve E J and ℘ E J = − d 2 du 2 log σ E J , which is the Weierstrass elliptic function. We consider the map Proof . Since the pullback of the holomorphic one form −dx/(2y) on E J with respect to ξ J is ds/(2t), we have We have 3), we obtain the statement of the proposition.
We consider the elliptic curve in the Legendre form

and the elliptic curve in the Weierstrass normal form
We have the isomorphism Let {a L , b L } be a canonical basis in the one-dimensional homology group of the curve E L . We define the periods by We define the period lattice Λ L = {2Ω L m 1 + 2Ω L m 2 | m 1 , m 2 ∈ Z} and consider the Jacobian variety Jac(E L ) = C/Λ L . Let σ E L (u) be the sigma function associated with the elliptic curve E L and ℘ E L = − d 2 du 2 log σ E L , which is the Weierstrass elliptic function. We consider the map The map I L is the isomorphism. Let u = I L (S) for S = (s, t) ∈ E L .
Proposition 2.5. We have Proof . Since the pullback of the holomorphic one form −dx/(2y) on E L with respect to We have From (2.4) and (2.5), we obtain the statement of the proposition.
3 The curve of genus 2 and the elliptic curves where e 1 , e 2 ∈ C satisfying In appendix, we give a proof of this theorem. 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 √ z. For α, β ∈ C satisfying α 2 , β 2 = 0, 1, we consider the nonsingular hyperelliptic curve of genus 2 1 This curve V is considered in [5,6,7,12,17,19,25]. For i = 2, 4, 6, 8, we define λ i such that the following relation holds: i.e., we set Conversely, given α, β ∈ C satisfying (3.3), the curve V is isomorphic to the curve H with The hyperelliptic curve of genus 2 defined by  . Given e 1 , e 2 ∈ C satisfying (3.2), an isomorphism from H to V with (3.5) is given by Furthermore, given α, β ∈ C satisfying (3.3), an isomorphism from V to H with (3.6) is given by The elliptic curve W 1 with (3.6) is isomorphic to the elliptic curve in Legendre form by the morphism The elliptic curve W 2 with (3.6) is isomorphic to the elliptic curve in Legendre form by the morphism We consider the maps Then the maps ϕ i , i = 1, 2, are described by (3.10) Remark 3.4. For P = (x, y) ∈ V \{∞} such that x = αβ, we have ϕ 1 (P ) = ∞. For P = (x, y) ∈ V \{∞} such that x = −αβ, we have ϕ 2 (P ) = ∞. Furthermore, we have ϕ i (∞) = (0, 0) for i = 1, 2. Let Remark 3.5. In [5, Section 7.1], [6, Section 3.1.1], [19,Chapter 6], [25], the elliptic curves are considered. We have the isomorphisms , , Remark 3.6. In [12, Lemma 1.7 and Proposition 1.8], for i = 1, 2, the elliptic curves i and the morphisms are considered. For i = 1, 2, we have the isomorphisms , 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 Λ = 2ω m 1 + 2ω m 2 | m 1 , m 2 ∈ Z 2 and consider the Jacobian variety Jac(V ) = C 2 /Λ. For i = 1, 2, let a (i) , b (i) 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 We define the periods by We define the period lattice Λ i = 2Ω i m 1 + 2Ω i m 2 | m 1 , m 2 ∈ Z and consider the Jacobian variety Jac(E i ) = C/Λ i . Let π : C 2 → Jac(V ) and π i : C → Jac(E i ) be the natural projections for i = 1, 2.
In [43], the curves of genus 2 defined by by the morphism where The curve C 48,1 is isomorphic to the curve V defined by (3.4) with Then the Jacobian variety Jac(V ) is isogenous to E 1 × 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 by the morphism The curve C 48,2 is isomorphic to the curve V defined by (3.4) with Then the Jacobian variety Jac(V ) is isogenous to E 1 × E 2 defined by (3.7) and (3.8) with (3.12).
For i = 1, 2, we consider the maps It is well known that the map J i is the isomorphism. We consider the maps For i = 1, 2, the map ψ i, * is the holomorphic map between the complex manifolds Jac(V ) and Jac(E i ), and the map ψ i, * is the homomorphism between the groups Jac(V ) and Jac(E i ).
Proposition 4.1. For i = 1, 2, we consider the maps Then we have ψ i, * • π = π i • ψ i, * , i.e., we have the following commutative diagram: Proof . Since ψ 1, * is holomorphic and the group homomorphism, it is well known that there exist complex numbers a 1 and b 1 such that ψ 1, * • π = π 1 • ψ 1, * , where (e.g., [8, Proposition 1.2.1]). We consider the following commutative diagram: We consider the holomorphic one forms du on Jac(E 1 ) and du 1 , du 3 on Jac(V ). The pullback of the holomorphic one form Ω 1 on E 1 with respect to J −1 1 is du. The pullback of du with respect to ψ 1, * is a 1 du 1 + b 1 du 3 . The pullback of a 1 du 1 + b 1 du 3 with respect to I 1 is a 1 ω 1 + b 1 ω 3 . Thus, the pullback of Ω 1 with respect to J −1 1 • ψ 1, * • I 1 is a 1 ω 1 + b 1 ω 3 . On the other hand, from the definition of ϕ 1 and the direct calculations, we find that the pullback of Ω 1 with respect to ϕ 1 is (1−αβ)ω 1 +αβ(1−αβ)ω 3 . Since ω 1 and ω 3 are linearly independent, we obtain a 1 = 1−αβ and b 1 = αβ(1−αβ). We can obtain the defining equation of the elliptic curve E 2 and the map ϕ 2 by changing β with −β in the defining equation of the elliptic curve E 1 and the map ϕ 1 (see (3.7), (3.8), (3.9), and (3.10)). On the other hand, when we change β with −β 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 β with −β in the expressions of a 1 and b 1 , respectively. Thus we have a 2 = αβ + 1 and b 2 = −αβ(αβ + 1).
Let Sym 2 (V ) be the symmetric square of V . Then Sym 2 (V ) is a complex manifold of dimension 2. We consider the holomorphic map Let σ(u), u = t (u 1 , u 3 ) ∈ C 2 , be the sigma function associated with the curve V , ℘ j,k = −∂ u k ∂ u j log σ, and ℘ j,k, = ∂ u ℘ j,k , where ∂ u j = ∂ ∂u j . For i = 1, 2, let σ E i (u) be the sigma function associated with the elliptic curve E i ,

which is the Weierstrass elliptic function, and ℘
) for u ∈ Jac(V ) and i = 1, 2. Then the functions f 1 (u) and f 2 (u) are the meromorphic functions on Jac(V ). When we regard f 1 (u) and f 2 (u) as meromorphic functions on C 2 , from Proposition 4.1, we have Theorem 4.3. The hyperelliptic function f 1 is expressed in terms of ℘ 1,1 , ℘ 1,3 , and ℘ 3,3 as follows: where u ∈ C 2 and Here for simplicity we denote ℘ j,k (u) by ℘ j,k .
Proof . Let U be the subset of Sym 2 (V ) consisting of elements (P, Q) such that P, Q ∈ V \{∞}, x 1 , x 2 = αβ, x 1 = x 2 , and x 1 x 2 = α 2 β 2 , where P = (x 1 , y 1 ) and Q = (x 2 , y 2 ). Then U is an open set in Sym 2 (V ). Let U = I(U ). Since the restriction of I to U is injective, U is an open set in Jac(V ). We take a point u ∈ U . Then there exists (P, Q) ∈ U such that u = I ((P, Q)).
Proof . In the same way as the proof of Proposition 4.1, we can obtain the expression (4.14) of f 2 by changing β with −β in the expression of f 1 given in (4.8).

The map from Jac(E i ) to Jac(V )
Since the degree of ϕ i is 2 for i = 1, 2, the preimage ϕ −1 i (S) consists of two elements with multiplicity for any S ∈ E i . We consider the morphisms We consider the maps Then we have Proof . From the definition of ϕ 1 and the direct calculations, we obtain the statement of the lemma.
Lemma 5.3. For i = 1, 2, the map ϕ * i is the holomorphic map between the complex manifolds E i and Jac(V ), and the map ϕ * i is the homomorphism between the groups E i and Jac(V ).
Proof . We consider the following point on E 1 .
From the definition of ϕ * 1 and Lemma 5.1, the map ϕ * 1 is holomorphic on E 1 . Since the map ϕ * 1 is bounded around the points (0, 0), D, χ 1 (D), and ∞, where 0 is the unit element of Jac(V ). Therefore the map ϕ * 1 is the homomorphism between the groups E 1 and Jac(V ). By changing β with −β, we can obtain the statement of the lemma for ϕ * 2 .
By changing β with −β, we can find Therefore we obtain the statement of the lemma.
We consider the maps For i = 1, 2, the map ψ * i is the holomorphic map between the complex manifolds Jac(E i ) and Jac(V ), and the map ψ * i is the homomorphism between the groups Jac(E i ) and Jac(V ). We consider the maps u → t (ψ 1, * (u), ψ 2, * (u)).
Then we have ψ * i • π i = π • ψ * i , i.e., we have the following commutative diagram: Proof . For i = 1, 2, since ψ * i is holomorphic and the group homomorphism, it is well known that there exist complex numbers c i and d i such that ψ * (e.g., [8, Proposition 1.2.1]). We consider the maps where a 1 , b 1 , a 2 , and b 2 are defined in Proposition 4.1. We consider the map Then we have the following commutative diagrams: Therefore we obtain the following commutative diagram: We have From Lemma 5.5, we obtain From (4.1), we obtain (5.1).
Theorem 5.7. For v ∈ C, the following relations hold: Proof . We consider the set E 1 defined in the proof of Lemma 5.3 and U = J 1 (E 1 ). Then U is an open set in Jac(E 1 ). We take a point S = (X, Y ) ∈ E 1 and set v = J 1 (S). We have where ϕ −1 1 (S) = {S 1 , S 2 }. From Proposition 5.6, we have ϕ * 1 (S) = π(k 1 v), where v ∈ C such that π 1 (v) = v. From Lemma 5.1 and S = (0, 0), we have S 1 , S 2 = ∞. Let S i = (x i , y i ) for i = 1, 2. From Lemma 5.1 and S ∈ E 1 , we have x 1 = x 2 , i.e., S 2 = χ(S 1 ). Therefore, from the well-known solution of the Jacobi inversion problem, we have Thus we have We have X = ℘ E 1 (v) and Y = −℘ E 1 (v)/2. Therefore, from Lemma 5.1 and the direct calculations, we find that ( Corollary 5.8. For v ∈ C, the following relations hold: Proof . In the same way as the proof of Proposition 4.1, we can obtain the expressions (5.8)-(5.13) by changing β with −β in the expressions (5.2)-(5.7), respectively.
Remark 5.12. It is well known that the functions ℘ 1,1 , ℘ 1,3 , and ℘ 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 ℘ E 1 and ℘ 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 E 1 and E 2 be the elliptic curves defined by For i = 1, 2, let { a (i) , b (i) } be a canonical basis in the one-dimensional homology group of the curve E i . For i = 1, 2, we define the periods We define the period lattice Λ i = 2 Ω i m 1 + 2 Ω i m 2 | m 1 , m 2 ∈ Z and consider the Jacobian variety Jac E i = C/ Λ i . For i = 1, 2, we take the canonical basis a (i) , b (i) in the onedimensional homology group of the curve E i such that the following relations hold in Jac E i For i = 1, 2, let τ i = Ω i −1 Ω i . We consider the Jacobi elliptic functions sn(u, τ i ), cn(u, τ i ), and dn(u, τ i ) associated with τ 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 τ i . Then we have m 2 i = κ 2 i (cf. [36]). In [6,19,25], the Jacobi elliptic functions with modulus κ i are considered. Since the Jacobi elliptic functions are determined by the second power of the modulus, the Jacobi elliptic functions sn(u, τ i ), cn(u, τ i ), and dn(u, τ i ) are equal to the ones considered in [6,19,25]. For u = t (u 1 , u 3 ) ∈ C 2 , let and Z 1 = sn(w 1 , τ 1 )sn(w 2 , τ 2 ), Z 2 = cn(w 1 , τ 1 )cn(w 2 , τ 2 ), Z 3 = dn(w 1 , τ 1 )dn(w 2 , τ 2 ).
Then we can define the morphism of degree 2 H → W : (s, t) → s 2 , t .
Therefore we obtain the statement of the theorem.