The Half-period Addition Formulae for Genus Two Hyperelliptic $\wp$ Functions and the Sp(4,$\mathbb{R}$) Lie Group Structure

In the previous study, by using the two-flows Kowalevski top, we have demonstrated that the genus two hyperelliptic functions provide the Sp(4,$\mathbb{R}$)/$Z_2$ $\cong$ SO(3,2) Lie algebra structure. In this study, by directly using the differential equations of the genus two hyperelliptic $\wp$ functions instead of using integrable models, we demonstrate that the half-period addition formula for the genus two hyperelliptic functions provides the order two Sp(4,$\mathbb{R}$) Lie group structure.

We expect there is a Lie group structure behind some non-linear differential equation, which is the reason why such non-linear differential equation has a series of infinite solutions. Owing to the addition formula of the Lie group structure, there is a series of infinitely many solutions. As the representation of the addition formula of the Lie group, the algebraic functions such as trigonometric/elliptic/hyperelliptic functions will emerge for solutions of special partial differential equations.
In this study, in order to examine the Lie group structure, instead of the Lie algebra structure, we use algebraic functions such as the elliptic functions and the genus two hyperelliptic functions directly rather than integrable models indirectly. Especially, by using the half-period addition formula, we have deduced that there is the order two Sp(4,R) Lie group structure for the genus two hyperelliptic ℘ functions.
The paper is organized as follows: In section 2, we demonstrate that the elliptic functions have SO(3) Lie group structure via the algebraic addition formula. In section 3, we briefly review the genus two Jacobi's inversion problem to explicitly present the genus two hyperelliptic ℘ function. Then we review the addition formula of the genus two sigma function, which is used in the next section. In section 4, we first review that the half-period addition formula of the ℘ function gives the order two Sp(2,R) Lie group structure. Next, we demonstrate that the half-period addition formulae of the genus two hyperelliptic ℘ functions give the order two Sp(4,R) Lie group structure. We devote the final section to the summary and the discussions.

2
The various addition formulae for the elliptic functions We investigate various types of addition formulae for the elliptic functions, classified into analytic, algebro-geometric, and algebraic ones. For the addition formula which includes derivative terms, we define the analytic addition formula.

2.1
The various addition formulae for the Weierstrass type and Jacobi type elliptic functions We first examine the SO(3)/SO(2,1) Lie group structure for the elliptic functions.
An analytic addition formula of the Weierstrass ℘ function is given by While, an algebro-geometric addition formula is given by 3) which constitutes the Mordell-Weil group in number theory. In addition, there is an algebraic addition formula, which will be discussed in the subsequent subsection.

Algebraic addition formulae for the Weierstrass' ℘ function
In order to obtain an algebraic addition formula for the ℘ function, we use relations between the ℘ function and the Jacobi's elliptic functions in the form [27] 1 Accordingly, we define x sn puq, x cnpuq , and x dnpuq functions as (2.20) They satisfy the relations x sn 2 puq`x cn 2 puq " 1 and k 2 x sn 2 puq`x dn 2 puq " 1.
By using the addition formulae of the Jacobi's elliptic functions, we obtain those of the x sn puq, x cnpuq and x dnpuq functions as follows . : Eqs We can express Eqs.(2.24)-(2.26) in the relation of the Lie group elements of the form

The Rosenhain's solution for the genus two Jacobi's inversion problem
The Weierstrass-Klein type approach to the Jacobi's inversion problem is quite useful to observe the whole structure of the Jacobi's inversion problem. However, it is difficult to obtain explicit expressions of the sigma function for higher genus hyperelliptic ℘ functions.

The Jacobi's inversion problem for the elliptic function
It is instructive to examine the Jacobi's inversion problem for the elliptic function in order to observe the genus two Jacobi's inversion problem. We adopt the elliptic curve and consider the problem of finding the function u " upxq u " Then, Jacobi's inversion problem of obtaining the function x " xpuq is solved by introducing the theta function in such a way as expressing x as a function of the ratio of the theta functions [27], i.e., x " xpuq " ℘puq " e 3`p e 1´e3 q¨ϑ with We must notice that the ℘puq is the quadratic function of the ratio of the theta functions instead of the linear function. Furthermore, the argument of the theta function becomes u{2ω 1 instead of the simple u. By introducing the sigma function as the potential of the ℘ function in the form σpuq " e η 1 u 2 {2ω 1 ϑ where η 1 " ζpω 1 q. The role of the factor e η 1 u 2 {2ω 1 is to shift the constant value of the ℘ function in such a way as ℘puq has no constant term in the Laurent expansion around u « 0 in the form ℘puq " 1{u 2`g 2 {20 u 2`g 3 {28 u 4`¨¨¨, which is equivalent to set λ 2 " 0 in the elliptic curve of the form

The genus two Jacobi's inversion problem
The genus two hyperelliptic ℘ functions were given by Göpel [30,31] and independently by Rosenhain [32,33] via the solution of the Jacobi's inversion problem . However, they are too complicated to derive the addition formula of the sigma function; which is used in the next section. Nowadays, Göpel and Rosenhain's results are little known. Hence, we sketch the Rosenhain's solution for the genus two Jacobi's inversion problem [34], which provides the explicit expressions of ℘ 22 pu 1 , u 2 q and ℘ 12 pu 1 , u 2 q by the theta functions. For the genus two case, we adopt Jacobi's standard form of the hyperelliptic curve in the form y 2 " xp1´xqp1ḱ 2 0 xqp1´k 2 1 xqp1´k 2 2 xq " f 5 pxq. By using three theta function identities, we can consistently parametrize as¨ϑ a k 2 0´k 2 2 , and k 12 " a k 2 1´k 2 2 . Combining any two of these five relations, we obtain ten different expressions for x 1`x2 and´x 1 x 2 . The other ten independent ratios of the theta functions are expressed by the symmetric function of x 1 , x 2 in such the form as with F 01 pxq " xp1´xq. Next, we differentiate Eqs. (3.5) and (3.6) and express the result with the theta functions by using the addition formulae of the theta functions. In the expression of that addition formulae, other ratios of the theta functions than those of Eqs.(3.5)-(3.9), i.e., Eq.(3.10) etc. come out. Hence, the function f 5 pxq naturally emerges in the Jacobi's inversion problem. In order to obtain the standard Jacobi's inversion problem, we can deduce the following equations from Eqs.(3.5) and (3.6) by denoting U 1 " ξ 1 u 1`ξ2 u 2 , U 2 " ξ 3 u 1`ξ4 u 2 , where ξ i , pi " 1, 2, 3, 4q are given by values of the various theta functions and their derivatives at u 1 " u 2 " 0, which take the rather complicated expressions. Then u 1 and u 2 are expressed as y using Eqs.(3.5) and (3.6), we obtain (3.14) Substituting the expressions of u 1 and u 2 in (3.13) into the right-hand side of Eqs.(3.14) and (3.15), we obtain the functional expression of ℘ 22 pU 1 , U 2 q and ℘ 22 pU 1 , U 2 q. The sigma function σpu 1 , u 2 q is guaranteed to exist as the potential of the ℘ function from the integrability conditions. However, it seems difficult to obtain an explicit form of the sigma function which is expressed by the theta functions.
For the practical use of the sigma function, it is useful to define the sigma function in the Taylor expansion form in such a way as the hyperelliptic ℘ functions satisfy the differential equations. Here, we adopt the genus two hyperelliptic curve in the Jacobi's standard form y 2 " λ 5 x 5`λ 4 x 4`λ 3 x 3`λ 2 x 2`λ 1 x`λ 0 with λ 5 " 4 and λ 0 " 0, because we can easily notice a dual symmetry in this case. The differential equations are given by [35] 1q where we set λ 0 " 0. There is the dual symmetry in the form Eq.

The half-period addition formulae
The half-period addition formula for the elliptic/hyperelliptic functions forms the order two group. We first examine the half-period addition formula for the elliptic function, which will be instructive to observe the half-period addition formula for the genus two hyperelliptic functions.

The half-period addition formula for the Weierstrass' ℘ function
For the genus one case, we adopt the Weierstrass elliptic curve of the form The Jacobi's inversion problem is to obtain x " ℘puq from Considering on the Riemann surface, if x reaches one of the branch points e i pi " 1, 2, 3q, u reaches the corresponding half-period ω i pi " 1, 2, 3q, The half-period addition formula of the ℘ function is given by and that of the cyclic permutation of tω 1 , ω 2 , ω 3 u and te 1 , e 2 , e 3 u. We have expressed a " e 1 , b " e 2 1`e 2 e 3 , c " 1, d "´e 1 in (4.3) and observe a matrix defined by as SL(2,R) -Sp(2,R) Lie algebra structure. Furthermore, we obtain M 2 "ˆa 2`b c ab`bd ac`cd bc`d 2˙" p2e 2 1`e2 e 3 q 1, which is equivalent to ℘pu`2ω 1 q " pa 2`b cq℘puq`pab`bdq pac`cdq℘puq`pbc`d 2 q " p2e 2 1`e 2 e 3 q℘puq p2e 2 1`e 2 e 3 q " ℘puq. Hence, the half-period addition formula (4.3) provides the order two SL(2,R) -Sp(2,R) Lie group structure in addition to the SL(2,R) -Sp(2,R) Lie algebra structure, which suggests that genus one Weierstrass' ℘ function has SL(2,R) -Sp(2,R) Lie group structure in the general case. By applying the half-period transformation twice, we obtain the identity transformation. Therefore, the half-period transformation forms the order two Lie group transformation. Thus, we first demonstrate the relation between the Lie algebra element and the order two Lie group element for the general Sp(2g,R) (g=1, 2,¨¨¨) Lie group. By using the almost complex structure J, which is skew symmetric real matrix with J 2 "´1, the Lie algebra element A and the order two Lie group element G satisfy JA`A T J " 0, G T JG " J, G 2 " 1. (4.5) By using G 2 " 1, we obtain G T J " JG from G T JG " J.
For the projective representation of any matrix M , (const.)ˆM is equivalent to M . Thus, in the right-hand side of G T J " JG, JG is equivalent to´JG, hence we obtain the Lie algebra relation G T J "´JG, which implies that the Lie algebra element A becomes also the order two Lie group element G. For the Sp(2,R) case, we adopt J "ˆ0 1 1 0˙a nd the Lie group transformation is given byˆx The projective representation is given by x 1 {y 1 " ax{y`b cx{y`d . For the constant multiplied group element λG "ˆλ a λb λc λd˙, the projective representation of the transformation is given by x 1 {y 1 " λax{y`λb λcx{y`λd " ax{y`b cx{y`d , i.e., λG is equivalent to G for the projective representation. The above M satisfies M T J`JM T " 0, M 2 " pconst.q1. This implies that M is not only the Sp(2,R) Lie algebra element but also the order two Sp(2,R) Lie group element.

The half-period addition formula for the genus two hyperelliptic ℘ functions
We adopt the genus two hyperelliptic curve in the form which gives λ 4 "´4pe 1`e2`e3`e4`e5 q,¨¨¨, λ 0 "´4e 1 e 2 e 3 e 4 e 5 . In the Riemann surface, there are six branching points e 1 , e 2 , e 3 , e 4 , e 5 and e 6 " 8. The cuts are drawn from e 2i´1 to e 2i pi " 1, 2, 3q. The Jacobi's inversion problem is given by and the genus two hyperelliptic ℘ functions are given by By setting x 1 " e i , x 2 " e j , the half-period is given by Ω " pω 1 , ω 2 q for u " pu 1 , u 2 q in the form which provides ℘ 22 pΩq " e i`ej , ℘ 22 pΩq "´e i e j , ℘ 11 pΩq " F pe i , e j q{4pe i´ej q 2 , where we use y 1 pΩq " 0 and y 2 pΩq " 0 because x 1 pΩq " e i and x 2 pΩq " e j .
In order to obtain the half-period addition formula for the hyperelliptic ℘ functions, we use the addition formula of the sigma function Eq. (3.22)   There are two types of the half-periods. Type I is given by setting x 1 pΩq " e i , x 2 pΩq " e j , pi ‰ j, 1 ď i, j ď 5q. Type II is given by setting x 1 pΩq " e i , x 2 pΩq " e 6 " 8, p1 ď i ď 5q. For the type I half-period addition formula, we consider the following example of e i " e 1 , e j " e 2 , and use the expression of Buchstaber et al.'s paper. In this case, we have the expression ℘ 22 pΩ I q " e 1`e2 , ℘ 22 pΩ I q "´e 1 e 2 , ℘ 11 pΩ I q " F pe 1 , e 2 q{4pe 1´e2 q 2 " e 1 e 2 pe 3è 4`e5 q`e 3 e 4 e 5 , which provides (4.17) One of the examples is given by a 1 " S 3´S1 s 2 , a 2 "´s 2´S1 s 1`S2 , a 3 "´s 1 , a 4 "´s 2 2`S2 s 2`S1 s 2 s 1 , b 1 "´S 3 s 1`S2 s 2 , b 2 "´a 1 , b 3 " s 2 , b 4 "´2S 3 s 2`S3 s 2 1´S 2 s 2 s 1 , c 1 " b 4 , c 2 "´a 4 , c 3 "´S 3´S1 s 2 , c 4 "´S 2 with s 1 " e 1`e2 , s 2 " e 1 e 2 , S 1 " e 3`e4`e5 , S 2 " e 3 e 4`e4 e 5`e5 e 3 , S 3 " e 3 e 4 e 5 . All type I half-periods are given by arranging te 1 , e 2 , e 3 , e 4 , e 5 u into two sets te i , e j u Y te p , e q , e r u. We can verify G 2 I " (const.)1 for all type I half-periods. For the type II half-period addition formula, we consider the following example of e i " e 1 , e j " e 6 " 8 and we use the expression of Buchstaber et al.'s paper. We take the most singular term and the limit e 6 Ñ 8 at the end. Thus, we have the expression ℘ 22 pΩ II q " e 6 , ℘ 22 pΩ II q "´e 1 e 6 , ℘ 11 pΩ II q " e 2 1 e 6 , which provides One of the examples is given bŷ with T 1 " e 2`e3`e4`e5 , T 2 " e 2 e 3`e2 e 4`e2 e 5`e3 e 4`e3 e 5`e4 e 5 , T 3 " e 2 e 3 e 4`e2 e 3 e 5è 2 e 4 e 5`e3 e 4 e 5 , T 4 " e 2 e 3 e 4 e 5 . All type II half-periods are given by arranging te 1 , e 2 , e 3 , e 4 , e 5 u into two sets te i u Y te k , e p , e q , e r u. We can verify G 2 II " (const.)1 for all type II half-periods. For Sp(4,Rq case, we adopt the representation of almost complex structure J with J 2 "´1 in the form 2 J "¨0´1 The bases of the Sp(4,R) Lie algebra, which satisfies JA`A T J " 0, is given by We have verified that G I and G II satisfies JG I`G T I J " 0, and JG II´G T II J " 0. We observe that Eq.(4.17) is the element of the ordinary Sp(4,R) Lie algebra; yet Eq.(4.18) is not that of the ordinary Sp(4,R) Lie algebra. However, the transformation of the half-period addition formula for the hyperelliptic ℘ functions provides the projective representation. Hence, in the projective representation, G I and G II are not only the elements of the Sp(4,R) Lie algebra but also the elements of the order two Sp(4,R) Lie group. This suggests that the genus two hyperelliptic functions have the general continuous Sp(4,R) Lie group structure.

Summary and Discussions
First, we have examined various types of addition formulae for the elliptic functions. The algebraic addition formula can be rearranged into the relation of the Lie group elements, which is called the Yang-Baxter's integrable condition. Second, we have reviewed the Rosehnain's approach to the genus two Jacobi's inversion problem. It is difficult to express the explicit form of the sigma function for the genus two case, thus we use the Taylor expansion form for the sigma function. We pointed out that addition formula of the sigma function depends on what kind of sigma function we adopt. Finally, we have obtained the order two addition formula of the genus two hyperelliptic ℘ functions by using the addition formula of a sigma function.
In the previous study, via the two flows Kowalevski top, we had demonstrated that the genus two hyperelliptic functions provide the Sp(4,R)/Z -SO(3,2) Lie algebra structure. In this study, by directly using the differential equations of the genus two hyperelliptic ℘ functions, we have demonstrated that the half-period addition formula for the genus two hyperelliptic ℘ functions provides the order two Sp(4,R) Lie group structure. This suggests that the genus two hyperelliptic ℘ functions have the general continuous Sp(4,R) Lie group structure. (A.10)