Algebraic construction of the sigma function for general Weierstrass curves

The Weierstrass curve $X$ is a smooth algebraic curve determined by the Weierstrass canonical form, $y^r + A_{1}(x) y^{r-1} + A_{2}(x) y^{r-2} +\cdots + A_{r-1}(x) y + A_{r}(x)=0$, where $r$ is a positive integer, and each $A_j$ is a polynomial in $x$ with a certain degree. It is known that every compact Riemann surface has a Weierstrass curve $X$ which is birational to the surface. The form provides the projection $\varpi_r : X \to {\mathbb{P}}$ as a covering space. Let $R_X := {\mathbb{H}}^0(X, {\mathcal{O}}_X(*\infty))$ and $R_{\mathbb{P}} := {\mathbb{H}}^0({\mathbb{P}}, {\mathcal{O}}_{\mathbb{P}}(*\infty))$. Recently we have the explicit description of the complementary module $R_X^{\mathfrak{c}}$ of $R_{\mathbb{P}}$-module $R_X$, which leads the explicit expressions of the holomorphic one form except $\infty$, ${\mathbb{H}}^0({\mathbb{P}}, {\mathcal{A}}_{\mathbb{P}}(*\infty))$ and the trace operator $p_X$ such that $p_X(P, Q)=\delta_{P,Q}$ for $\varpi_r(P)=\varpi_r(Q)$ for $P, Q \in X\setminus\{\infty\}$. In terms of them, we express the fundamental 2-form of the second kind $\Omega$ and a connection to the sigma functions for $X$.


Introduction
In Weierstrass' elliptic function theory, the algebraic properties associated with an elliptic curve of Weierstrass' standard equation y 2 = 4x 3 − g 2 x − g 3 are connected with the transcendental properties defined on its Jacobi variety via the σ function since ℘(u) = − d 2 du 2 log σ(u), d℘ (u) du is identical to a point (x, y) in the curve [1,2]. These algebraic and transcendental properties are equivalently obtained by the identity, and play the central role in the elliptic function theory. Via the equivalence, the elliptic function theory affects several fields in mathematics, science, and technology. In other words, in Weierstrass' elliptic function theory, the equivalence between the algebraic objects of the curve and the transcendental objects on its Jacobi variety is crucial.
Weierstrass himself extended the picture to general algebraic curves [3], but it failed due to difficulties. Some of the purposes in mathematics in the XX-th century were to overcome the difficulties which were achieved. We have studied the generalization of this picture to algebraic curves with higher genera in the series of the studies [4,5,6] following Mumford's studies for the hyperelliptic curves based on the modern algebraic geometry [7,8], i.e., the unification of the theory of algebraic curves in the XIX-th century with the modern one.
The elliptic theta function was generalized by Riemann for an Abelian variety. In contrast, its equivalent function Al was defined for any hyperelliptic curve by Weierstrass, which was refined by Klein using only the data of the hyperelliptic Riemann surface and Jacobi variety as a generalization of the elliptic sigma function [9]. Baker re-constructed Klein's sigma functions by using the data of hyperelliptic curves algebraically [10]. Buchstaber, Enolskii, and Leykin extend the sigma functions to certain plane curves, so-called (n, s) curves, based on Baker's construction which we call the EEL construction due to work by Eilbeck, Enolskii, and Leykin [11] [ [12] and its references.]. For the (n, s) curves with the cyclic symmetry, the direct relations between the affine rings and the sigma functions were obtained as the Jacobi inversion formulae [13,14]. Additionally, we generalized the sigma functions and the formulae to a particular class of the space curves using the EEL-construction [15,4,6].
We have studied further generalization of the picture in terms of the Weierstrass canonical form [16,17]. The Weierstrass curve X is a normalized curve of the curve given by the Weierstrass canonical form, y r + A 1 (x)y r−1 + A 2 (x)y r−2 + · · · + A r−1 (x)y + A r (x) = 0, where r is a positive integer and each A j is a polynomial in x of a certain degree (c.f. Proposition 2.7) so that the Weierstrass non-gap sequence at ∞ ∈ X is given by the numerical semigroup H X whose generator contains r as its minimal element. It is known that every compact Riemann surface has a Weierstrass curve X, which is birational to the surface. We also simply call the Weierstrass curve W-curve.
It provides the projection ̟ r : X → P as a covering space. Let R X := H 0 (X, O X ( * ∞)) and R P := H 0 (P, O P ( * ∞)). In [17], we have the explicit description of the complementary module R c X of R P -module R X , which leads the explicit expressions of the holomorphic one form except ∞, H 0 (P, A P ( * ∞)).
Recently D. Korotkin and V. Shramchenko [18] and Nakayashiki [19] defined the sigma function of every compact Riemann surface as a generalization of Klein's sigma function transcendentally. Every compact Riemann surface can be characterized by the Weierstrass nongap sequence, which is described by a numerical semigroup H called Weierstrass semigroup. Nakayashiki defined the sigma function for every compact Riemann surface with Weierstrass semigroup H [19] based on Sato's theory on the universal Grassmannian manifolds (UGM) [20,21].
In this paper, we use our recent results on the complementary module of the W-curve [17] to define the trace operator p X such that p X (P, Q) = δ P,Q for ̟ r (P ) = ̟ r (Q) for P, Q ∈ X \ {∞}. In terms of them, we express the fundamental 2-form of the second kind Ω algebraically in Theorem 3.34, and finally obtain a connection to Nakayashiki's sigma function by modifying his definition in Theorem 4.9. It means an algebraic construction of the sigma function for every W-curve as Baker did for Klein's sigma function following Weierstrass' elliptic function theory.
Contents are as follows: Section 2 reviews the Weierstrass curves (W-curves) based on [17]; 1) the numerical semigroup in Subsection 2.1, 2) Weierstrass canonical form in Subsection 2.2, 3) their relations to the monomial curves in Subsection 2.3, 4) the properties of the R P -module R X in Subsection 2.4, 5) the covering structures in W-curves in Subsection 2.5, and 6) especially the complementary module R c X of R X in Subsection 2.6; the explicit description of R c X is the first main result in [17]. Section 3 provides the first and the second theorems in this paper on the W-normalized Abelian differentials on X. In Subsection 3.1, we review the second main result in [17] on the W-normalized Abelian differentials H 0 (X, A X ( * ∞)), which contains the Abelian differentials of the first kind. Further we extend the trace operator p ∈ R X ⊗ R P R X introduced in [17] to R X ⊗ C R X in Subsection 3.2. The trace operator p enables us to define a proper one-form Σ and its differential dΣ in Subsection 3.3. After investigating dΣ, we find the W-normalized (1) N(n) − n ≤ g for every n ∈ N 0 , (2) N(n) − n = g for N(n) ≥ c X = N(g) or n ≥ g, (3) N(n) − n < g for 0 ≤ N(n) < c X or n < g, Proof. 1-3 and 5 are obvious. Noting #H c = g, 4 means that what is missing must be filled later for H c . 6 is left to [22].
The length r Λ of the diagonal of the Young diagram Λ is called the rank of Λ. The number of boxes below and to the right of the i-th box of the diagonal from lower right to upper left are assumed a r Λ −i+1 and b r Λ −i+1 respectively. The Young diagram is represented by (a r Λ , . . . , a 2 , a 1 ; b r Λ , . . . , b 2 , b 1 ), which is known as Frobenius representation or characteristics of Λ. Then ℓ i := a i + b i + 1 is called the hook length of the characteristics. For Λ = (6, 3, 3, 2, 1, 1, 1, 1) associated with H = 5, 7, 11 , it is (0, 2, 7; 0, 1, 5) and rank of Λ is three. For Λ = (6, 3, 3, 3, 1, 1, 1, 1) associated with H = 5, 6, 14 , it is (1, 2, 7; 0, 1, 5) and rank of Λ is three. We show their examples of the Young diagram: 5, 7, 11 5, 6, 14 . (2.2) We show the properties of the Young diagram associated with the numerical semigroup H in the following lemma, which is geometrically obvious: (1) If we put the number on the boundary of the Young diagram Λ from the lower to upper right as in (2.3), each number in the right side box of the i-th row corresponds to the gap number in N c (g − i) ∈ H c or Λ i + g − i, i.e., N c (g − i) ∈ H c = Λ i + g − i, for 0 < i ≤ g, and each number in the top numbered box of the i-th column corresponds to N(i) for 0 < i < Λ 1 .
Further by letting |Λ| := In this paper, we mainly consider the r-numerical semigroup, H. We introduce the tools as follows: We have the following elementary but essential results [17, Lemma 2.8, 2.9]: The following is obvious: For the generators r and s in the numerical semigroup H, there are positive integers i s and i r such that i s s − i r r = 1.
Proposition 2.7. [25,24] For a pointed curve (X, ∞) with Weierstrass semigroup H X := H(X, ∞) for which r min (H X ) = r, and e i ∈ E H X , (i ∈ Z × r ) in Definition 2.4, and we let s := min i∈Z × r {e i ∈ E H X | (e i , r) = 1} and s = e ℓs . (X, ∞) is defined by an irreducible equation, for a polynomial f X ∈ C[x, y] of type, where the A i (x)'s are polynomials in x, A 0 = 1, A i = ⌊is/r⌋ j=0 λ i,j x j , and λ i,j ∈ C, λ r,s = −1.
In this paper we call the curve in Proposition 2.7 a Weierstrass curve or a W-curve. The Weierstrass canonical form characterizes the W-curve, which has only one infinity point ∞. [26]. Since every compact Riemann surface of the genus, g(> 1), has a Weierstrass point whose Weierstrass gap sequence with genus g [27], it characterizes the behavior of the meromorphic functions at the point, and thus there is a Weierstrass curve which is bi-rationally equivalent to the compact Riemann surface.
Further Proposition 2.7 is also applicable to a pointed compact Riemann surface (Y, P ) of genus g whose point P is non-Weierstrass point rather than the Weierstrass point; its Weierstrass gap sequence at P is H c (Y, P ) = {1, 2, . . . , g}. Even for the case, we find the Weierstrass canonical form f X and the W-curve X with H c X = {1, 2, . . . , g} which is bi-rational to Y .
y]/(f X (x, y)) for (2.4) and its normalized ring be R • X if X • := Spec R • X • is singular. R • X is the coordinate ring of the affine part of X \ {∞} and we identify R • X with R X = H 0 (X, O X ( * ∞)). Then the quotient field C(X) := Q(R X ) of R X is considered as an algebraic function field on X over C.
By introducing R P := H 0 (P, O P ( * ∞)) = C[x] and its quotient field C(x) := Q(R P ), Q(R X ) is considered a finite extension of Q(R P ). We regard R X as a finite extended ring of R P of rank r, e.g., R • X • = R P [y]/(f X (x, y)) as mentioned in Subsection 2.5 [28]. For the local ring R X,P of R X at P ∈ X, we have the ring homomorphism, ϕ P : R X → R X,P . We note that R X,∞ plays crucial roles in the Weierstrass canonical form. We let the minimal generator M X = {r 1 , r 2 , . . . , r m X } of the numerical semigroup H X = H(X, ∞). The Weierstrass curve admits a local cyclic C r = Z/rZ-action at ∞ c.f., Subsection 2.3. The genus of X is denoted by g X , briefly g and the conductor of H X is denoted by c X := c H X ; the Frobenius number c X − 1 is the maximal gap in H(X, ∞). We let, H c X := Z \ H X .
2.3. The monomial curves and W-curves. This subsection shows the monomial curves and their relation to W-curves based on [5,6,16]. For a given W-curve X with the Weierstrass semigroup H = H X , and its generator M X = {r = r 1 , r 2 , . . . , r m X }, the behavior of singularities of the elements in R X at ∞ is described by a monomial curve X Z H . For the numerical semigroup H = M X , the numerical semigroup ring R H is defined as R H := C[z r 1 , z r 2 , · · · , z rm X ].
Following a result of Herzog's [29], we recall the well-known proposition for a polynomial ring C[Z] := C[Z r 1 , Z r 2 , · · · , Z rm X ].
and a positive integer Sending Z r to 1/x and Z r i to 1/y r i , the monomial ring R Z H determines the structure of gap sequence of X at ∞ [29,30]. Bresinsky showed that k X can be any finitely large number if m X > 3 [31].
Let X H := Spec R Z H ,which we call a monomial curve. We also define the ring isomorphism A monomial curve is an irreducible affine curve with G m -action, where G m is the multiplicative group of the complex numbers; Z a → g a Z a for g ∈ G m , and it induces the action on the monomial ring R Z H . The following cyclic action of order r plays a crucial role in this paper.
Lemma 2.10. The cyclic group C r of order r acts on the monomial ring R Z H ; the action of the generator ζ r ∈ C r on Z a is defined by sending Z a to ζ a r Z a , where ζ r is a primitive r-th root of unity. By letting r * i := (r, r i ), r i := r/r i * , and r i := r i /r * i , the orbit of Z r i forms C r i ; especially for the case that (r, r i ) = 1, it recovers C r .
Thus in R Z H , . For a ring R, let its quotient field be denoted by Q(R). Further we obviously have the Corresponding to the standard basis of H X in Definition 2.4, we find the monic monomial Z e i ∈ C[Z] such that ϕ Z H (Z e i ) = z e i , and the standard basis {Z e i |i ∈ Z r }; Z e 0 = 1.
is given by we have the identity To construct our curve X from R H or Spec R H , we could follow Pinkham's strategy [30] with an irreducible curve singularity with G m action, though we will not mention it in this paper. Pinkham's investigations provide the following proposition [30][17, Proposition 3.7]: Proposition 2.13. For a given W-curve X and its associated monomial ring, Definition 2.14.
(1) Recalling Lemma 2.6, we define arithmetic local parameter at ∞ by The degree at Q(R X,∞ ) as the order of the singularity with respect to t is naturally defined by which is called Sato-Weierstrass weight [23]. so that it belongs to the ideal, The weight of y r i is given by Lemma 2.15. We have the decomposition of R X as a C-vector space, Further by assigning a certain weight on each coefficient λ i,j in (2.4) so that (2.4) is a homogeneous equation of weight rs, we also define another weight, (2.14) wt λ : R X → Z.
. .} by the basis of R X as in (2.13).
Lemma 2.17. Let t be the arithmetic local parameter at ∞ of R X .
(1) By the isomorphism (2) There is a surjection of ring ϕ ∞ : , the existence of g is obvious. 2.4. R P -module R X . R X is an R P -module, and its affine part is given by the quotient ring of We recall Definition 2.4 and Lemma 2.11, and apply them to W-curves and then we obtain the following [17, Proposition 3.11].
Proposition 2.18. For e i ∈ E H X , we let y e i be the monic monomial in R X whose weight is −e i , (y e 0 = 1) satisfying R P y e i = y e 0 , y e 1 , . . . , y e r−1 R P with the relations, where a ijk ∈ R P , a ijk = a ijk , especially a 0jk = a j0k = δ jk .

2.5.
The covering structures in W-curves. We will follow [28,33] to investigate the covering structure in W-curves.
2.5.1. Galois covering. As mentioned in Remark 2.8, let us consider the Riemann sphere P and R P = H 0 (P, O P ( * ∞)). We identify R P with its affine part R • P = C[x] and its quotient field is denoted by C(x) = Q(R P ). The quotient field Q(R X ) = C(X) of R X is an extension of the field C(x).
Following the above description, we consider the W-curve X. The covering ̟ r : X → P ((x, y • ) → x) is obviously a holomorphic r-sheeted covering. When we have the Galois group on X, i.e., Gal(Q(R X )/Q(R P )) = Aut(X/P) = Aut(̟ r ), it is denoted by G X . The ̟ r is a finite branched covering. A ramification point of ̟ r is defined as a point of such that is not biholomorphic at the point. The image ̟ r of the ramification point is called the branch point of ̟ r . The number of the finite ramification points is denoted by ℓ B .
We basically focus on the holomorphic r-sheeted covering ̟ x = ̟ r : X → P. G x denotes the finite group action on ̟ −1 r (x) for x ∈ P, refered to as group action at x in this paper:

2.5.2.
Riemann-Hurwitz theorem. Let us consider the behaviors of the covering ̟ r : X → P, including the ramification points. The Riemann-Hurwitz theorem [28], where e B i is the ramification index at B i , shows the following: Corollary 2.20. The divisor of dx is given by, There is a projection ̟ r,r 2 : X → X • . Thus we can find the subring R X X (x, y r i ) of R X as its normalized ring, as we prove this fact in [17,Proposition 3.14] precisely. The image of f Further we also define the tensor product of these rings Moreover we also define the direct product of these rings R ∂y , as an R X -analog of (2.9), let us consider The following is obvious: However, some parts in its numerator and denominator are canceled because they belong to We have the expression of h R X (x, y • , y ′ • ) following [17,Proposition 4.16,Lemma 4.20].
+ lower weight terms with respect to −wt as an R P -module. Here y ′ e 0 = 1, and each Υ i has the following properties (2) Υ i =ẙ e i + lower weight terms with respect to −wt, whereẙ e i = δ i (x)y e * ℓ,i with an element ℓ ∈ Z r , and a monic polynomial such that y e 0 =ẙ e i y e i + lower weight terms with respect to −wt We introduce more convenient quantities y e i (i = 0, 1, . . . , r − 1): , y e i =ẙ e i + certain terms, and the number of the terms is minimal satisfying the relations as R P -modules, Υ 1 , . . . , Υ r−1 R X = y e 1 , . . . , y e r−1 R X = y e 0 , . . . , y e r−1 R P , τ h ( y e i ) = 1 for i = 0, 0 otherwise.
Since some of f (j) X,y (P ) = 0 at P = B i ∈ B X \ {∞}, h X (x, y • ) ∈ R X vanishes only at the ramification point B i ∈ X and the construction of h X , we have the following lemma from Dedekind's different theorem Proposition 4.27 in [17].
Definition 2.27. The effective divisor, K X > 0 and let k X := Lemma 2.28. The divisor of dx h X is expressed by (2g−2+k X )∞−K X , and 2g−2+k X = d h −r−1 From Corollary 2.20, we note that these K X and k X play crucial roles in the investigation of the differentials on X.
Proposition 2.29. k X is equal to zero if d h is symmetric whereas k X is not zero otherwise.
We recall e i in Definition 2.4 in the standard basis in Lemma 2.11 and Proposition 2.18, and y e i in Definition 2.25.
From Proposition 4.32 in [17] we have the properties of 3. W-normalized Abelian differentials on X 3.1. W-normalized Abelian differentials H 0 (X, A X ( * ∞)). Following K. Weierstrass [3], H. F. Baker [10], V.M. Buchstaber, D.V. Leykin and V.Z. Enolskii [12], J.C. Eilbeck, V.Z. Enolskii and D.V. Leykin [11] and our previous results [4,6,16], we construct the Abelian differentials of the first kind and the second kind H 0 (X, A X ( * ∞)) on X for more general W-curves based on Proposition 2.30 [17]. We consider the Abelian differentials of the first kind on a W-curve. Due to the Riemann-Roch theorem, there is the i-th holomorphic one-form whose behavior at ∞ is given by where N c (i) ∈ H c X (i = 1, 2, . . . , g) satisfying N c (i) < N c (i + 1), and t is the arithmetic local parameter at ∞. We call this normalization the W-normalization. Similarly we find the differentials or the basis of H 0 (X, A X ( * ∞)) associated with H c X . The W-normalized holomorphic one-forms are directly obtained from Proposition 2.30: Lemma 3.1. For x k y e i in Proposition 2.30, we have the relation, By re-ordering x k y e j with respect to the weight at ∞, we define the ordered set { φ i }: (1) Let us define the ordered subset S X of R X by such that φ i is ordered by the Sato-Weierstrass weight, i.e., −wt φ i < −wt φ j for i < j, and S X is equal to {x k y e i | i ∈ Z r , k ∈ N 0 } as a set. (2) Let R X be an R X -module generated by S X , i.e., R X := S X R X ⊂ R X .
(3) Recalling K X and k X in Definition 2.27, we let N (n) := −wt ( φ n ) − k X , H X := {−wt ( φ n ) |n ∈ N 0 }, and we define the dual conductor c X as the minimal integer satisfying c X + N 0 ⊂ H X − k X . (4) We define S (g) X := { φ 0 , φ 1 , . . . , φ g−1 }, and the W-normalized holomorphic one form, or W-normalized Abelian differentials of the first kind ν I i as the canonical basis of X, We note that at ∞, ν I i behaves like ν I i = (t N c (g−i−1)−1 (1 + d >0 (t)))dt for the arithmetic local parameter t at ∞, and further they are W-normalized Abelian differentials. We summarize them: By the Abel-Jacobi theorem [26], K X in Definition 2.27 can be divided into two pieces, which are related to the spin structure in X.
Definition 3.4. Let K s and K c X be the effective divisors which satisfy K X − k X ∞ ∼ 2K s − 2k s ∞, K X + K c X − (k X + k c X )∞ ∼ 0 as the linear equivalence, where k s and k c X are the degree of K s and K c X respectively.

3.2.
Extension p ∈ R X ⊗ R P R X to R X ⊗ C R X . As we have h R X in Lemma 2.22, we extend it in R X ⊗ R P R X to an element in R X ⊗ C R X , though the extension is not unique; there are two different h(x, y • , x ′ , y ′ • ) and h ′ (x, Since they are quasi-isomorphic, we select one of them, and thus, it is welldefined in the meaning of Proposition 3.18 4. and 3.20 4. Definition 3.8. Using Υ i in Proposition 2.24, for a point (P = (x, y • ), P ′ = (x ′ , y ′ • )) ∈ X × X, we define a polynomial h X (x, y • , y ′ • ) ∈ R X ⊗ C R X by h X (x, y • , y ′ • ) := Υ 0 · 1 + Υ 1 y ′ e 1 + · · · + Υ r−1 y ′ e r−1 . Then the following lemma is evident from Lemma 2.22.
Definition 3.10. Using h X (x, y • , y ′ • ) for a point (P = (x, y • ), P ′ = (x ′ , y ′ • )) ∈ X ×X, we define It is obviously that p ̟ (P, Q) belongs to h X,P R X,P as a function of P at P ∈ X and thus Proposition 2.30 whose origin is the Dedekind's different theorem [17,Proposition 4.27] shows the proposition: Further the direct computations provide the following proposition: Remark 3.13. We should remark that p(P, Q) ∈ R X ⊗ C R X which holds the relations in Proposition 3.12 is not unique. The problem comes from the fact that there are infinitely many different p ′ (P, Q) from p(P, Q) such that p(P, Q) = p ′ (P, Q) for ̟ r (P ) = ̟ r (Q).
Proposition 3.14. For a point Q ∈ X \ B X , div P (p ̟ (P, Q)) = ζ∈G X,̟x(Q) ,ζ =e Further, this relation is extended to the condition Q ∈ B X \ {∞} by considering the multiplicity of the action G ̟r(Q) .
Proof. At the ramification point B i of ̟ x : X → P, Proposition 3.11 shows the third term. We note that p ̟ (∞, Q) = 1/r. From the Riemann-Hurwitz theorem (2.16), there exist the first and the second terms.
For an element f ⊗ f ′ in R X ⊗ C R X , we define the weight wt(f ⊗ f ′ ) by wt(f ) + wt(f ′ ).
and an element, This h H (Z, Z ′ ) is uniquely defined in the meaning of Proposition 3.18 4. and 3.20 4.
(2) Σ(P, Q) is holomorphic over X except Q and ∞ as a function of P such that (a) at Q, in terms of the local parameters t Q (P ) = 0 and t P (P ) = 0, it behaves like (b) at ∞, the local parameter t P (t P (∞) = 0), it behaves like (3) Σ(P, Q) as a function of Q is singular at P and ∞ such that (a) at P , in terms of the local parameters t Q (P ) = 0 and t P (P ) = 0, it behaves like (4) Let h ′ X (x P , y •P , y •Q ) be an element in R X ⊗ C R X satisfying the conditions in Lemma 3.9, i.e., h ′ X (x P , y •P , y •Q ) = h X (x P , y •P , y •Q ) at ̟ x (P ) = ̟ x (Q), and let .
Then Σ ′ P, Q − Σ P, Q belongs to the set, . Proof. Lemma 3.9 1 shows 1. Noting the properties in Proposition 3.14 and Corollary 2.20, the numerator of Σ is zero with the first order at the points which P = Q and P ∈ ̟ −1 x (̟ x (Q)) and thus, Σ behaves like finite one-form there. At P = Q, the numerator is equal to one and thus we have Σ = dx P /(x P − x Q ), which means Σ = dt P /t P (1 + d >0 (t P )) in 2.a and 3.a.
Recalling ϕ ∞ : R X → R H in Lemma 2.17, let t P and t Q be the local parameters at ∞ corresponding to P and Q respectively. Proposition 3.12 shows lim ). The maximum of {e i − r} i∈Zr is e r−1 − r, which is equal to c X − 1 due to Lemma 3.7 4, and thus ϕ ∞ dx y e r−1 = t . We obtain 3.b.
Let us consider 4. Since both Σ ′ P, Q and Σ P, Q satisfy these properties 1-3, their difference is holomorphic over X with respect to P , and over X \ {∞} with respect to Q. It shows 4.

3.3.2.
The one-form Σ at ∞. Noting Lemma 2.17, we consider a derivation in Q(R Z H ⊗ R Z H ) for the monomial curve X H × X H instead of R X,∞ using surjection ϕ ∞ .
In order to investigate the behavior of Σ at ∞, we consider the differential in monomial curve and Lemmas 2.17, 2.12, and 3.15, we define as an element in Q(R H ⊗ C R H ) using isomorphism ϕ Z H and the parameters t P = 1/z P and t Q = 1/z Q ; Σ H is regard as a derivation in Q(R X,∞ ⊗ C R X,∞ ).
Then the direct computations lead the following results: (1) ϕ ∞ (Σ(P, Q)) = Σ H t P , t Q with the local parameters t P and t Q at ∞. and it shows the behavior at ∞, i.e., for the local parameters at ∞, t P (∞) = t Q (∞) = 0.

3.3.3.
The differential dΣ on X. In order to define Ω, we consider the derivative dΣ in this subsubsection.
(3.6) d Q Σ P, Q := dx Q ∂ ∂x Q Σ P, Q has the following properties: (1) d Q Σ P, Q as a function of P is holomorphic over X except Q such that (a) at Q, in terms of the local parameter t P (Q) = 0, it behaves like (b) at ∞, in terms of the arithmetic parameters t P and t Q , it behaves like and d Q Σ P, Q is holomorphic at ∞. (2) d Q Σ P, Q as a function of Q is holomorphic over X except P and ∞ such that (a) at P , in terms of the local parameter t Q (P ) = 0, it behaves like (b) at ∞, using the arithmetic parameter t Q (∞) = 0, it behaves like  We use the facts: It means that there exist F dΣ (P, Q) ∈ R X ⊗ C R X such that

3.3.4.
The differential dΣ H on X H . Noting Lemma 2.17, we also consider dΣ as a two-form in Q(R Z H ⊗ C R Z H ) and the monomial curve X H × X H instead of R X,∞ . We consider the differential of (3.4), We recall the correspondence between z of R H and t of R X,∞ in Lemma 2.17. Using (3.5), we have Lemma 3.21.
(1) We have the equality: (2) For the case |t Q | < |t P |, we have the expansion, (3) For the case |t P | < |t Q |, we have the expansion, Proof. 1: Using the relation, The relation rt r P + e i (t r Q − t r P ) = e i t r Q − (e i − r)t r P enables us to obtain 1.
When t P = t Q + ε, it vanishes for the limit ε → 0.
Proof. The direct computations show them.
3.3.5. W-normalized differentials of the second kind. We introduce the W-normalized differentials of the second kind using this d Q Σ P, Q .
Definition 3.27. We consider a sufficiently small closed contour C ε at ∞. Let D ε be the inner side of C ε = ∂D ε including ∞ and ε ′ be a point in D ε such that ε ′ = ∞. For differentials ν and ν ′ in H 0 (X, A( * ∞)), we define a pairing: The following is obtained from the primitive investigation of complex analysis on a compact Riemann surface [26].
Definition 3.29. We define the pre-normalized differentials ν II i ∈ H 0 (X, A X ( * ∞)) of the second kind (i = 1, 2, . . . , g), which satisfies the relations (if they exist) It is obvious that from Lemma 3.3 2 and Lemma 3.26, ν II j with j ′ in Lemma 3.26 is expressed Noting Proposition 3.20 3.b, we have the following relations: dx, wt ν II g = −2, and wt ν II 1 = −c X .
Proof. −wt( φ g ) + wt( φ g−1 ) = 2 and thus On the other hand, −wt( φ 0 ) = d h − r + 1 − c X in Lemma 3.7 means Hence wt ν II 1 = −c X and we obtain Theorem 3.31. There exist the pre-normalized differentials ν II j (j = 1, 2, · · · , g) of the second kind such that they have a simple pole at ∞ and satisfy the relation, where the set of differentials {ν II 1 , ν II 2 , ν II 3 , · · · , ν II g } is determined modulo the linear space spanned by ν I j j=1,...,g and dR X . We call these ν II i 's W-normalized differentials of the second kind.
The embedding j is realized by j : . Due to the Riemann-Roch theorem, for every ℓ in H c X , there is an element k in (−N)\(H dR X +1) such that ℓ + k = 0. It shows the relations.
is holomorphic over X \ {P 2 } as a function of P 1 and is holomorphic over X \ {P 1 } as a function of P 2 .
Proof. From Proposition 3.20, d P 2 Σ(P 1 , P 2 ) is holomorphic over X \ {P 2 } as a function of P 1 whereas d P 2 Σ(P 1 , P 2 ) is holomorphic over X \ ({P 2 } ∪ {∞}) as a function of P 2 . The order of the singularity at Q = ∞ is 2g and thus, which can be canceled by ν II i (P 2 ). Since the numerator of d P 2 Σ(P 1 , P 2 ) in (3.7) consists of the elements in R X ⊗ C R X dx 1 ⊗ dx 2 , from Proposition 3.20, there is no term whose weight is −1 in d P 2 Σ(P 1 , P 2 ) as a function in Q. Noting the homogeneous property of h X from Lemma 3.17, we have the result. (1) The one-form, ν III P 1 ,P 2 (P ) := Σ(P, P 1 ) − Σ(P, P 2 ), is a differential of the third kind whose only (first-order) poles are P = P 1 and P = P 2 , with residues +1 and −1 respectively.
We note the W-normalized differentials of the first kind and the second kind, in Definitions 3.2 and 3.29.

Definition 3.35.
(1) ν III P 1 ,P 2 is called the W-normalized differential of the third kind (2) Ω(P, Q) is the W-normalized fundamental differential of the second kind and when it is expressed by where F Ω is called Klein fundamental form in R X ⊗ C R X .
Lemma 3.36. We have Proof. The expression (3.13) and Corollary 3.30 give the result.
For the connection of these algebraic tools with the sigma functions. we define Π (3.16) It has the properties.
Definition 3.38. We define the complete Abelian integrals of the first kind and the second kind by and the Jacobian (Jacobi variety) by J X := C g /Γ X with κ J : C g → J X and Γ X := ω ′ , ω ′′ Z . Let ν III• Q 1 ,Q 2 be the normalized Abelian differential of the third kind, i.e., [26]. The following lemma corresponds to Corollary 2.6 (ii) in [34].
where 1 g is the unit g × g matrix.
The following matrix satisfies the generalized Legendre relation: Proof. It is the same as Proposition 5.1 in [4].
Proof. Due to the definition of Sp(2g, Z), we have t ρ ζ −1 g 1 g t ρ ζ = −1 g 1 g and thus, (1) N c (g − k − i) − N(i − 1) (i = 1, 2, . . . , n k ) is an element of the gap sequence H c X , and thus let N c (L For a Young diagram Λ = (Λ 1 , Λ 2 , . . . , Λ n ), the Schur function s Λ is defined by the ratio of determinants of n × n matrices [14], When Λ associated with the semigroup H as in Subsection 2.1, it can be also regarded as a function of T = t (T Λ 1 +n−1 , . . . , T Λn ) [14], and thus, we express it by We recall the truncated Young diagrams Λ We also write the decomposition, Proposition 4.6. For ♮ X,k in Definition 4.5, Following Nakayashiki's results in [19], we state the Riemann-Kempf theorem [27,35] of the W-curves. X |, and u ∈ Θ k X , Proof. See Corollary 3 in [19].

4.3.
Sigma function and W-curves. We now define the sigma function following Nakayashiki [19,Definition 9]. We remark that due to the shifted Riemann constant, our definition differs from Nakayashiki's so that our sigma function has the natural properties, including the parity and Galois action and the fact that the point of expansion by Schur polynomials is also shifted as mentioned in Theorem 4.9. In other words, we employ some parts of the definition of the sigma function by Korotkin and Shramchenko [18] who defined the several sigma functions with spin structures based on Klein's transcendental approaches.
Then we have the following theorem, i.e., Theorem 4.9.
It is worthwhile noting that the following (4.6) obviously leads the Jacobi inversion formulae on the Jacobian J X and its strata as mentioned in [4,6,13,14]; though we omit the inversion formulae for the reason of space, we can easily obtain them as its corollary following [13,14]. Since Π P,Q P i ,P ′ j in (4.6) can be expressed in terms of R X in (3.16), we can represent the elements of R X by using the differentials of the sigma functions. More explicitly, since the Jacobi inversion formulae on J X provide that the multi-variable differentials of the sigma are equal to the meromorphic functions of R X as predicted in [16], they imply that if the formulae are integrable, the sigma function is, in principle, obtained by integrating the meromorphic functions on S g X; since the integrability is obvious, the sigma function for every W-curve can be, in principle, algebraically obtained like the elliptic sigma function in Weierstrass' elliptic function theory. They also show the equivalence between the algebraic and transcendental properties of the meromorphic functions on X. The sigma function is defined for every compact Riemann surface by Nakayashiki following Klein's construction of his sigma functions [9]. Klein defined his sigma functions using only the data of hyperelliptic Riemann surfaces, following Riemann's approach. On the other hand, Weierstrass criticized Riemann's approach and insisted on the algebraic ways, associated with Weierstrass curves. 4 Unifying Klein's and Weierstrass' views, Baker reformulated Klein's sigma functions after defining explicit algebra curves, and connected the sigma functions and the meromorphic functions of the curves like Weierstrass' elliptic function theory [10]. Thus we emphasize that the following theorem implies completing Weierstrass' program by succeeding Baker's approaches. Theorem 4.9. σ(u) has the following properties: (1) it is modular invariant, (2) it obeys the translational formula; for u, v ∈ C g , and ℓ (= 2ω , the following holds (4.5) σ(u + ℓ) = σ(u) exp(L(u + 1 2 ℓ, ℓ))χ(ℓ), (3) its divisor is κ −1 J Θ X,s ⊂ C g , where Θ X,s := w s (S g−1 X) ⊂ J X , (4) it satisfies the Jacobi-Riemann fundamental relation, For (P, Q, P i , P ′ i ) ∈ X 2 × (S g (X) \ S g 1 (X)) × (S g (X) \ S g 1 (X)), u := w s (P 1 , . . . , P g ), v := w s (P ′ 1 , . . . , P ′ g ), = σ( w(P ) − w s (P 1 , · · · , P g ))σ( w(Q) − w s (P ′ 1 , · · · , P ′ g )) σ(( w(Q) − w s (P 1 , · · · , P g ))σ( w(P ) − w s (P ′ 1 , · · · , P ′ g )) , (4.6) which generates the Jacobi inversion formulae for S k X, (5) the leading term in the Taylor expansion of the σ function associated with X, with normalized constant factor c, is expressed by the Schur function of Λ X σ(u + w(ι X K s )) = S Λ X (T )| T Λ i +g−i =u i + |wg(α)|>|Λ X | a α u α , where a α ∈ C[λ ij ], α = (α 1 , ..., α g ), u α = u α 1 1 · · · u αg g , and w g (α) = α i wt(u i ). Here S Λ (T ) is the lowest-order term in the w-degree of the u i ; σ(u) is homogeneous of degree |Λ X | with respect to wt λ , (6) σ(−u) = ±σ(u), and 4 see Weierstrass's words in a letter to Schwarz (Werke II, 235) cited by Poincare [36]:"Plus je réfléchis aux principes de la théorie des fonctions -et c'est ce que je fais sans cesse -plus je suis solidement convaincu qu'ils sont bâtis sur le fondement des vérités algébriques et que, par conséquent, ce n'est pas le véritable chemin, si inverse ment ou fait appel au transcendant pourétablir les théorèmes simples et fondamentaux de l'Algébre ; et cela reste vrai, quelque pénétrantes que puissent paraître au premier abord les considérations par lesquelles Riemann a découvert tant d'importantes propriétés des fonctions algébriques."("The more I think about the principles of function theory -and I do continuously -the more I am convinced that this must be built on the foundations of algebraic truths [my emphasis], and that it is consequently not correct to resort on the contrary to the 'transcendent', to express myself briefly, as the basis of simple and fundamental algebraic propositions. This view seems so attractive at first sight, in that through it Riemann was able to discover so many important properties of algebraic functions."[37]) (7) If ζ ∈ G X satisfies ζ ℓ = id, and ζ[σ(u+ ℓ)/σ(u)] = σ(u+ ℓ)/σ(u) for ℓ ∈ Γ X and u ∈ C g , the action gives the one-dimensional representation such that ζσ(u) = ρ ζ σ(u), where ρ ℓ ζ = 1. Proof. 1 and 5 are obtained by Theorem 13 in [19] by noting the difference of the definition of our sigma function in Definition 4.8 from Nakayashiki's [19,Definition 9]. 3 is due to Proposition 4.3. 2 is standard and can be obtained by the direct computations [13]. 4 is the same as Proposition 4.4 in [13]. 6 and 7 are the same as Lemma 3.6, and Lemma 4.1 in [35].

Conclusion and Discussions.
We have considered the Weierstrass curves (W-curves), which are algebraic expression of compact Riemann surfaces; The set of W-curves represent the set of compact Riemann surfaces.
By using the algebraic tools we constructed in this paper,we have a connection between the sigma function for W-curve X and the meromorphic functions on X as in Theorem 4.9. Since the Jacobi inversion formulae via Theorem 4.9 4 are given by the differential identity, by integrating it, it, in principle, provides that the sigma function is constructed by an integral formula of the meromorphic functions on the W-curve X. In other words, we give an algebraic construction of the sigma function, or so-called the EEL-construction [11] in this paper.
It is noted that this construction is based on our recent result on the trace structure of the affine ring R X [17].
Further we also discuss mathematical meaning of our result as follows. We also note that for an ordinary point P in every W-curve Y with H Y at ∞ ∈ Y of genus g, the Weierstrass gap sequence at P is given by the numerical semigroup H c = {1, 2, · · · , g} and there is a Wcurves X which is birationally equivalent to X such that ∞ ∈ X corresponds to P ∈ Y and H c X = H c . Then there appear two sigma functions σ Y for the W-curve Y with non-trivial Weierstrass semigroup H Y and σ X for the W-curve X with H X = H. By some arguments on the both Jacobians J X and J Y , we find that σ X and σ Y , due to the translational formulae and so on, are the same functions, and the both sifted Abelian integrals agree. Then the above theorem (5) means that we have the expansions of the sigma function at the Abelian image of the ordinary point P in Y . It means that the problem of finding the expansion of the sigma function for a point u in Jacobian J Y is reduced to the problem that we should find the birational curves associated with the preimage of the Abelian integral. Since in Weierstrass' elliptic function theory, we often encounter the reductions of the transcendental problems to the algebraic problems, we also remark that this reduction has the same origin, i.e., the equivalence between algebraic objects and transcendental objects in the Abelian function theory.
With Theorem 4.9, we recognize that this theorem is the goal Weierstrass had in mind, and at the same time, with it, we also recognize that we finally reached the starting point for the development of the Weierstrass program to construct an Abelian function theory for every W-curve X like his elliptic function theory.