ALGEBRAIC CONSTRUCTION OF THE SIGMA FUNCTIONS FOR GENERAL WEIERSTRASS CURVES

. 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 each A j is a polynomial in x of degree ≤ js/r for certain coprime positive integers r and s ( r < s ); the Weierstrass non-gap sequence at ∞ ∈ X is given by the numerical semigroup H X whose generator contains r and s . Every compact Riemann surface has a Weierstrass curve X which is birational to the surface. 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 ( ∗∞ )). Recently 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 ( ∗∞ )) and 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 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 Jacobian via the σ function since ℘(u) = − d 2 du 2 log σ(u), d℘(u) du is identical to a point (x, y) of the curve [W3,WW]. 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 [W2], 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 [KMP1,KMP2,KMP3] following Mumford's studies for the hyperelliptic curves based on the modern algebraic geometry [Mu1,Mu2], 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 Jacobian as a generalization of the elliptic sigma function [K]. Baker re-constructed Klein's sigma functions by using the data of hyperelliptic curves algebraically [Ba1]. 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 EEL construction due to work by Eilbeck, Enolskii, and Leykin [EEL] [ [BEL] 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 [MP1,MP2]. Further, we generalized the sigma functions and the formulae to a particular class of the space curves using the EEL-construction [MK,KMP1,KMP3].
We have studied further generalization of the picture in terms of the Weierstrass canonical form [KM,KMP4]. 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 each A j is a polynomial in x of degree ≤ js/r for certain coprime positive integers r and s (r < s); the Weierstrass non-gap sequence at ∞ ∈ X is given by the numerical semigroup H X whose generator contains r and s. Every compact Riemann surface has a Weierstrass curve X which is birational to the surface. 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 [KMP4], 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 ( * ∞)). We also simply call the Weierstrass curve W-curve.
Recently D. Korotkin and V. Shramchenko [KS] and Nakayashiki [N] 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, which is called Weierstrass semigroup. Nakayashiki defined the sigma function for every compact Riemann surface with Weierstrass semigroup H [N] based on Sato's theory on the universal Grassmannian manifolds (UGM) [SN, SW].
In this paper, we use our recent results on the complementary module of the W-curve [KMP4] 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 [KMP4]; 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 [KMP4]. 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 [KMP4] 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 [KMP4] to R X ⊗ C R X in Subsection 3.2. The trace operator p enables us to define the essential one-form Σ and its differential dΣ in Subsection 3.3. After investigating dΣ, we find the Abelian differentials of the second kind in Theorems 3.31 and 3.32. Further, in Theorem 3.34, we mention our results on the fundamental differential of the second kind Ω in our W-curves and the Abelian differentials of the third kind. We obtain the generalized Legendre relation in Proposition 3.41. Using them, we show the connection of the sigma functions for X with R c X in the W-curves X in Section 4. As studied in [KMP2], we introduce the shifted Abelian integrals in Subsection 4.1. Subsection 4.2 shows the properties of the Riemann theta functions of W-curves and its Riemann-Kempf theorem as in Proposition 4.7. In Subsection 4.3, we define the sigma function for W-curves X by modifying the definition of Nakayashiki [N,Definition 9], and show its properties in Theorem 4.9 as our main results in this paper. We obviously have the following proposition: Definition 2.2. For given Young diagram Λ = (Λ 1 , Λ 2 , · · · , Λ g ), we define 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 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 .
(1) Let Z r := {0, 1, 2, . . . , r − 1} and Z × r : The following is obvious: Lemma 2.6. 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. [Ka, CK] 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 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 ∞. The infinity point ∞ is a Weierstrass point if H c X = H c (X, ∞) = {N(i)} differs from {1, 2, · · · , g} [FK]. Since every compact Riemann surface of the genus, g(> 1), has a Weierstrass point whose Weierstrass gap sequence with genus g [ACGH], 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 an ordinary 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 . Remark 2.8. Let R • X • := C[x, 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 [Ku2]. 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, ∞) appearing in the proof of Proposition 2.7. 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 . 2.2.1. Projection from X to P. There is the natural projection, (2.6) ̟ r : X → P, (̟ r (x, y r 2 , . . . , y rm X ) = x = y r ) such that ̟ r (∞) = ∞ ∈ P. Let {y • } := {y s = y r 2 , y r 3 , . . . , y rm X } and C[x, y • ] := C[x, y s = y r 2 , y r 3 , . . . , y rm X ].
2.3. The monomial curves and W-curves. This subsection shows the monomial curves and their relation to W-curves based on [KMP2,KMP3,KM]. 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 ], and a polynomial ring Following from a result of Herzog's [H], we recall the well-known proposition: 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 [H, Pi]. Bresinsky showed that k X can be any finitely large number if m X > 3 [Br].
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 .
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. Lemma 2.11. The C[Z r ]-module C[Z] is given by Lemma 2.12. By defining an element in To construct our curve X from R H or Spec R H , we could follow Pinkham's strategy [Pi] 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 [Pi][KMP4, 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 (2) The degree at Q(R X,∞ ) as the order of the singularity with respect to t is naturally defined by so that it belongs to the ideal, 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, Definition 2.16. We define S X := {φ i | i = 0, 1, 2 . . .} 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 ∞ : Proof. By letting g = ϕ Z H • ϕ X H (f ), 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 [KMP4,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 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 [Ku2,St] 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 the Galois covering, and thus we have the Galois group on X, i.e., Gal(Q(R X )/Q(R P )) = Aut(X/P) = Aut(̟ r ), which 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 Galois covering ̟ x = ̟ r : X → P: 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 [Ku2], shows the following: Corollary 2.20. The divisor of dx is given by, X (x, y r i ) of R X as its normalized ring, as we prove this fact in [KMP4,Proposition 3.14] precisely. The image of f Further we also define the tensor product of these rings R Moreover we also define the direct product of these rings R ∂y , as an R X -analog of (2.9), let us consider X as an element of Q(R X ⊗ R P R X ). The following is obvious: However, some parts in its numerator and denominator are canceled because they belong to R P . Thus we introduce an element h( We have the expression of h R X (x, y • , y ′ • ) following [KMP4,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 δ i (x) ∈ C[x] whose weight is −δ i r, (especially, y e 0 = δ 0 (x)y e * ℓ,0 = δ 0 (x)y e ℓ ) 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): Definition 2.25. For i ∈ Z r , we define a truncated polynomial y e i of Υ i such that the weight −wt of Υ i − y e i is less than −wt(ẙ e i ), i.e., 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 [KMP4].
Definition 2.27. The effective divisor, K X > 0 and let k X := 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 [KMP4] 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 [W2], H. F. Baker [Ba1], V.M. Buchstaber, D.V. Leykin and V.Z. Enolskii [BEL], J.C. Eilbeck, V.Z. Enolskii and D.V. Leykin [EEL], 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 [KMP4]. 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 , 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 }: Definition 3.2.
(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 [FK], 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 ∞ ∼ 0 as the linear equivalence, where k s and k c X are the degree of K s and K c X respectively.
We note that there are h s and h c X in Q(R X ) and their behaviors at ∞ are given by ) using the arithmetic parameter t in Definition 2.14.
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 [KMP4,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 , Further this relation is extended to the condition Q ∈ B X \{∞} by considering the multiplicity of the action G X .
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.

W-normalized differentials of the second and the third kinds.
3.3.1. The one-form Σ on X. We construct an algebraic representation of the fundamental Wnormalized differential of the second kind in [F, Corollary 2.6], namely, a two-form Ω(P 1 , P 2 ) on X × X which is symmetric and has quadratic singularity as in Theorem 3.34. Following K. Weierstrass [W2], H. F. Baker [Ba1], V.M. Buchstaber, D.V. Leykin and V.Z. Enolskii [ [BEL] and therein], J.C. Eilbeck, V.Z. Enolskii, and D.V. Leykin [EEL], we have Ω using a meromorphic one-form Σ(P, Q) on X × X for the hyperelliptic curves and plane Weierstrass curves (W-curves). In this subsection, we extend it to more general W-curves based on Definition 3.10 and Proposition 3.12 to introduce Σ(P, Q) on X × X.
(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 Σ P, Q = − dt P t P (1 + d >0 (t P )).
(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 near ∞ corresponding to P and Q respectively. Proposition 3.12 shows lim )dt P , and thus we have 2.b. On the other hand, the following Lemma 3.19 3 shows that Σ(P, Q) behaves ). 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 . 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: Lemma 3.19.
(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.
(2) for the case |t P | < |t Q |, 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 (3) We have the following relations, (4) For Σ ′ P, Q defined in Proposition 3.18 4, we have Proof. 1 (a), 2 (a), and finite part of 3 (a) are directly obtained from Proposition 3.18. The properties of dΣ on X H × X H determines the behavior at ∞ in the following subsection, Subsection 3.3.4. We use the facts: dx Q ∂ ∂x Q = dt Q ∂ ∂t Q , and dt Q ∂ ∂t Q t ℓ Q = ℓ t ℓ−1 Q dt Q . Lemmas 3.21 and 3.19 show 1 (b), 2 (b) and the behavior at ∞ of 3 (a). Lemma 3.19 3 shows 3 b.
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.
For the case |t Q | < |t P |, noting ϕ ∞ (h X (P )) = r t −d h P , and ϕ ∞ (dx P ) = r t −r−1 P dt P , (3.8) in Lemma 3.21 leads the expression, 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, ν II j is expressed like Noting Proposition 3.20 3.b, we have the following relations: 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 1 . Proof. Noting Proposition 3.40, Lemmas 3.24 and 3.25 show the fact.
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.
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.
We note the W-normalized differentials of the first kind and the second kind, in Definitions 3.2 and 3.29.
(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 Lemma 3.36. We have (3.15) lim 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.
The following lemma corresponds to Corollary 2.6 (ii) in [F].
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 [KMP1].
From Definitions 3.27 and 3.29, we have the following corollary, which is the dual of the homological relations (3.17): The Galois action on the basis of the Homology H 1 (Z, Z) shows the actions of these period matrices (ω ′ , ω ′′ ) geometrically: Lemma 3.43. For the Galois action ζ ∈ G X , i.e., ζ : X → X, its associated element ρ ζ of Sp(2g, Z) acts on (ω ′ , ω ′′ ) and (η ′ , η ′′ ) by and the generalized Legendre relation (3.18) is invariant for the action.

Sigma functions for W-curves
4.1. W-normalized shifted Abelian integrals. Since the non-symmetric W-curves have the non-trivial R X -module, R X in Definition 3.2, (and properties in Proposition 3.5), the Abel-Jacobi map [FK] for R X naturally appears. Let X be the Abelian universal covering of X, which is constructed by the path space of X; κ X : X → X. Thus recalling Definitions 2.27, and 3.4, we introduce the shifted Abelian integral w s and the Abel-Jacobi map w s as an extension of the Wnormalized Abelian integral w : S k X → C g w(γ 1 , · · · , γ k ) = i γ i ν I , w • := 1 2 ω ′−1 w, and the Abel-Jacobi map w : S k X → J X := C g /Γ X w(P 1 , · · · , P k ) = i P i ∞ ν I , w • := 1 2 ω ′−1 w as mentioned in [KMP2]: Definition 4.1. We define w s and w s by The following holds There is a θ-characteristic δ X of a half period which represents the shifted Riemann constant ξ X,s , i.e., θ[δ X ] 1 2 ω ′−1 w s • ι X (P 1 , . . . , P g−1 ) = 0, i.e., δ X = The following comes from the investigation of the truncated Young diagram and the Schur polynomials in [MP2]; though we did not consider the Young diagram associated to plane curve in the paper [MP2], the investigation is easily generalized to general Young diagrams associated with any numerical semigroups (c.f. Lemma 2.3). Thus we state the facts without proofs.
Proposition 4.4. For the Young diagram Λ X associated with the numerical semigroup H X of genus g, an integer k (0 ≤ k < g), and the characteristics of the partition of Λ [k] X = (Λ k+1 , Λ k+2 , . . . , Λ g ) = (a 1 , a 2 , . . . , a n k ; b 1 , b 2 , . . . , b n k ), (n k := r Λ [k] X , the rank of r Λ [k] X ), the following holds: (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 every i = 1, . . . , n k , and (2) 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 [MP2], 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 ) [MP2], 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 [N], we state the Riemann-Kempf theorem of the W-curves.
Proposition 4.7. For u ∈ C g and a multi-index J ⊂ Index(g, ℓ), we define ∂ I := j∈J ∂ ∂u j . Let Λ X be the Young diagram of the W-curve X and for given k ∈ {0, 1, . . . , g}, let Λ X = Λ 4.3. Sigma functions and W-curves. We now define the sigma function following Nakayashiki [N,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 [KS] who defined the several sigma functions based on Klein's transcendental approaches.
Then we have the following theorem. 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 [KMP1,KMP3,MP1,MP2]; though we omit the inversion formulae for the reason of space, we can easily obtain them as its corollary following [MP1,MP2]. 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 [KM], 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 [K]. 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 [Ba1]. Thus we emphasize that the following theorem implies completing Weierstrass' program by succeeding in Baker's approaches.
(1) and (5) are obtained by Theorem 13 in [N] by noting the difference of the definition of our sigma function in Definition 4.8 from Nakayashiki's [N,Definition 9]. (3) is due to Proposition 4.3. (2) is standard and can be obtained by the direct computations [MP1]. (4) is the same as Proposition 4.4 in [MP1]. (6) and (7) are the same as Lemma 3.6, and Lemma 4.1 in [O1].
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 W-curves 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 X is reduced to the problem that we should find the birational curves associated with the preimage of the Abelian integral. We also remark that we often encounter the reductions of the transcendental problems to the algebraic problems in Weierstrass' elliptic function theory. This reduction has the same origin that is 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.