On Regularization of Second Kind Integrals

We obtain expressions for second kind integrals on non-hyperelliptic $(n,s)$-curves. Such a curve possesses a Weierstrass point at infinity which is a branch point where all sheets of the curve come together. The infinity serves as the basepoint for Abel's map, and the basepoint in the definition of the second kind integrals. We define second kind differentials as having a pole at the infinity, therefore the second kind integrals need to be regularized. We propose the regularization consistent with the structure of the field of Abelian functions on Jacobian of the curve. In this connection we introduce the notion of regularization constant, a uniquely defined free term in the expansion of the second kind integral over a local parameter in the vicinity of the infinity. This is a vector with components depending on parameters of the curve, the number of components is equal to genus of the curve. Presence of the term guarantees consistency of all relations between Abelian functions constructed with the help of the second kind integrals. We propose two methods of calculating the regularization constant, and obtain these constants for $(3,4)$, $(3,5)$, $(3,7)$, and $(4,5)$-curves. By the example of $(3,4)$-curve, we extend the proposed regularization to the case of second kind integrals with the pole at an arbitrary fixed point. Finally, we propose a scheme of obtaining addition formulas, where the second kind integrals, including the proper regularization constants, are used.


Introduction
In this paper we consider second kind integrals on algebraic curves, which are second kind 0-forms r(x, y; [γ]) obtained from second kind differential forms dr r(x, y; [γ]) = (x,y) [γ] dr(x,ỹ), (1.1) where [γ] denotes a class of homotopically equivalent paths from a fixed basepoint to (x, y), by (x,ỹ) we denote a point which serves as a dummy parameter of integration. Definitions of second kind integrals were discussed in [11], where the theory of second kind differentials was developed from the theory of stacks adapted for use in the theory of functions of several complex variables. Here we define a second kind differential form as a locally exact meromorphic form, whose all singularities are poles of order greater than one.
In what follows we deal with a particular class of curves called (n, s)-curves, possessing a Weierstrass point at infinity, which is the branch point where all sheets of the curve come together. We assign the infinity to the basepoint of Abel's map on the curve. An accurate definition of the (n, s)-curve is given below in Section 2.1. Throughout the paper we consider arXiv:1709.10167v2 [math.CV] 21 Jul 2018 second kind differential forms with the only singularity at the infinity, except Section 5.1 where the singularity is located at an arbitrary point. We call the antiderivative of a second kind 1-form dr with the basepoint at the infinity a second kind integral r, namely, r(x, y) = (x,y) ∞ dr(x,ỹ), (1.2) and define the latter on the fundamental domain of a genus g (n, s)-curve. Suppose the curve has a homology basis of 2g cycles {a i , b i | i = 1, . . . , g}, all disjoint except for each pair a i and b i intersecting at one point. The fundamental domain is the one-connected domain obtained by cutting the curve along all 2g cycles of the homology basis and g paths connecting the basepoint with g intersection points. Evidently, the integral (1.2) requires regularization at the basepoint, which is the only singularity point of the integrand function.
The idea of regularization is adopted from the definition of Weierstrass zeta-function. On the Weierstrass normal form of elliptic curve 0 = f (x, y) = −y 2 + 4x 3 − g 2 x − g 3 , which is the simplest (n, s)-curve, the second kind integral is defined by Here and throughout the text we denote ∂f /∂y by ∂ y f . Let be a parametrization in the vicinity of infinity, namely (x, y) → ∞ as u → 0, where u serves as a local parameter. With the standard Abel's map A(x, y) = (x,y) ∞ (∂ỹf ) −1 dx the second kind integral transforms into zeta function ζ A(x, y) = r(x, y). The textbook definition of Weierstrass zeta-function, see, e.g., [14, Chapter XX, Section 20.4], serves as a regularization of the second kind integral In the present paper we extend this regularization to a wider collection of curves. In this connection, a constant arises to be added to the regularized second kind integral, we call it a regularization constant. As shown in [1,Section 193], where the notion of a regularization constant arose, these constants vanish in the case of hyperelliptic curves. This can explain why it was not revealed before. In the non-hyperelliptic case, the regularization constant does not vanish, and plays an important role when the second kind integral is used to obtain relations between Abelian functions defined on the Jacobi variety of the curve. In particular, for this purpose we employ the primitive function ψ(x, y) = exp − (x,y) ∞ r(x,ỹ) t du(x,ỹ) (1.3) introduced in [6], where {du(x, y), dr(x, y)} form a special cohomology basis, and r(x, y) is the antiderivative (1.2) of the dr(x, y). In general, the primitive function ψ(x, y; [γ]) also depends on a path γ from the basepoint to a point (x, y), and the second kind integral is defined by (1.1). As above, [γ] denotes a class of homotopically equivalent paths. The paper is organized as follows. In Section 2 we briefly recall the notions of (n, s)-curve and primitive function, we explain how to construct the special cohomology basis, and specify the curves under consideration. Section 3 is devoted to the idea of regularization of the second kind integral (1.2), also the special cohomology bases on the curves under consideration are specified.
In Section 4 we show how to produce relations between Abelian functions defined on the Jacobian of V (3,4) with the help of equality (4.14), which involves the primitive function ψ. Similar computations in the case of (3, 5)-curve are shown in Appendix D. We emphasize that this method gives consistent relations between Abelian functions only with a correct choice of regularization constant in the definition of r(x, y). Therefore, we could say that the regularization constant is unique, and consistent with the structure of Abelian function field.
The following proposition summarizes results obtained in the paper. Each regularization constant c (n,s) relates to the second kind integral on (n, s)-curve with the special cohomology basis.
(1−t) −1 −(1−t ns )(1−t n ) −1 (1−t s ) −1 . Further consider meromorphic 1-forms with poles only at infinity. Up to globally exact forms any such 1-form can be represented as a linear combination of 2g differentials x i y j (∂ y f (n,s) ) −1 dx with 0 i s − 2 and 0 j n − 2, which are holomorphic on V (n,s) \∞, and the condition in + js 2g singles out differentials that actually have a pole at infinity. One can form a vector dr of wgt dr(ξ) = (w 1 , . . . , w g ) t which is subject to condition This condition completely determines the principle part of dr(ξ). Though the holomorphic part of dr(ξ) is inessential in what follows, it is worth to note that it can be chosen so that dr(x, y) and du(x, y) form an associated system, see [1,Section 138] and [10, p. 131]. Below in actual calculations we use the first A(x, y) and second kind integrals A * (x, y) obtained from the chosen basis of 1-forms, namely, So A : V → Jac(V) denotes the standard Abel's map. The meromorphic map A * has a pole at infinity, and requires regularization. The regularization constants c (n,s) computed in this paper relate to these particular second kind integrals.

Primitive function
The primitive function introduced by (1.3) is employed by the both methods of computing the regularization constant. More accurately, we define it as In the definition an antiderivative A * (x, y) of the second kind differential is supposed to contain the regularization constant when parametrized. In terms of local parameter ξ the following representation of (2.3) can be obtained where due to condition (2.2) the integrand is a holomorphic function of ξ. Evidently, the primitive function is entire with a zero at ξ = 0 of order g, where g denotes the genus of a curve. The primitive function ψ(ξ) coincides with a certain derivative of sigma function σ(u) at u = A(ξ) up to a constant factor, see Remark 4.3, and also with a certain modification of the prime form arisen from [12]. Following [5], we define the sigma function of g variables, which we call a multivariate sigma function, as a solution of a system of heat equations with a Schur-Weierstrass polynomial as an initial condition. Here we apply the approach to constructing multivariate sigma functions on (n, s)-curves after [5,7]. The most complete survey of the theory of multivariate sigma is given in [4].

Regularization of second kind integral in general
Now we define the regularization more accurately. Consider a second kind integral r(x, y) in the vicinity of its pole. Let ξ be a local coordinate, and dr(ξ) = r (ξ)dξ, where r(ξ) is 0-form. Then r(ξ) is decomposed into singular and regular parts: Essentially, this decomposition is a textbook regularization, and c denotes the regularization constant. We define the regularized second kind integral as follows The regularization constant c(λ) is a vector with g components, which are polynomials in λ with rational coefficients, and wgt c(λ) = wgt r(ξ) = (w 1 , . . . , w g ) t . Note that representation (3.1) is essentially connected to parameterization in the vicinity of infinity, and holds true within the fundamental domain. The regularization constant should not be confused with a constant of integration. The latter vanishes in order to keep ψ satisfying the functional equation (4.2). The regularization constant appears only when the second kind integral has the pole at infinity and is parameterized in the vicinity of this pole. If one needs to define the regularization constant, we say that the correct choice of this constant makes the primitive function ψ at ξ equal, up to a rational factor, to the first nonvanishing derivative of sigma-function on the Abel's image of ξ. We also recall the method of producing relations between Abelian functions, which involves the primitive function ψ. The correct choice of the regularization constant is necessary for obtaining consistent relations. In what follows, we use both properties of the primitive function ψ to compute the regularization constants announced in Introduction.

Basis differentials and integrals on V (3,4)
On the curve V (3,4) punctured at infinity we fix a basis of holomorphic differentials Introduce a local coordinate ξ in the vicinity of infinity. Then we have a parameterization In terms of the local coordinate the basis holomorphic differentials acquire the form The first kind integral A(ξ) = ξ 0 du(ξ) is a well defined holomorphic function of local parameter ξ. On the curve V (3,4) the regularized second kind integral A * (ξ), defined by (3.1), has the following singular part The integral on the right hand side of (3.1) is regular, and dA * (ξ)/dξ = dr(ξ)/dξ. Basis differentials of the first and second kinds defined on V (3,5) , V (3,7) , and V (4,5) are given in Appendix A, and singular parts of the second kind integrals in terms of a local coordinate in the vicinity of infinity can be found in Appendix B.
On one hand, we have From the series expansion On the other hand, using notation c(λ) = (c 1 , c 2 , c 5 ) t , from (2.3) we obtain log ψ(ξ) = 3 log ξ + c 1 ξ + 1 6 By Lemma 4.2 the series (4.4) and (4.5) are equal. Thus, we come to a system of linear equations, namely and find (4.1). This solution is unique.
The primitive function ψ(ξ) coincides, up to a factor which is a rational number, with the first non-vanishing derivative of sigma function on the Abel's image of ξ. The first nonvanishing derivative of sigma function has Sato weight equal to −g, the corresponding differential operator has Sato weight − wgt σ(u) − g. Recall that Sato weight of sigma function is negative and calculated by the formula −(n 2 − 1)(s 2 − 1)/24 for (n, s)-curve, see [2].  [5,7]. In [9, ref. 15] the reader can find a reference to page http://www.ma.hw. ac.uk/Weierstrass/, where an expansion for the sigma function related to (3,4)-curve with some extra parameters is presented, the expansion is obtained by J.C. Eilbeck. The case of cyclic (3, 4)-curve, when λ 2 , λ 5 and λ 8 vanish, is considered in [9], where a series expansion for the cyclic (3, 4)-sigma function is proposed. Taking into account the difference in the number of parameters and signs between equations of (3, 4)-curves, we compared (4.3), and also higher terms which are not presented here, with the two mentioned expansions, and found that they coincide.
Proof 2. Let u and v be the Abel's map images of two non-special divisors on V (3,4) , namely, at that f (x i , y i ) = 0 and f (z i , w i ) = 0, i = 1, 2, 3, and σ(u) = 0, σ(v) = 0. By the residue theorem we have Direct calculation using (3.4a) and (3.1) gives where Here we apply the relation (4.10a), and use the following notation Lemma 4.5. The following relations for Abelian functions on Jacobian of V (3,4) hold Remark 4.6. The complete list of relations between Abelian functions on Jacobian of genus 3 trigonal curve are obtained in [8], the curve is defined by an equation slightly different from (2.4).
Proof . Differentiating (4.8) over x 1 we obtain which produces three relations with rational functions R k of order k on V (3,4) For brevity we omit argument u of Abelian functions. Clearly, the functions (4.11) vanish on (x 2 , y 2 ) and (x 3 , y 3 ) as well.
Next, denote by similar reason.
In the case of V (3,4) we have the equality, for more detail see [6], where ( Applying (4.10) we find (4.1).
Remark 4.7. The equality (4.14) produces bilinear Hirota type equations, which give relations between Abelian functions. Note that only correct choice of the regularization constant guarantees consistency of relations. So the Hirota type equations carry information about the regularization constant.

Second kind integrals with poles at an arbitrary point
Suppose (z, w) is a fixed point on the curve V (n,s) , which is characterized by ∂ i y f (x, y)| (x,y)=(z,w) = 0, i = 0, . . . , k − 1 and ∂ k y f (x, y)| (x,y)=(z,w) = 0 with k ∈ N. Then a local parameterization near (z, w) is defined as follows Points with k = 1 are regular. When k > 1, the point (z, w) is a branch point where k sheets join. On (n, s)-curves we have k n < s. Consider the following function defined on the curve f (x, y) = 0 Given finite values of (x, y) the function R(x, y) has a single simple pole at (z, w): Assume (z, w) is a regular point, that is the case k = 1. With the help of R(x, y) we construct the function having a simple pole at (z, w) and vanishing at infinity, namely, where r(x, y) = r 1 (x, y), r 2 (x, y), r 5 (x, y) is the second kind integral containing regularization constant c(λ), and u(x, y) = u 1 (x, y), u 2 (x, y), u 5 (x, y) is the first kind integral. The function I 1 (x, y) has zero of order 2g = 6 at ∞.
Differentiating I 1 (x, y) over the parameter z, we obtain an integral of the second kind with a pole of any order = 2, 3, . . . at the point (z, w), actually Applying parameterization (5.1), an expansion in the vicinity of the pole (z, w) can be found Clearly, I (x, y) has zero of order 2g = 6 at infinity. In a similar way, taking into account (5.1) for k > 1, integrals I (x, y) can be constructed when the pole (z, w) is a branch point.

Further, introduce
where the argument u of Abelian functions ℘ i,j (u), ℘ i,j,k (u) is omitted for brevity.
Remark 5.3. In fact, all α k are symmetric with respect to u and v. Indeed, the expression for (α 1 , α 2 , α 3 ) t is explicitly symmetric. From (5.4) we find Remark 5.4. The function of the form (5.3) described by Theorem 5.2 has 3g = 9 zeros on V (3,4) . Besides the sets ( Then by Abel's theorem u + v + w = 0. Finally, the addition law on Jacobian of V (3,4) can be written in the following natural way This form is natural in the sense that it is a direct analogue of the famous elliptic relation Introduce even and odd parts of The following addition formulas hold on Jacobian of V (3,4) Proof . We need the following assertion.
Lemma 5.6. Under the assumptions of Theorem 5.2 we have Proof follows immediately from Riemann vanishing theorem and properties of ψ(ξ), for more detail see [6].
Remark 5.7. In general, equalities of the form (5.6) produce trilinear relations which serve as addition formulas. Note that the equality contains the primitive function ψ(ξ), requiring the regularized second kind integral with the correct regularization constant. The correct regularization constant makes the formulas (5.5) consistent with the 'natural' addition formulas, cf. Remark 5.4.
6 Regularization of second kind integral on (3, 5)-curve Theorem 6.1. In the defifnition of regularized second kind integral (3.1) on V (3,5) , with the basis first and second kind differentials given by (A.1), the regularization constant is equal to We briefly consider the two methods of proof.
In the case of V (3,5) we have the equality where the local parametrization in the vicinity of infinity is applied, for more details see [6]. Again the both left and right hand sides are rational functions on the curve, and vanish at 2g = 8 points which are Abel's map preimages of u and −u. Comparing the leading terms of expansions in the vicinity of ξ = 0 we see that the functions are equal. Applying (6.6) to the expansion of (6.10) given in Appendix D, we find (6.1).

Regularization of second kind integrals on (3, 7)-and (4, 5)-curves
Theorem 7.1. In the definition of regularized second kind integrals (3.1) on V (3,7) and V (4,5) , with the basis first and second kind differentials given by (A .2) and (A.3), respectively, the regularization constants are equal to The first method of computing regularization constants, see Section 4 or 6, requires the series expansion for sigma function. Computational complexity increases with genus, one needs 2g terms with coefficients of Sato weights from 0 to 2g − 1, which are polynomials in λ. The total number of terms needed to be computed grows exponentially. In the case of (3, 7)-and (4, 5)-curves we omit this method.
The second method uses relations between Abelian functions, the number of which also increases. Actually we need 2 relations for V (3,4) , then 7 for V (3,5) , 16 for V (3,7) and 18 for V (4,5) . Auxiliary lemmas, introducing the relations in the cases of (3, 7)-and (4, 5)-curves, are given in Appendices E and F. To prove Theorem 7.1 we again use the equality, similar to the one for hyperelliptic case from [6], where the rational function φ 2g x(ξ), y(ξ); u can be found with the help of (4.7) as well as the relations between Abelian functions.

Concluding remarks
In the paper we introduce regularization procedure for the second kind integral defined on a plane algebraic curve. This regularization is an extension of the standard textbook regularization, known for the elliptic case, to a wider collection of curves, namely (n, s)-curves. The proposed regularization is combined with parameterization in the vicinity of infinity, which is a special point on an (n, s)-curve where all sheets come together. We prove the importance of a correct regularization constant in the definition of the second kind integral (3.1) parameterized near infinity. The regularization constant does not serve as an integration constant, it arises only under parameterization and corresponds to the case of zero integration constant. The choice of the constant is significant in connection to the definition of the primitive function ψ, see (2.3), as a function of the complex parameter ξ near infinity. As conjectured in [6] and proven for some particular cases in the present paper, the primitive function coincides with a certain derivative of sigma-function on the Abel's image of ξ, see Remark 4.3. This is true only for the correct choice of the regularization constant in the definition of the second kind integral. Furthermore, the primitive function occurs in polylinear equalities of the forms (4.14) and (5.6), introduced in [6] with regard to hyperelliptic curves. We call the equalities polylinear, in general, or bilinear and trilinear in the mentioned cases, since the left hand sides are results of the action of polylinear operators, for more details see [6]. A bilinear equality of the type (4.14) allows to obtain relations between Abelian functions, as seen from Proofs 2 of Theorems 4.1 and 6.1. Such equalities are used in the proof of Theorem 7.1. In the paper we compare the relations between Abelian functions, obtained from the bilinear equality relating to a curve under consideration, with the same relations obtained independently, and compute a regularization constant. Conversely, when the regularization constant of the second kind integral on the curve is known, the bilinear equality produces all relations between Abelian functions. A trilinear equality of the type (5.6) leads to addition formulas, as we show in Section 5.2. So the primitive function provides a new way to obtain relations between Abelian functions, and addition laws. And the correct regularization constant is essential for obtaining consistent relations and true addition laws.
In addition to these results, we suggest a new technique of obtaining relations between Abelian functions, that is to derive the relations from (4.7) where the second kind integral on a divisor is computed by the residue theorem. In the paper we consider the difference of the second kind integrals on two divisors in order to cancel the regularization constant. This technique is displayed by the examples of proving Lemmas 4.5 and 6.3, and was used to prove Lemmas E.1 and F.1. The technique is applicable to any curve, and easy to use, unlike the known way using Klein's formula. The latter demands the knowledge of Klein's fundamental 2-form, which is another problem, rather difficult for non-hyperelliptic case. Recently, in [13] it was solved for a wide class of curves including (n, s)-curves. In the case of trigonal curves the problem was solved before in [3].

A First kind and associated second kind differentials
According to the scheme proposed in [7] we compute the second kind differentials associated to the standard first kind differentials. The both collections are holomorphic on the corresponding curve punctured at infinity.
The proof is similar to the proofs of Lemmas 4.5 and 6.3.