Modular Form Representation for Periods of Hyperelliptic Integrals

To every hyperelliptic curve one can assign the periods of the integrals over the holomorphic and the meromorphic differentials. By comparing two representations of the so-called projective connection it is possible to reexpress the latter periods by the first. This leads to expressions including only the curve's parameters $\lambda_j$ and modular forms. By a change of basis of the meromorphic differentials one can further simplify this expression. We discuss the advantages of these explicitly given bases, which we call Baker and Klein basis, respectively.


Introduction
Expressions of periods of hyperelliptic integrals in terms of θ-constants have a long history, starting with Rosenhain (1851) and Thomae (1866). Such a representation of second-kind differentials was a part of the program by Felix Klein (1886Klein ( , 1888 of constructing Abelian functions in terms of multi-variable σ-functions. Leaving aside considerations of expressions for higher genera analogues of the elliptic periods 2ω in terms of θ-constants, in this note we focus on expressions for the higher genera periods 2η, which in the Weierstrass theory is of the form η = − 1 12ω We present below an analogue of such an expression for hyperelliptic curves. This problem is interesting both for theoretical reasons and computational viewpoints for developing black box calculation methods for these quantities. In recent times, the σ-function has come into focus in many applications, e.g., in theoretical physics and mathematics (see, e.g., [5] and references therein).

The method
Consider a genus-g hyperelliptic curve C : f (x) − y 2 = 0, with f (x) = 2g+2 i=0 λ i · x i , λ i ∈ C, and a canonical homology basis (a 1 , . . . , a g , b 1 , . . . , b g ), a i • a j = ∅, b i • b j = ∅, a i • b j = δ i,j (see Fig. 1). Then fix a basis of 2g differentials in the form B = 1 y x i−1 · dx 1≤i≤2g , of which the first half are holomorphic and the second half are meromorphic. Though partly meromorphic, we refer to this basis as a cohomology basis. (The holomorphic part is in fact dual to the cycle basis.) This basis can be normalized in a form especially due to Baker [1]: of which again the first half are the (holomorphic) differentials of the first kind and the second half are the (meromorphic) differentials of the second kind. Defining now 2ω, 2ω as the aand b-periods of the integrals over the holomorphic differentials, and 2η, 2η as the respective periods of the integrals over the meromorphic differentials, these matrices fulfill the generalized Legendre relation which make this setup as a natural generalization of the theory of elliptic functions by Weierstrass. Analogously to the Weierstrass theory, we introduce Via κ we will calculate in the following the periods of second kind integrals Using the Riemann period-matrix τ = ω −1 · ω , we define the Riemann-θ-function with half-integer characteristics We call a characteristic even (resp. odd), if 4ε T ε mod 2 = 0 (resp. 1); the associated θ-function inherits the parity of the characteristic. We also can define all θ-constants: which is the number of non-singular even characteristics.
We can clarify the role of the characteristics more explicitly by pointing out their connection to the Abelian images of the branching points e i , which are in the hyperelliptic case the zeroes of f . That is, there exists a characteristic [ε j ], such that the following relation holds A j = (e j ,0) R 0 u =: 2ωε j + 2ω ε j , j = 1, . . . , 2g + 2.
We use the notation [ε j ] = [A j ], and hence the notion of even or odd branching points. Now consider a partition of the branching points, The divisors of zeros and poles, respectively, are linear equivalent and hence the Abel maps of these divisors coincide. Now, we have the tools to introduce the bi-differential Ω(P, Q) on C × C (P, Q), which is called the canonical bi-differential of the second kind if it is • symmetric Ω(P, Q) = Ω(Q, P ), • normalized at a-periods: a k Ω(P, Q) = 0, k = 1, . . . , g, • and has the only pole of the second order along the diagonal, namely it has the following expansion: Ω(P, Q) = dξ(P )dξ(Q) (ξ(P ) − ξ(Q)) 2 + 1 6 S(R) + higher order terms, where ξ(P ) and ξ(Q) are local coordinates of points P = (x, y) and Q = (z, w) in the vicinity of a point R, ξ(R) = 0 respectively.
The quantity S(R) is called the holomorphic projective connection (see [7, p. 19]; note that this object is sometimes called Bergman projective connection, but here we adopt the notion of Fay). Our purpose is now to express S(R) in two different ways, one containing κ, equate them and solve for κ.

Two representations of Ω(P, Q)
The canonical bi-differential is uniquely defined by the given conditions. But it has several representations, whose derivations are described in particular in [4]. We restrict this inspection to the Fay-representation and the Klein-Weierstrass-representation. Note that Fay is using a monic polynomial f (see [7, p. 20]), so in the following we set λ 2g+2 = 1. But this is no loss of generality, because we can rescale f as necessary after calculating κ, respectively all related objects. (In especially, for f → λ 2g+2 · f we have κ → λ 2g+2 · κ, η → λ 2g+2 · η and ω → ω/ λ 2g+2 .) Introducing the normalized holomorphic differentials v = (2ω) −1 u and non-singular odd characteristics [δ], we present two realizations The symmetric polynomial F (x, z) is the Kleinian 2-polar, which was introduced by Klein to represent the second kind bi-differential in the above given form.
From the first and second expression of equation (3.1) we can derive two different representations, which we will cite in the following propositions: where {·, ·} is the Schwartzian derivative The next result is taken from [4, p. 307], where the projective connection was constructed for a larger class of curves: Write the curve C as y 2 = f (x), so that a prime means differentiation with respect to x. Then all differentials evaluated at the point R = (x, y).
As stated above, S F ay and S KW coincide, so we can solve linearly for κ. We see that the Schwartzian derivatives cancel and as can be seen in particular in the term u T κu, only entries of κ along the anti-diagonals share the same order of x. Therefore we will solve order by order and for a first insight we will do so by expanding x in terms of a local coordinate ξ.

The results
Consider a partition I 0 ∪ J 0 = {1, . . . , 2g + 2}, and denote with S j (I), j = 0, . . . , g + 1 the elementary symmetric function of order j built in the branching points e i with indices taken from I, namely S 1 (I) = i∈I e i , S 2 (I) = i,k∈I; i≤k e i e k , etc. Example 4.1. For any even g = 2 hyperelliptic curve, κ is given as with ε according to the partition J 0 = {i, j, 6}. We observe that κ splits nicely into a transcendental part consisting of various θ-constants, and a rational part, which will be inspected more closely below.
After defining the column-vectors (V 1 , . . . , V g ) = (2ω) −1 we use a shorter notation by setting To avoid establishing the correspondence between branching points and characteristics in the given homology basis, we sum over all 2g+1 g possible partitions. The calculation time may grow, but as a result κ is expressible in terms of the parameters λ j of the curve: The same technique is applicable for g = 3, but with the difference, that the elements of the anti-diagonal share the same order in ξ. So at first, we only can derive the sum of the diagonal: Example 4.2. For any even g = 3 hyperelliptic curve, κ is given as Again, we can sum over all allowed [ε]. If we do so, we find that the symmetric functions 2X +Y will sum to 20λ 4 . Examining different (even, g = 3 and hyperelliptic) curves numerically, it is reasonable to assume that each entry depends linearly on a single λ i , i.e., there are no additional constants present. (Please note that this is at this point only an assumption, but the sum of 20λ 4 forbids many other possibilities.) This means that the sum over the entries X or Y are multiples of λ 4 . If so, the respective prefactors of λ 4 for the X-and Y -sums are independent of the specific curve (of this type). Hence we find the right separation of 20λ 4 by a numerical investigation of a concrete curve. That way we get up to numerical accuracy integer numbers In especially the two "partition numbers" at X and Y are 1 and 18. This summed version of κ relies on two numerically justified assumptions (linearity in λ i and integer solutions). The examination of higher genera will reveal a number pattern, which adds further indication to that structure. But a full prove will be given below in Section 6, where we develop a method to disentangle these partition numbers. For that we first need some statements about the structure of κ for arbitrary genus.

The general pattern
We want to generalize the representation of κ as it is in equations (4.2)  Summed over all N g non-singular even characteristics (given in (2.2)) and divided by their number we get symmetric expressions independent of the special characteristic and fixed homology basis. In that sense, the following theorem mainly states the splitting of κ into a modular part and a "residual" matrix Λ g , which we will get rid of in the last section.
Theorem 5.1. Let C be a hyperelliptic curve of genus g and introduce Baker's 2g-dimensional basis for the singular cohomology (equation (2.1)). Let 2ω, 2ω , 2η, 2η be period g × g matrices satisfying the generalized Legendre relation. Then κ = η(2ω) −1 is of the form We conjecture that the matrix Λ g consists of the parameters λ j of the curve, together with integer coefficients, the "partition numbers", as it was the case in genus 2 and (up to numerical uncertainty) in genus 3.
An anonymous referee pointed out a way to straighten the claims about Λ g . We give his idea in form of a lemma: Lemma 5.2. The before defined matrix Λ g is expressible as Defining the vector X = 1, x, . . . , x g−1 T changes the left-hand side of equation (5.3) to X T Λ g Xdx 2 and so it is clear, that the differentials cancel.
Proof . We can combine the proof of Lemma 5.2 and Theorem 5.1. From the comparison of equations (3.2) and (3.3) it is clear, that Λ g includes the following terms Using the elementary identities , and the equality we find after summation over all partitions I 0 ∪ J 0 , Furthermore, it is clear that Plugging these parts into the right-hand side of (5.4) we get the expression Because all of the f (x) and most of the f (x) cancel, this expression shrinks to Next, we investigate the term Further inspection of this double-sum leads to Plugging equation (5.6) into equation (5.5) and multiplying with f (x) = y 2 gives equation (5.3), but summed up to 2g + 2. The additional terms for k = 2g + 1 and k = 2g + 2 are zero, hence the statement of Lemma 5.2 follows. With the help of equation (5.3) we can now read off the sum of each anti-diagonal of Λ g by comparing the order of x, and we find that these anti-diagonal sums are integer multiples of a single λ j . Corollary 5.3. For given genus g let κ be of the form (5.2). Then the (k − 1)th anti-diagonal sum of Λ g is given as Σ g;k λ k with the integer number Partly, the structure of Λ g is obvious: Apparently, the matrix is symmetric along the main diagonal (because κ is, too). Furthermore the coefficients of the λ j are distributed symmetric also along the main anti-diagonal: Σ g;k is symmetric along k = g + 1.
So far, these statements concern the anti-diagonal sums of Λ g . As soon as it comes to the single entries we need another method to arrive at reliable claims. Such a method will be derived in Section 6 and exemplified there in a number of examples. Statements for arbitrary genus are under inspection right now, but still we can extrapolate the structure of Λ g : The above assumption of linearity with respect to a single λ j leads to the occurrence of the λ j (without partition numbers) in a Hankel-type structure. The distribution of the partition numbers on the other hand is not trivial. We believe that they are given in terms of combinatorial expressions, but by now their pattern is only partly revealed.
Assuming this structure, we can solve for the partition numbers for a given hyperelliptic curve (and thus for all of fixed genus). The next cases are Further inspection of the matrices lead to the observation: The kth entry of the first row (column) is 2g+1−k g−k λ 1+k and this row's coefficients sum to 2g+1 g−1 . We remark that for odd curves it was shown in [5] on examples g = 2, 3, 4 that this fact follows from the solution of the Jacobi inversion problem in terms of Klein-Weierstrass ℘ i,j -functions.
6 Formulae for entries of Λ g So far the full description of the matrix Λ g for arbitrary genus is still open. The main problem within the method presented here is that in equation (3.3) the vectors u, which appear in the quadratic form u T κu, are evaluated at the same point and hence don't lead to a full system of equations. In [5,Lemma 4.2] linear equations were derived for Λ [ε] g;i,j based on Baker's construction of Abelian functions in terms of Klein-Weierstrass functions ℘ i,j (documented in [1,2] and more recently by Buchstaber, Enolski and Leykin in [3] and others). In the derivation the constants Λ [ε] g;i,j appear as values of the ℘ i,j -functions for even non-singular halfperiods relevant for a partition I 0 .
A general formula for the integer entries of the matrix Λ g was not found there, but it was understood that finding of each such integer in higher genera using the above mentioned equations is an extremely time consuming procedure. Therefore we suggest to combine the derived formula for the anti-diagonal sums and the aforementioned formulae, which permit us to compute only a part of the entries to Λ g and substantially speed up the whole calculations because of that.
Here we present the derivation of a solvable system of linear equation for Λ [ε] g;i,j which is independent of the multi-variable σ-functions of Baker's theory. Namely we will prove the following Proposition 6.1. Let a hyperelliptic curve of genus g be realized in the form ) T , i ∈ I 0 , be g-vectors. Then the following formulae are valid Here, F (x, z) is the Kleinian 2-polar, whilst S Remark that (6.1) is analogous to the case of odd curves stated in [5] and is backed up with many computer experiments. We omit here the proof of the second identity and place emphasis on the proof of the first identity, which will be given below.
Writing equations (6.1) for all possible pairs {i, j} ∈ {i 1 , . . . , i g+1 } we get g(g +1)/2 equations with respect to g(g+1)/2 quantities Λ [ε] g;i,j , 1 ≤ i < j ≤ g. It is convenient to introduce g(g+1)/2vectors, Due to the symmetry of Λ g only includes the entries with ordered indices. Now the above described equations can be written in the short form g is a g(g + 1)/2 × g(g + 1)/2-matrix of the following form: Each row's entry M g;i,j . Now take the matrix Then we have M g .
When Λ [ε] g is found according to (6.3) it remains to sum over all partitions by g + 1 elements of {1, . . . , 2g + 2} to find the partition numbers of the anti-diagonal sums we were looking for.
Proof . Now to the proof of Proposition 6.1, which accords with the anonymous referee's suggestion to implement Corollary 2.12 of [7, p. 28]. This corollary represents a remarkable relation between the canonical bi-differential Ω(P, Q) (see equation (3.1)) and the Szegö kernel R (P, Q|e): Here, P = (x, y), Q = (z, w) are points on the curve, e is a vector for which θ(e) = 0, and with E(P, Q) being the Schottky-Klein prime-form, the Szegö kernel is given as Our purpose demands e to be a non-singular even half period associated to the partition I 0 = {i 1 , . . . , i g+1 }; we will denote further [e] = [ε]. Fay states [7, p. 13] an algebraic representation of the Szegö kernel of a hyperelliptic curve associated to non-singular even half-periods: Note that in (6.4) all terms are evaluated at two points P and Q and therefore presumably deliver a full system. Moreover, for even half periods the second logarithmic derivative of θ reduces to θ /θ. Together, (6.4) takes the form where the algebraic representation of the canonical bi-differential was used.
This is the "unsummed", characteristic dependent representation (5.1) of κ, hence the matrix Λ g has the index [ε], indicating it consists of symmetric functions like it was in equations (4.1) and (4.3).
Because of the representation of κ in this form, the last two terms on the right-hand side of equation (6.5) are reduced to g;ij u i (P )u j (Q).
To complete the calculations it is enough to find the integers X and Y . To calculate X fix a partition, e.g., I 0 = {1, 2, 3, 4, 5, 6}, and denote with s j the elementary symmetric functions of the branching points with indices from the set I 0 and denote s j the same functions from the complementary set. Solving equations (6.2) for this partition we find X = 2s 6 + 2 s 6 + s 1 s 5 + s 5 s 1 .

Choice of basis
In this paper we used a basis due to Baker for a 2g-dimensional space of singular cohomologies. Our results were derived and numerically tested within this basis. But there is a certain freedom of choice for this basis, namely we are able to add a linear combination of holomorphic differentials to the meromorphic differentials without changing the Legendre relation. As a result κ changes by a certain matrix (see [1, p. 328]): If r → r − C · u with a symmetric g × g matrix C, then κ → κ + C/2. Consequently, if we fix C = − 1 4 1 Ng Λ g , our main result will change to κ = − Of course, for this new basis of meromorphic differentials, one has to know explicitly the matrix Λ g . The change of basis is in that sense only a reformulation of Theorem 5.1. But it is important to us to point out the existence of such a basis. Formula (7.1) first appears in F. Klein [8,9], then it was recently revisited in a more general context by Korotkin and Shramchenko [10]. Let us call a cohomology basis B = {u 1 , . . . , u g , r 1 , . . . , r g } a Klein basis, if the associated κ-matrix has a representation of the form (7.1).
Our development results in the following