Klein's Fundamental 2-Form of Second Kind for the $C_{ab}$ Curves

In this paper, we derive the exact formula of Klein's fundamental 2-form of second kind for the so-called $C_{ab}$ curves. The problem was initially solved by Klein in the 19th century for the hyper-elliptic curves, but little progress had been seen for its extension for more than 100 years. Recently, it has been addressed by several authors, and was solved for subclasses of the $C_{ab}$ curves whereas they found a way to find its individual solution numerically. The formula gives a standard cohomological basis for the curves, and has many applications in algebraic geometry, physics, and applied mathematics, not just analyzing sigma functions in a general way.


Introduction
Since Abel, Jacobi, Poincaré, and Riemann established its framework, theories of Abelian and modular functions associated with algebraic curves have been of crucial importance in algebraic geometry, physics, and applied mathematics. Algebraic curves in this paper are intended as compact Riemann surfaces.
The study of the hyper-elliptic curves goes back to the beginning of the 20th century, and these appear in much detail in advanced text-books such as Baker [2] and Cassels and Flynn [9]. However, little has been considered for more general curves than the hyper-elliptic curves. In this paper, we consider a class of curves (C ab curves) in the form i,j c i,j x i y j = 0, (1.1) i = 1, 2, . . . , g for the hyper-elliptic curve with c i,0 = c i , c i,1 = 0, i = 0, . . . , 2g + 1.
For example, for the hyper-elliptic curve (1.3), such a 2-form can be obtained using the canonical meromorphic differentials and 1-form It is known that the normalized symmetric bilinear R((x, y), (z, w)) can be expressed using (1.4) for C ab [17] and telescopic curves [1] although Ω((x, y), (z, w)) needs more extensions. However, for the canonical meromorphic differentials {dr i } g i=1 , no exact formula has been given. For example, C 34 and C 35 curves and C 45 curves were studied in [6] and [12], respectively. Recently, the reference [7] addressed C 3,g+1 curves with genus g. These results dealt with a specific class of curves and failed to obtain the formula for general C ab curves with arbitrary mutually prime a, b. As a result, still many researchers are either numerically calculating or algorithmically computing {dr i } g i=1 given {c i,j } and {du i } g i=1 . This paper extends (1.5) and obtains the general formula in a closed form for the C ab curves. It contains all the existing results and has many applications including algebraic expression of sigma functions [17], defining equations of the Jacobian varieties, etc. [5,10]. This paper is organized as follows: Section 2 sets up the holomorphic differentials and gives background of this paper. Section 3 states and proves the main result (formula) and gives two typical examples both of which extend the previous cases. In the last section, we raise an open problem.

Background
Let a, b be mutually prime positive integers, and C the curve defined by with a unique point O at which the zero orders of x and y are a and b, respectively, where (i, j) range over and c i,j are constants such that either We consider the set of 1-forms [17] We know by the general theory that for g variable points (x 1 , y 1 ), . . . , (x g , y g ) on C, the sum of integrals from O to those g points fill the whole space C g , where the weights of the variables u i,j are ab − a(i + 1) − b(j + 1). If we regard the weight of each coefficient c i,j in (1.1) is ai + bi − ab, the weights assigned to the differentials render F (x, y) homogeneous. 2) it has its only pole along the diagonal of C × C, and 3) in the vicinity of each point, it is expanded in power series as where t x,y and t z,w are local coordinates of points (x, y) and (z, w), respectively.
We shall look for a realization of R((x, y), (z, w)) in the form where G((x, y), (z, w)) is a symmetric polynomial in its variables.
Example 2.3 (hyper-elliptic curves [3,4]). For the hyper-elliptic curve (1.3), in which a = 2 and b = 2g + 1, where g is the genus of the curve C, then (1.1) expresses a hyper-elliptic curve, the meromorphic function on C × C to the numerator of (2.1), we obtain G((x, y), (z, w)) = 2 k : even for l = 1, 2, . . . , i/2 has been applied ( k/2 = k/2 and (k − 1)/2 when k is even and odd, respectively), which means that by choosing 3 The fundamental 2-form of the second kind for the C ab curves , H z := ∂H ∂z , and H w := ∂H ∂w . We find symmetric R((x, y), (z, w)) for the meromorphic function To this end, if we note dw dz = − Fz Fw , we have for i, j = 0, . . . , a. Then, from (3.2) and that the arithmetic is modulo which coincides with the numerator of (3.1).

8)
and Proof . First of all, we prove u r s c r,u c s,v t (12) In fact, we see where (r, s) range over s ≥ i + 1 in D. Thus, from the first and second terms in (3.3), we obtain (3.4) and (3.7). Similarly, where (r, s) range over r ≥ i + 1 in D. Thus, from the third and fourth terms in (3.3), we obtain (3.5) and (3.8).
On the other hand, we claim where p is a unique integer such that In fact, we see Similarly, we obtain where q is a unique integer such that From (3.11) and (3.13), we have two possibilities: p + q = r + s and p + q + 1 = r + s. For the former case, (3.10) and (3.12) are −(v − u)z q x p and −(v − u)z p x q , respectively; and for the latter case, (3.10) and (3.12) are

respectively.
Since we have (3.9), which means we obtain (3.6) as well v−1 k=u+1 r s c r,u c s,v ∆ where we have chosen for each (i, j) ∈ J(a, b), i = h−1 and j = u+v −k −1, so that 1 ≤ i+1 ≤ p and u Hence, (u, v) and (s, r) need to satisfy respectively. This completes the proof.
Proof . Let f u := 0≤ar+bu≤ab,r≥0 c r,u x r . Then, G u,v , G u,v , and G and converges to For the first part, without loss of generality, we assume a < b. For each 0 ≤ i ≤ b − 1, let f (i) be the number of integers j's such that Since for i = 0, . . . , b − 2, and f (b − 1) = 0, where x is the largest integer no more than x, we have Since m ≤ n, we have I(m, n) with i = n and j = m for i ≥ j + 1 and I(m, n) with i = m and j = n for i ≤ j − 1, so that (3.3) is obtained.