Abstract
In this paper, we propose a method of solving the Jacobi inversion problem in terms of multiply periodic \(\wp \) functions, also called Kleinian \(\wp \) functions. This result is based on the recently developed theory of multivariable sigma functions for (n, s)-curves. Considering (n, s)-curves as canonical representatives in the corresponding classes of bi-rationally equivalent plane algebraic curves, we claim that the Jacobi inversion problem on plane algebraic curves is solved completely. Explicit solutions on trigonal, tetragonal and pentagonal curves are given as an illustration.
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Notes
A solution of the Jacobi inversion problem is supposed to be a divisor of degree equal to the genus of a curve. If a divisor of degree greater than the genus of a curve is required, then the problem of inverting Abel’s map is called the extended Jacobi inversion problem.
The given formulas are obtained with the following differentials of the first and second kinds
$$\begin{aligned} \textrm{d}u_1&= \frac{x\,\textrm{d}x}{-2y},\quad \textrm{d}r_3 = \frac{x^2 \textrm{d}x}{-2y},\\ \textrm{d}u_3&= \frac{\textrm{d}x}{-2y},\quad \textrm{d}r_1 = (3x^3 +\lambda _4 x) \frac{\textrm{d}x}{-2y}. \end{aligned}$$A \((2,2g+1)\)-curve serves as a canonical form of hyperelliptic curves of genus g.
We say that n points \((a,b_k)\), \(k=1\), ..., n, are connected by involution of an (n, s)-curve \(f(x,y)=0\) if \(b_k\) give all solutions of f(a, y)=0.
The reader can find a solution of the regularization problem for integrals of the second kind on non-hyperelliptic curves in [5], in particular, \(c_1 = 0\), \(c_2 = -\lambda _2/3\) on \(\mathcal {V}_{(3,4)}\), and \(c_1 = 0\), \(c_2 = -2\lambda _2/3\) on \(\mathcal {V}_{(3,7)}\).
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Bernatska, J., Leykin, D. Solution of the Jacobi inversion problem on non-hyperelliptic curves. Lett Math Phys 113, 110 (2023). https://doi.org/10.1007/s11005-023-01726-3
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DOI: https://doi.org/10.1007/s11005-023-01726-3
Keywords
- Multiply periodic function
- Multivariable sigma function
- (n
- s)-curve
- Trigonal
- Tetragonal and pentagonal curves