ABSTRACT (original article): Numerical tools for computation of ℘-functions, also known as Kleinian, or multiply periodic, are proposed. In this connection, computation of periods of the both first and second kinds is reconsidered. An analytical approach to constructing the Reimann surface of a plane algebraic curves of low gonalities is used. The approach is based on explicit radical solutions to quadratic, cubic, and quartic equations, which serve for hyperelliptic, trigonal, and tetragonal curves, respectively. The proposed analytical construction of the Riemann surface gives a full control over computation of the Abel image of any point on a curve. Therefore, computation of ℘-functions on given divisors can be done directly. An alternative computation with the help of the Jacobi inversion problem is used for verification. Hyperelliptic and trigonal curves are considered in detail, and illustrated by examples. CITATION (original article): Julia Bernatska, Computation of ℘-functions on plane algebraic curves, arXiv:2407.05632. https://doi.org/10.48550/arXiv.2407.05632
In this discussion 2-, 3-, and 4-torsion divisors are computed directly from points of the Jacobian variety of a genus 4 hyperelliptic curve in the canonical form. Computation of $\wp$-functions, and a solution of the Jacobi inversion problem are employed. The proposed technique works for $n$-torsion points with arbitrary $n$.
Preliminaries
The curve $\mathcal{C}$ from discussion "Uniformization of a genus 4 hyperelliptic curve with arbitrary complex branch points" is used. Periods and first kind integrals between branch points are previously computed.
Let $D$ be a positive divisor on the symmetric product $\mathcal{C}^4$. Such a divisor, which contains $4$ points, finite or infinite, and no points in involution, is called a reduced divisor. Every class of equivalence has a representative among reduced divisors. Depending on the degree, $\deg D_k = k$, we distinguish 4 types of non-trivial divisors:
- $D_4=P_1+P_2+P_3+P_4$, all four points are finite,
- $D_3=P_1+P_2+P_3+\infty$,
- $D_2=P_1+P_2+2\infty$,
- $D_1=P_1+3\infty$.
These types cover all special and non-special divisors on a curve of genus $4$. Divisor $\mathcal{O}=\{4\infty\}$ serves as neutral on $\mathcal{C}^4$, and $\deg \mathcal{O} = 0$.
Definition. A divisor $D$ is $n$-torsion if $n D \sim \mathcal{O}$, and there exists no $\widetilde{n} < n$ such that $\widetilde{n} D \sim \mathcal{O} $.
We start with a point $u[\varepsilon]$ of the Jacobian variety $\mathrm{Jac}(C)=\mathbb{C}^4 / \{\omega,\omega'\}$, which is defined by a characteristic $[\varepsilon]$ as follows $$ u[\varepsilon] = \omega\, \varepsilon + \omega' \varepsilon'. $$ Here $\omega$, and $\omega'$ are not normalized first kind period matrices along $\mathfrak{a}$-, and $\mathfrak{b}$- cycles, correspondingly. A characteristic $[ \varepsilon]= (\varepsilon', \varepsilon)^t$ is a $2\times 4$ matrix with entries within the interval $[0,1)$. Thus, $u[\varepsilon]$ is located within the fundamental domain of $\mathrm{Jac}(C)$.
If $\sigma(u[\varepsilon]) \neq 0$, or $\theta[K](\omega^{-1}u[\varepsilon]) \neq 0$, which is the same, then the Abel pre-image $\mathcal{A}^{-1} (u[\varepsilon])$ is a non-special divisor, and can be found from the solution of the Jacobi inversion problem
$$\mathcal{R}_8(x;u) \equiv x^4 - x^3 \wp_{1,1}(u) + x^2 \wp_{1,3}(u) + x \wp_{1,5}(u) + \wp_{1,7}(u) =0,$$
$$\mathcal{R}_9(x,y;u) \equiv 2y + x^3 \wp_{1,1,1}(u) + x^2 \wp_{1,1,3}(u) + x \wp_{1,1,5}(u) + \wp_{1,1,7}(u) =0.$$
On the curve under consideration, n-torsion divisors with $n>2$ are non-special. Special divisors arise among $2$-torsion divisors, which are composed of branch points. It is easy to construct special $2$-torsion divisors from the correspondence between characteristics $[\varepsilon]+[K]\;(\text{mod } 2)$ and partitions of the set of indices of finite branch points, see [Fay, Theta functions on Riemann surfaces, Lectures Notes in Math. (Berlin), vol. 352, Springer, 1973].
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