ABSTRACT: 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
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
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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