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

Introduction

An efficient criteria to single out the characteristic of vector of Riemann constants on an algebraic curve is suggested. It perfectly works on non-hyperelliptic and hyperelliptic curves, and allows to select the unique characteristic which corresponds to the vector of Riemann constants.

Preliminaries

Let $\mathcal{C}$ be an algebraic curve of genus $g$, with the Weierstrass gap sequence $\mathfrak{W} = \{\mathfrak{w}_1, \mathfrak{w}_2, \dots, \mathfrak{w}_g\}$. Note that $\mathfrak{w}_1=1$. Let $\mathrm{d} u = (\mathrm{d}u_1, \mathrm{d}u_{\mathfrak{w}_2},\dots, \mathrm{d}u_{\mathfrak{w}_g})^t$ be not normalized differentials of the first kind on $\mathcal{C}$. The Abel map $\mathcal{A}: \mathcal{C} \to \mathrm{Jac}(\mathcal{C}) $ is defined with respect to these not normalized differentials: $$ \mathcal{A}(P) = \int_{P_0}^P \mathrm{d}u. $$

Let $u=(u_1, u_{\mathfrak{w}_2},\dots, u_{\mathfrak{w}_g})^t$ be coordinates (not notmalized) on $\mathrm{Jac}(\mathcal{C}) = \mathbb{C}^g / \mathfrak{P}$.

The period lattice $ \mathfrak{P}$ is formed by first kind periods $$\omega_{\ast,i} = \int_{\mathfrak{a}_i} \mathrm{d} u,\qquad \omega'_{\ast,i} = \int_{\mathfrak{b}_i} \mathrm{d} u, $$ computed on a canonical homology basis $\{\mathfrak{a}_i, \mathfrak{b}_i \mid i=1,\dots g\}$. Then $\omega_{\ast,i}$ are columns of $\mathfrak{a}$-period matrix $\omega$, and $\omega'_{\ast,i}$ are columns of $\mathfrak{b}$-period matrix $\omega'$.

The Riemann period matrix is $\tau = \omega^{-1} \omega'$, and normalized coordinates on $\mathrm{Jac}(\mathcal{C})$ are $v=\omega^{-1} u$.

Let $[\varepsilon] = (\varepsilon',\varepsilon)^t$ be a characteristic such that $\varepsilon'$ and $\varepsilon$ are $g$-component vectors composed from numbers located within the interval $[0,1)$. Let $\theta$-function with characteristic $[\varepsilon]$ be defined as follows $$\theta [\varepsilon] (v ; \tau) = \sum_{n\in \mathbb{Z}^g} \exp\big(\imath \pi (n+\varepsilon')^t \tau (n+\varepsilon') + 2\imath \pi (n+\varepsilon')^t (v+\varepsilon) \big). $$ In computation we use the partial sum with $n_i \leqslant 5$. In what follows, we consider $\theta$ as a function of not normalized coordinates $u$, and write $\theta[\varepsilon](\omega^{-1} u,\omega^{-1} \omega)$ instead of $\theta [\varepsilon] (v ; \tau)$.

Criteria

$\theta[K](\omega^{-1} u,\omega^{-1} \omega)$ has the maximal order of vanishing at $u =0$, that is

$$\forall \mathfrak{i} < \mathfrak{d} \quad \partial^{\mathfrak{i} }_{u_1} \theta [K] (0; \tau) = 0, \qquad \partial^{\mathfrak{d}}_{u_1} \theta [K] (0; \tau) \neq 0, $$

where $\ \mathfrak{d} = -\text{wgt}\ \sigma = \frac{1}{24} (n^2-1)(s^2-1)$ is the weighted order of vanishing of $\theta[K]$.

For definition of $\text{wgt}\ \sigma$, see p.106, Section 7.3 in arXiv:1208.0990 or [BEL,Funct.\,An.\,and Its Appl.,33(2),1999]

Note, that derivatives of $\theta$ are taken with respect to not normalized coordinate $u_1$, that is $$\partial^{\mathfrak{i} }_{u_1} \theta [\varepsilon] (0; \tau) = \lim_{u \to 0} \frac{\partial^{\mathfrak{i}}}{\partial u_1^{\mathfrak{i}}} \theta [\varepsilon] (\omega^{-1} u,\omega^{-1} \omega). $$ The proposed criteria allows to single out the unique characteristic which corresponds to the vector of Riemann constants. A unique characteristic can be found on either non-hyperelliptic, or hyperelliptic curve. Regarding hyperelliptic curves, the proposed criteria finds out $[K]$ in the case of even genera.

Below, two examples are presented with curves considered previously:

(i) a trigonal curve of genus $3$,

(ii) a hyperelliptic curve of genus $4$.