ABSTRACT: 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.

This discussion is related to https://doi.org/10.48550/arXiv.2407.05632

CITATION (original article): Julia Bernatska, Computation of ℘-functions on plane algebraic curves, arXiv:2407.05632.

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 $4$.