Vector of Riemann constants

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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 [2407.05632] Computation of $\wp$-functions on plane algebraic curves

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 .

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