Abstract

On a given curve, we obtain division polynomials for 3- and 4-torsion divisors. We also find all 3- and 4-torsion divisors by means of (i) solving the Jacobi inversion problem, and (ii) solving the division polynomials, and compare these results. This post serves as a supplementary material to arXiv:2412.10284

Preliminaries

Let a genus two curve $\mathcal{C}$ be defined by the equation $$f(x,y;\lambda) \equiv - y^2 + x^5 + \lambda_2 x^4 + \lambda_4 x^3 + \lambda_6 x^2 + \lambda_8 x + \lambda_{10}.\qquad (C25Eq) $$ Coefficients $\lambda \equiv \{\lambda_2,\lambda_4,\lambda_6,\lambda_8,\lambda_{10}\}$ serve as parameters of the curve. The curve has the form of a $(2,5)$-curve. The Sato weights are $\mathrm{wgt}\ x=2$, $\mathrm{wgt}\ y=5$, and the indices of $\lambda$ show their weights.

The curve possesses canonical cycles $\mathfrak{a}_1$, $\mathfrak{b}_1$, $\mathfrak{a}_2$, $\mathfrak{a}_2$, and a system of associated first and second kind differentials:

$\begin{split} &\mathrm{d} u= \begin{pmatrix} \mathrm{d} u_1 \\ \mathrm{d} u_3 \end{pmatrix} = \begin{pmatrix} x \\ 1 \end{pmatrix} \frac{\mathrm{d}x}{\partial_y f(x,y;\lambda)},\quad \mathrm{d} r = \begin{pmatrix} \mathrm{d} r_1 \\ \mathrm{d} r_3 \end{pmatrix} = \begin{pmatrix} x^2 \\ 3 x^3 +2 \lambda_2 x^2 + \lambda_4 x \end{pmatrix} \frac{\mathrm{d}x}{\partial_y f(x,y;\lambda)}. \end{split} $

Indices of the differentials indicate their weights: $\mathrm{wgt}\ u_i = - i$, $\mathrm{wgt}\ r_i = i$.

First kind integrals along $\mathfrak{a}$-, and $\mathfrak{b}$-cycles form a lattice of periods $\{\omega, \omega'\}$; and we define the Jacobian variety of $\mathcal{C}$ as $\mathrm{Jac}(\mathcal{C}) = \mathbb{C}^2 / \{\omega, \omega'\}$. Similarly, integrals of the second kind along $\mathfrak{a}$-, and $\mathfrak{b}$-cycles form two matrices $\eta$, and $\eta'$.

The curve $\mathcal{C}$ is equipped with the modular-invariant, entire $\sigma$-function, see [2, Chap. 3], which has the weight $\mathrm{wgt}\ \sigma = - 3$. With the help of $\sigma$-function abelian functions are generated by $$ \wp_{i,j}(u) = -\frac{\partial^2}{\partial u_i \partial u_j} \log \sigma(u),\qquad \wp_{i,j,k} (u) = -\frac{\partial^3}{\partial u_i \partial u_j \partial u_k} \log \sigma(u) ,\quad \text{etc.} $$ According to the Riemann vanishing theorem $\sigma(u) = 0$ if and only if the Abel pre-image of $u$ is a special divisor. Let $(\sigma)_0 = \{u \in \mathrm{Jac}(\mathcal{C}) \mid \sigma(u)=0\}$ be the theta-divisor in $\mathrm{Jac}(\mathcal{C})$.

Uniformization of the curve $\mathcal{C}$

is given by two polynomial functions of weights $4$ and $5$:

$\tag{C25Unif} \begin{split} &\mathcal{R}_4(x;u)= x^2 - x \wp_{1,1}(u) - \wp_{1,3}(u),\\ &\mathcal{R}_5(x,y;u)= y + \tfrac{1}{2} x \wp_{1,1,1}(u) + \tfrac{1}{2} \wp_{1,1,3}(u). \end{split} $

Actually, the Abel pre-image of $u \in \mathrm{Jac}(\mathcal{C}) \backslash (\sigma)_0$ is the common divisor of zeros of the two functions $\mathcal{R}_4$, $\mathcal{R}_5$ defined by (C25Unif). This divisor is non-special.

On the other hand, the two polynomial functions $\mathcal{R}_4$, $\mathcal{R}_5$ can be considered as elements of the ring $\mathbb{C}[x,y]/f(x,y;\lambda)$, namely

$\tag{JacCoords} \begin{split} &\mathcal{R}_4(x;u) = x^2 + \alpha_2 x + \alpha_4,\\ &\mathcal{R}_5(x,y;u) = y + \beta_3 x + \beta_5, \end{split} $ and $\alpha_2$, $\alpha_4$, $\beta_3$, $\beta_5$ serve as coordinates on $\mathrm{Jac}(\mathcal{C})$, and satisfy the equations which define the Jacobian variety, see [1, Theorem 3.2]:

$\tag{JacEqs} \begin{split} &J_8(\alpha_2,\alpha_4,\beta_3,\beta_5;\lambda) \equiv 2\beta_3 \beta_5 - \alpha_2^2 \alpha_4 - \alpha_4^2 + \lambda_4 \alpha_4 -\lambda_8 \\ &\qquad - \alpha_2 \big(\beta_3^2 + \alpha_2^3 - 4 \alpha_2 \alpha_4 + \lambda_2 (2 \alpha_4 - \alpha_2^2) + \lambda_4 \alpha_2 - \lambda_6 \big) = 0, \\ &J_{10}(\alpha_2,\alpha_4,\beta_3,\beta_5;\lambda) \equiv \beta_5^2 - 2 \alpha_2 \alpha_4^2 + \lambda_2 \alpha_4^2 - \lambda_{10} \\ &\qquad - \alpha_4 \big(\beta_3^2 + \alpha_2^3 - 4 \alpha_2 \alpha_4 + \lambda_2 (2 \alpha_4 - \alpha_2^2) + \lambda_4 \alpha_2 - \lambda_6 \big) = 0. \end{split} $

We call $\alpha_2$, $\alpha_4$, $\beta_3$, $\beta_5$ the Mumford coordinates of a non-special divisor.

Divisors

Every class of equivalent divisors on $\mathcal{C}$ has a representative $P_1 + P_2 - 2 \infty$, and $P_1$, $P_2$ are not in involution, that is $P_2 \neq -P_1$. Such a representative is called a reduced divisor. Since the poles are located at infinity, which serves as the basepoint, it is convenient to define every reduced divisor by its positive part, as follows

• non-special $D_2 = (x_1,y_1)+(x_2,y_2)$ of degree $2$,
• special $D_1=(x_1,y_1) + \infty$ of degree $1$,
• neutral $O = 2\infty$ of degree $0$, $u(2\infty)=0$.

Addition and duplication on $\mathcal{C}$

is described in detail in arXiv:2412.10284, Sect.4. We will use final formulas in terms of the Mumford coordinates.

$n$-Torsion divisors

Let $D \in \mathcal{C}^2$ be a reduced divisor, special or non-special. We say that $D$ is an $n$-torsion divisor, if $n D \sim O$, where $O = 2\infty$ denotes the neutral divisor, and $D$ generates a cyclic group of order $n$: $C_n = \langle O, D \rangle$. This definition implies the following criterion.

Theorem. A divisor $D \in \mathcal{C}^2$ is $n$-torsion when the following holds

(i) $(k+1) D \sim - k D$, if $n = 2k+1$; or

(ii) $k D \sim - k D$, if $n=2k$.

Remark. $n$-Torsion divisors are the Abel pre-images of points of order $n$ on $\text{Jac}(\mathcal{C})$, that is, of $u[\varepsilon] \in \text{Jac}(\mathcal{C})$ computed from characteristics $[\varepsilon]$ of order $n$ by

$u[\varepsilon] = \omega \varepsilon + \omega' \varepsilon'.$

Condition (i) in Theorem implies $$ \alpha_2^{[(k\,{+}\,1)D]} = \alpha_2^{[k D]},\qquad \alpha_4^{[(k\,{+}\,1)D]} = \alpha_4^{[k D]}. \qquad (OddTorsCond) $$ Relations for $\beta$-coordinates in this case are trivial.

Condition (ii) implies that $k D$ is $2$-torsion, that is

$$ \beta_3^{[k D]} = - \beta_3^{[k D]} =0,\qquad \beta_5^{[k D]} = -\beta_5^{[k D]} =0,\qquad (EvenTorsCond) $$

if $k D$ is non-special. If $k D$ is special, then $k D \sim$ $\,(x_{k D}, y_{k D}) $, and instead of (EvenTorsCond) we have

$$ y_{k D} = -y_{k D} =0. \qquad\qquad (EvenTorsCond') $$ Relations for $\alpha$-coordinates in case (ii) are trivial. Equations (OddTorsCond) and (EvenTorsCond) written in terms of $\alpha_2^{[D]}$, $\alpha_4^{[D]}$, $\beta_3^{[D]}$, $\beta_5^{[D]}$ can be considered as division polynomials in Mumford coordinates.

$2$-Torsion divisors

are defined by the condition $y_i = 0$, which implies that $x_i$ are $x$-coordinates of branch points. That is, $2$-torsion divisors are Abel pre-images of half-periods on $\text{Jac}(\mathcal{C})$. Among fifteen $2$-torsion divisors, ten are composed of two finite branch points, and correspond to even characteristics. The remaining five $2$-torsion divisors are composed of infinity and one finite branch point, and correspond to odd characteristics.

$3$-Torsion divisors

There are exist $80$ characteristics of order $3$, and the corresponding divisors are all non-special. Let $D_{2Q} = (x_1,y_1)+(x_2,y_2)$ be a $3$-torsion divisor, which means that $2 D_{2Q} \sim$ $\,- D_{2Q}$. From (OddTorsCond) we find the criterion $$\alpha_2^{[2Q]} = \alpha_2^{[Q]},\qquad \alpha_4^{[2Q]} = \alpha_4^{[Q]}. $$ At the same time, $\beta_3^{[2Q]} = - \beta_3^{[Q]}$, $\beta_5^{[2Q]} = - \beta_5^{[Q]}$. Therefore, $80$ $3$-torsion divisors split into $40$ pairs $D_{2Q} $, $-D_{2Q} $. In each pair $\alpha$-coordinates are the same.

Applying the duplication law we obtain the equations which define $3$-torsion divisors in terms of Mumford coordinates ( $\alpha_2^{[Q]} \equiv \alpha_2$, $\alpha_4^{[Q]} \equiv \alpha_4$): $$ 3\alpha_2 = 2\gamma_2 - \gamma_1^2, $$ $$ 3 \alpha_4 = 3 \alpha_2^2 - 2 \alpha_2 \big(2\gamma_2 - \gamma_1^2\big) + 2 \gamma_4 + \gamma_2^2 - \lambda_2 \gamma_1^2, $$

where parameters $\gamma_k$ are expressed in terms of $\alpha_2$, $\alpha_4$, $\beta_3$, $\beta_5$.

Expressions in terms of $x$-, $y$-coordinates are derived below.

$4$-Torsion divisors

There exist $240$ characteristics of order 4 excluding half-integer characteristics. Let $\mathfrak{E}$ denote the set of these $240$ characteristics. Each characteristic of $\mathfrak{E}$ produces a $4$-torsion divisor $D_{2Q}=(x_1,y_1)+(x_2,y_2)$, which is non-special. The $240$ characteristics of $\mathfrak{E}$ split into $15$ collections of $16$ characteristics each, such that for all $[\varepsilon]$ is a collection $[2\varepsilon]$ is a particular half-integer characteristic. Recall, that half-integer characteristics produce five special and ten non-special divisors. We split $\mathfrak{E}$ into two parts:

• $\mathfrak{E}_{\text{non-spec}} = \{[\varepsilon] \in \mathfrak{E} \mid \mathcal{A}^{-1}(u[2\varepsilon]) \,{=}\, D_2,\, \deg D_2 \,{=}\, 2\}$ of cardinality $160$, the divisor $D_2 \sim 2 D_{2Q}$ is characterized by its Mumford coordinates $\alpha_2^{[2Q]}$, $\alpha_4^{[2Q]}$, $\beta_3^{[2Q]}$, $\beta_5^{[2Q]}$;

• $\mathfrak{E}_{\text{spec}} = \{[\varepsilon] \in \mathfrak{E} \mid \mathcal{A}^{-1}(u[2\varepsilon]) \,{=}\, D_1,\, \deg D_1 \,{=}\, 1\}$ of cardinality $80$, the divisor $D_1 \sim 2 D_{2Q}$ is characterized by $x$-, $y$-coordinates: $D_1 = (x_{2Q},y_{2Q})$. \end{itemize}

Divisors $D_{2Q}$ produced from characteristics of $\mathfrak{E}_{\text{non-spec}}$ satisfy the conditions $$ \beta_3^{[2Q]} = 0,\qquad \beta_5^{[2Q]} = 0, $$ obtained from (EvenTorsCond). In the Mumford coordinates of $D_{2Q}$ the equalities acquire the form ( $\alpha_2^{[Q]} \equiv \alpha_2$, $\alpha_4^{[Q]} \equiv \alpha_4$) $$- 2 \alpha_4 - \alpha_2^2 + \big(2 \alpha_2 - \gamma_2 + \gamma_1^2\big) \big(\gamma_2 - \gamma_1^2\big) + \gamma_1^2 \big(\gamma_2 - \lambda_2\big) + \gamma_4 = 0, $$ $$ \big({-} 2 \alpha_4 + (3 \alpha_2 - \gamma_2)(\alpha_2 - \gamma_2) + \gamma_1^2 (2 \alpha_2 - \lambda_2) + 2 \gamma_4 \big) \big(2 \alpha_2 - \gamma_2 + \gamma_1^2\big) = \gamma_6, $$ where $\gamma_k$ are expressed in terms of $\alpha_2$, $\alpha_4$, $\beta_3$, $\beta_5$.

Divisors $D_{2Q}$ produced from characteristics of $\mathfrak{E}_{\text{spec}}$ satisfy the condition $y_{2Q} \,{=}\, 0$, which follows from (EvenTorsCond'), and in terms of the Mumford coordinates of $D_{2Q}$ acquires the form $$\big(\gamma_1 (2 \alpha_2 + \gamma_1^2 - \lambda_2) + \gamma_3\big) \big(2 \alpha_2 + \gamma_1^2 - \lambda_2 \big) + \gamma_5 = 0, $$ where $\gamma_k$ are expressed in terms of $\alpha_2$, $\alpha_4$, $\beta_3$, $\beta_5$.

References

[1] Buchstaber, V. M., Enolskii, V. Z., Leykin, D. V., Hyperelliptic Kleinian Functions and Applications, preprint ESI 380 (1996), Vienna

[2] Buchstaber, V. M., Enolskii, V. Z., Leikin, D. V., Multi-dimensional sigma-function, 2012, arXiv 1208.0990