Abstract

Below, a method of constructing cyclic $n$-isogenies on a genus two curve is implemented, and parameters of the image curve are computed in the cases of $n=2$, $3$, $4$, $5$, $6$, $7$. This 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. In fact, only four of then $\{\lambda_4,\lambda_6,\lambda_8,\lambda_{10}\}$ are essential, since any equation of a genus two curve is transformed bi-rationally to the form (C25Eq) with $\lambda_2=0$. As we will see from transformations of the parameters under $n$-isogenies, $\lambda_2$ does not change, which means it is not involved into such transformations.

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: $\mathrm{wgt}\ u_i = - i$, $\mathrm{wgt}\ r_i = i$. Integrals of the first kind differentials 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 differentials along $\mathfrak{a}$-, and $\mathfrak{b}$-cycles are composed in 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 $\Sigma = \{u \in \mathrm{Jac}(\mathcal{C}) \mid \sigma(u)=0\}$ 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 $u \in \mathrm{Jac}(\mathcal{C}) \backslash \Sigma$ 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} $

Main result

Let $D_{2P} = (x_1,y_1) + (x_2,y_2)$ be a non-special divisor from $\mathcal{C}^2$, which is defined by coordinates $\alpha_2^{[P]}$,
$\alpha_4^{[P]}$, $\beta_3^{[P]}$, $\beta_5^{[P]}$, according to (JacCoords) and (C25Unif). Let $D_{2Q}$ be an $n$-torsion, non-special divisor,
defined by coordinates $\alpha_2^{[Q]}$, $\alpha_4^{[Q]}$, $\beta_3^{[Q]}$, $\beta_5^{[Q]}$. Let $D_{2Q}$ generate a cyclic group

$$C_n = \langle O,D_{2Q}\rangle,$$ where $O$ denotes the neutral element in $\mathcal{C}^2$, which is $O=2\infty$. Let $C_n' \equiv C_n \backslash \{O\}$

A cyclic $n$-isogeny $\phi_n$ with such $C_n$ as a kernel is produced by the following analogue of Velu's formulas in genus two: $$\begin{split} &\tilde{\alpha}_2 = \alpha_2^{[P]} + \sum_{Q \in C'_n} \big( \alpha_2^{[P+Q]} - \alpha_2^{[Q]} \big) \\ &\tilde{\alpha}_4 = \alpha_4^{[P]} + \sum_{Q \in C'_n} \big( \alpha_4^{[P+Q]} - \alpha_4^{[Q]} \big) \end{split}\qquad \begin{split} &\bar{\beta}_3 = \beta_3^{[P]} + \sum_{Q \in C'_n} \beta_3^{[P+Q]} ,\\ &\bar{\beta}_5 = \beta_3^{[P]} + \sum_{Q \in C'_n} \beta_5^{[P+Q]}. \end{split} $$ Here $\tilde{\alpha}_2$, $\tilde{\alpha}_4$ serve as new $\alpha$-coordinates on the image Jacobian variety $\mathrm{Jac} (\tilde{\mathcal{C}})$. At the same time, the given expressions for $\bar{\beta}_3$, $\bar{\beta}_5$ do not provide new $\beta$-coordinates on $\mathrm{Jac} (\tilde{\mathcal{C}})$, since the corresponding homomorphism of Jacobian varieties $\hat{\phi}_n$ is not trivial. The new $\beta$-coordinates are obtained by

$$\begin{split} &\tilde{\beta}_3 = \bar{\beta}_3 \frac{\partial u_1}{\partial U_1} + \bar{\beta}_5 \frac{\partial u_3}{\partial U_1},\\ &\tilde{\beta}_5 = \bar{\beta}_3 \frac{\partial u_1}{\partial U_3} + \bar{\beta}_5 \frac{\partial u_3}{\partial U_3}, \end{split} $$ where $(u_1, u_3)\mapsto$ $(U_1,U_3$ $)$ under $\hat{\phi}_n$.

We use the fact, that on the boundary $(x_2,y_2) \to$ $(x_1,- y_1$ $)$, which turns $D_{2P}$ into a pair of points in involution, and places the Abel image of $D_{2P}$ on the Kummer variety, the homomorphism $\hat{\phi}_n$ is trivial, and $\tilde{\beta}_3 = \bar{\beta}_3$, $\tilde{\beta}_5 = \bar{\beta}_5$. Thererfore, parameters $\mu$ of the image curve $\tilde{\mathcal{C}}$:

$$ f(X,Y;\mu) = - Y^2 + X^5 + \mu_2 X^4 + \mu_4 X^3 + \mu_6 X^2 + \mu_8 X + \mu_{10} = 0, $$ such that $\tilde{\mathcal{C}}^2 = \mathcal{C}^2 / C_n$, are obtained from expansions of $\tilde{J}_8(\tilde{\alpha}_2, \tilde{\alpha}_4, \bar{\beta}_3, \bar{\beta}_5; \mu$ $)$ and $\tilde{J}_{10}$ $(\tilde{\alpha}_2, \tilde{\alpha}_4, \bar{\beta}_3,\bar{\beta}_5; \mu$ $)$ in the local parameter $\xi$ in the vicinity of infinity on $\mathcal{C}$, introduced by $$x(\xi) = \xi^{-2},\quad y(\xi) = \xi^{-5} \big( 1 + \tfrac{1}{2} \lambda_2 \xi^2 + \tfrac{1}{2} (\lambda_4 - \tfrac{1}{4} \lambda^2_2 ) \xi^4 + \mathrm{O}(\xi^6)\big). $$

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