Listing superspecial curves of genus three by using Richelot isogeny graph

Ryo Ohashi, Hiroshi Onuki, Momonari Kudo,
Ryo Yoshizumi and Koji Nuida

Abstract.

In algebraic geometry, superspecial curves are important research objects. While the number of superspecial genus-3 curves in characteristic $p$ is known, the number of hyperelliptic ones among them has not been determined even for small $p$ . In this paper, in order to compute the latter number, we give an algorithm for computing the Richelot isogeny graph of superspecial abelian threefolds by using theta functions. Our algorithm enables us to efficiently list superspecial genus-3 curves, and we succeeded in counting hyperelliptic curves among them when $11\leq p<100$ by executing our algorithm in Magma.

1. Introduction

Throughout this paper, by a curve we mean a non-singular projective variety of dimension one and isomorphisms between two curves are ones over an algebraically closed field. A curve $C$ over an algebraically closed field $k$ of characteristic $p>0$ is called superspecial if the Jacobian of $C$ is isomorphic to the product of supersingular elliptic curves. In recent years, superspecial curves have been used for isogeny-based cryptography (e.g. [18], [6], [2] and so on), and studies on such curves are becoming more important.

For given integers $g$ and $p$ , it is known that the number of isomorphism classes of superspecial curves of genus $g$ in characteristic $p$ is finite, and determining that number is an important problem. This problem was already solved for genus $g\leq 3$ ; Deuring [12] found the number of all supersingular elliptic curves (i.e. $g=1$ ), and the number for $g=2$ is given by Ibukiyama, Katsura, and Oort [24]. For the case of $g=3$ , Oort [41] and Ibukiyama [23] showed there exists a superspecial curve of genus $3$ in arbitrary characteristic $p\geq 3$ , and the number of such curves was also computed by Brock [4]. However, the number of a superspecial hyperelliptic curve of genus 3 is not known, and in fact, even the existence of such a curve in arbitrary characteristic $p\geq 3$ has not been shown.

Our aim is to clarify the above problems for superspecial hyperelliptic curves of genus 3 in small characteristic $p$ . For this, we utilize the connectivity of the Richelot isogeny graph (see Section 2.4 for definitions) of superspecial principally polarized abelian varieties of dimension $g$ , denoted by $\mathcal{G}_{g}(2,p)$ . While the graph $\mathcal{G}_{2}(2,p)$ can be computed by using known formulae (cf. [6, $\S 3$ ]), this is not the case where $g\geq 3$ (partial formulae such as those in [50] exist for $g=3$ , but they are not sufficient to compute the entire graph). In this paper, we will describe an explicit algorithm for computing $\mathcal{G}_{3}(2,p)$ by using theta functions. Theta functions enable us to compute Richelot isogenies between abelian varieties as outlined in [10, Appendix F], and the case of $g=2$ is implemented by [11] indeed. One of our contributions is explicitly formulating and implementing this algorithm in the case of $g=3$ .

Our overall strategy for listing superspecial genus-3 curves is as follows: we start from one principally polarized superspecial abelian threefold, and compute vertices connected to it via edges on $\mathcal{G}_{3}(2,p)$ , repeating this process to generate all vertices. An important point is that the computation of Richelot isogenies, the verification of whether the codomain is isomorphic to the Jacobian of a genus-3 curve, and the reconstruction of those curve can be carried out through theta constants. We note that this process is derived based on discussions over $\mathbb{C}$ , but it also works correctly in characteristic $p>7$ . Consequently, we obtain the following main theorem:

Theorem 1.1.

There exists an algorithm (Algorithm 3.16) for listing superspecial genus-3 curves in characteristic $p>7$ with $\widetilde{O}(p^{6})$ operations over $\mathbb{F}_{p^{4}}$ .

Our algorithm allows us to restore the explicit defining equations of all superspecial curves of genus 3 efficiently, compared to the methods using the Gröbner basis (this complexity would be exponential with respect to $p$ in worst case).

Executing our algorithm, we succeeded in proving the existence of superspecial hyperelliptic curves of genus 3 in small $p$ as follows:

Theorem 1.2.

The number of isomorphism classes of superspecial genus-3 hyperelliptic curves of genus 3 for $11\leq p<100$ is summarized in Table 1 of Section 4. Moreover, there exists such a curve in any characteristic $p$ with $7\leq p<10000$ .

We note that the upper bounds on $p$ in Theorem 1.2 can be increased easily. From our computational results, we obtain a conjecture that there exists a superspecial hyperelliptic curve of genus 3 in arbitrary characteristic $p\geq 7$ .

The rest of this paper is organized as follows: Section 2 is dedicated to reviewing some facts on analytic theta functions. In Section 2.4, we recall the definition and properties of superspecial Richelot isogeny graph $\mathcal{G}_{g}(2,p)$ . We then explain how to compute the graph $\mathcal{G}_{3}(2,p)$ in Section 3, and finally Algorithm 3.16 gives its explicit construction. In Section 4, we state computational results about the existence and the number of superspecial hyperelliptic curves of genus 3 in small $p$ , obtained by our implementation. At last, we give a concluding remark in Section 5.

2. Preliminaries

In this section, we summarize background knowledge required in later sections. Abelian varieties and isogenies between them before Section 2.3 are defined over $\mathbb{C}$ , but in Section 2.4 we consider those over a field of odd characteristic.

2.1. Abelian varieties and their isogenies

In this subsection, we shall review the theory of complex abelian varieties and isogenies. A complex abelian variety of dimension $g$ is isomorphic to $\mathbb{C}^{g}/(\mathbb{Z}^{g}+\varOmega\mathbb{Z}^{g})$ , where $\varOmega$ is an element of the Siegel upper half-space

\mathcal{H}_{g}\coloneqq\{\varOmega\in{\rm Mat}_{g}(\mathbb{C})\,\mid\,^{t}\varOmega=\varOmega,\,{\rm Im}\,\varOmega>0\}.\vspace{1mm}

Such $\varOmega\in\mathcal{H}_{g}$ is called a (small) period matrix of the abelian variety. First, we recall a characterization of isomorphisms between abelian varieties:

Definition 2.1.

For a commutative ring $R$ and an integer $g\geq 1$ , we define a group

{\rm Sp}_{2g}(R)\coloneqq\{M\in{\rm GL}_{2g}(R)\,\mid\,ME\hskip 0.85358pt^{t}M=E\},\quad E\coloneqq\begin{pmatrix}0&{\mathbf{1}}_{g}\\ -{\mathbf{1}}_{g}&0\end{pmatrix}.\vspace{0.7mm}

We say that a matrix $M\in{\rm GL}_{2g}(R)$ is symplectic over $R\hskip 0.56905pt$ if $M\in{\rm Sp}_{2g}(R)$ .

The following lemma helps determine whether $M$ is a symplectic matrix or not:

Lemma 2.2 ([32, Lemma 8.2.1]).

For any $(2g\times 2g)$ -matrix $M=\begin{pmatrix}\alpha&\beta\\[-0.85358pt] \gamma&\delta\end{pmatrix}\in{\rm GL}_{2g}(R)$ with $\alpha,\beta,\gamma,\delta\in{\rm Mat}_{g}(R)$ , the following three statements are equivalent:

(1)

The matrix $M$ is symplectic over $R$ .
(2)

Two matrices $\alpha\hskip 0.56905pt^{t}\beta,\,\gamma\hskip 0.85358pt^{t}\delta$ are both symmetric and $\alpha\hskip 0.85358pt^{t}\delta-\beta\hskip 1.42262pt^{t}\gamma={\mathbf{1}}_{g}$ .
(3)

Two matrices ${}^{t}\alpha\hskip 0.85358pt\gamma,\hskip 1.42262pt^{t}\beta\hskip 1.13809pt\delta$ are both symmetric and ${}^{t}\alpha\hskip 0.85358pt\delta-\hskip 0.28453pt^{t}\gamma\beta={\mathbf{1}}_{g}$ .

For a matrix $M=\begin{pmatrix}\alpha&\beta\\[-0.85358pt] \gamma&\delta\end{pmatrix}\in{\rm Sp}_{2g}(\mathbb{Z})$ , the action on the Siegel upper half-space

\displaystyle\begin{split}{\rm Sp}_{2g}(\mathbb{Z})\times\mathcal{H}_{g}&\longrightarrow\mathcal{H}_{g}\\[-1.42262pt] (M,\varOmega)&\longmapsto(\alpha\varOmega+\beta)(\gamma\varOmega+\delta)^{-1}\eqqcolon M.\varOmega\end{split}

is well-defined. Moreover, the map

\mathbb{C}^{g}\longrightarrow\mathbb{C}^{g}\,;\,z\longmapsto^{t}(\gamma\varOmega+\delta)^{-1}z\eqqcolon M.z\vspace{1mm}

induces an isomorphism $\mathbb{C}^{g}/(\mathbb{Z}^{g}+\varOmega\mathbb{Z}^{g})\rightarrow\mathbb{C}^{g}/(\mathbb{Z}^{g}+M.\varOmega\mathbb{Z}^{g})$ .

Proposition 2.3 ([32, Proposition 8.1.3]).

The principally polarized abelian varieties of dimension $g$ with period matrices $\varOmega,\varOmega^{\prime}\in\mathcal{H}_{g}$ are isomorphic to each other if and only if there exists a symplectic matrix $M\in{\rm Sp}_{2g}(\mathbb{Z})$ such that $\varOmega^{\prime}=M.\varOmega$ .

An isogeny between abelian varieties is a surjective homomorphism with a finite kernel. If there exists an isogeny $A\rightarrow B$ , then the dimensions of $A$ and $B$ are equal to each other, and we say that $A$ and $B$ are isogenous. Two isogenies $A\rightarrow B$ with the same kernel are equivalent up to an automorphism of $B$ , thus we identify them.

Proposition 2.4.

For an abelian variety $A$ of dimension $g$ and a prime integer $\ell$ , the number of maximal isotropic subgroups $G$ of $A[\ell]$ is equal to

N_{g}(\ell)\coloneqq\prod_{k=1}^{g}(\ell^{k}+1).\vspace{0.5mm}

Proof.

See [9, Lemma 2] or [5, Lemma 2], for example. ∎

An isogeny whose kernel is such a group $G$ the above is called an $(\overbrace{\ell,\dots,\ell}^{g})$ -isogeny. In particular, one can see that the $(\ell,\ldots,\ell)$ -isogeny

\displaystyle\begin{split}\mathbb{C}^{g}/(\mathbb{Z}^{g}+\varOmega\mathbb{Z}^{g})&\longrightarrow\mathbb{C}^{g}/(\mathbb{Z}^{g}+\ell\varOmega\mathbb{Z}^{g})\\[-1.42262pt] z&\longmapsto\ell z\end{split}

has the kernel $\frac{1}{\ell}\mathbb{Z}^{g}/\mathbb{Z}^{g}$ .

2.2. Theta functions

The theta function of characteristics $a,b\in\mathbb{Q}^{g}$ is given as

\theta\bigl{[}\begin{smallmatrix}a\\ b\end{smallmatrix}\bigr{]}(z,\varOmega)\coloneqq\sum_{n\in\mathbb{Z}^{g}}\exp\bigl{(}\pi i\hskip 1.42262pt^{t}(n+a)\hskip 0.56905pt\varOmega\hskip 0.56905pt(n+a)+2\pi i\hskip 1.42262pt^{t}(n+a)(z+b)\bigr{)}\vspace{0.5mm}

for $z\in\mathbb{C}^{g}$ and $\varOmega\in\mathcal{H}_{g}$ . All the characteristics can be considered modulo $\mathbb{Z}^{g}$ , since one can check that

\theta\bigl{[}\begin{smallmatrix}a+\alpha\\ b+\beta\end{smallmatrix}\bigr{]}(z,\varOmega)=\theta\bigl{[}\begin{smallmatrix}a\\ b\end{smallmatrix}\bigr{]}(z,\varOmega)\exp(2\pi i\hskip 1.42262pt^{t}a\beta)\vspace{0.3mm}

for all $\alpha,\beta\in\mathbb{Z}^{g}$ . In the following, we shall consider only theta functions $\theta\bigl{[}\begin{smallmatrix}a\\ b\end{smallmatrix}\bigr{]}(z,\varOmega)$ for characteristics $a,b\in\frac{1}{2}\mathbb{Z}^{g}/\mathbb{Z}^{g}$ .

Lemma 2.5.

For all characteristics $a,b\in\frac{1}{2}\mathbb{Z}^{g}/\mathbb{Z}^{g}$ , the function $\theta\bigl{[}\begin{smallmatrix}a\\ b\end{smallmatrix}\bigr{]}(z,\varOmega)$ is even with respect to $z$ if and only if $4\hskip 0.56905pt^{t}ab$ is an even integer; otherwise $\theta\bigl{[}\begin{smallmatrix}a\\ b\end{smallmatrix}\bigr{]}(z,\varOmega)$ is odd.

Proof.

The first assertion follows from that

\theta\bigl{[}\begin{smallmatrix}a\\ b\end{smallmatrix}\bigr{]}(-z,\varOmega)=(-1)^{4\hskip 0.56905pt^{t}ab}\theta\bigl{[}\begin{smallmatrix}a\\ b\end{smallmatrix}\bigr{]}(z,\varOmega).

The second assertion follows from that $4\hskip 0.56905pt^{t}ab$ is an integer for all $a,b\in\frac{1}{2}\mathbb{Z}^{g}/\mathbb{Z}^{g}$ . ∎

The following two formulae are very fundamental and generate many relations:

Theorem 2.6 (Riemann’s theta formula).

Let $m_{k}:=\begin{pmatrix}a_{k}\\ b_{k}\end{pmatrix}$ with $a_{k},b_{k}\in\frac{1}{2}\mathbb{Z}^{g}/\mathbb{Z}^{g}$ , and we define

\begin{pmatrix}n_{1}\\ n_{2}\\ n_{3}\\ n_{4}\end{pmatrix}\coloneqq\hskip 0.85358pt\frac{1}{2}\begin{pmatrix}{\mathbf{1}}_{2g}&\hskip 7.68222pt{\mathbf{1}}_{2g}&\hskip 7.68222pt{\mathbf{1}}_{2g}&\hskip 7.68222pt{\mathbf{1}}_{2g}\\ {\mathbf{1}}_{2g}&\hskip 7.68222pt{\mathbf{1}}_{2g}&-{\mathbf{1}}_{2g}&-{\mathbf{1}}_{2g}\\ {\mathbf{1}}_{2g}&-{\mathbf{1}}_{2g}&\hskip 7.68222pt{\mathbf{1}}_{2g}&-{\mathbf{1}}_{2g}\\ {\mathbf{1}}_{2g}&-{\mathbf{1}}_{2g}&-{\mathbf{1}}_{2g}&\hskip 7.68222pt{\mathbf{1}}_{2g}\end{pmatrix}\begin{pmatrix}m_{1}\\ m_{2}\\ m_{3}\\ m_{4}\end{pmatrix}.

Then, we have the equation

\prod_{k=1}^{4}\theta[m_{k}](0,\varOmega)=\frac{1}{2^{g}}\sum_{\alpha,\beta\in\frac{1}{2}\mathbb{Z}^{g}/\mathbb{Z}^{g}}(-1)^{4\hskip 0.56905pt^{t}a\beta}\prod_{k=1}^{4}\theta[n_{k}+\gamma_{\alpha,\beta}](0,\varOmega)\vspace{0.7mm}

where $\gamma_{\alpha,\beta}$ denotes $\begin{pmatrix}\alpha\\ \beta\end{pmatrix}$ with $\alpha,\beta\in\frac{1}{2}\mathbb{Z}^{g}/\mathbb{Z}^{g}$ .

Proof.

Apply [26, Chapter IV, Theorem 1] for $z_{1}=z_{2}=z_{3}=z_{4}=0$ . ∎

Theorem 2.7 (Duplication formula).

For all $a,b\in\frac{1}{2}\mathbb{Z}^{g}/\mathbb{Z}^{g}$ , we have

\theta\bigl{[}\begin{smallmatrix}a\\ b\end{smallmatrix}\bigr{]}(z,\varOmega)^{2}=\frac{1}{2^{g}}\hskip 0.56905pt\sum_{\beta\in\frac{1}{2}\mathbb{Z}^{g}/\mathbb{Z}^{g}}(-1)^{4\hskip 0.56905pt^{t}a\beta}\hskip 0.85358pt\theta\bigl{[}\begin{smallmatrix}0\\ b\end{smallmatrix}\bigr{]}(z,\varOmega/2)\theta\bigl{[}\begin{smallmatrix}0\\ b+\beta\end{smallmatrix}\bigr{]}(z,\varOmega/2).

Proof.

Apply [26, Chapter IV, Theorem 2] for $m_{1}=m_{2}$ and $z_{1}=z_{2}=z$ . ∎

Especially if a matrix $\varOmega\in\mathcal{H}_{g}$ is written as

\varOmega=\begin{pmatrix}\varOmega_{1}&0\\ 0&\varOmega_{2}\end{pmatrix}\ \,{\rm where}\ \,\varOmega_{1}\in\mathcal{H}_{g_{1}},\ \varOmega_{2}\in\mathcal{H}_{g_{2}}

with $g_{1}+g_{2}=g$ , then we have the following lemma:

Lemma 2.8 ([10, Lemma F.3.1]).

With the above notations, we write $z\coloneqq(z_{1},z_{2})$ where $z_{1}\in\mathbb{C}^{g_{1}},\,z_{2}\in\mathbb{C}^{g_{2}}$ . Then, we have the equation

\theta\bigl{[}\begin{smallmatrix}a_{1}&a_{2}\\ b_{1}&b_{2}\end{smallmatrix}\bigr{]}(z,\varOmega)=\theta\bigl{[}\begin{smallmatrix}a_{1}\\ b_{1}\end{smallmatrix}\bigr{]}(z_{1},\varOmega_{1})\hskip 0.85358pt\theta\bigl{[}\begin{smallmatrix}a_{2}\\ b_{2}\end{smallmatrix}\bigr{]}(z_{2},\varOmega_{2})\vspace{1mm}

for all characteristics $a_{1},b_{1}\in\frac{1}{2}\mathbb{Z}^{g_{1}}/\mathbb{Z}^{g_{1}}$ and $a_{2},b_{2}\in\frac{1}{2}\mathbb{Z}^{g_{2}}/\mathbb{Z}^{g_{2}}$ .

Let us return to cases with a general (not necessarily diagonal) matrix $\varOmega\in\mathcal{H}_{g}$ . For any symplectic matrix $M=\begin{pmatrix}\alpha&\beta\\[-0.85358pt] \gamma&\delta\end{pmatrix}\in{\rm Sp}_{2g}(\mathbb{Z})$ and characteristics $a,b\in\frac{1}{2}\mathbb{Z}^{g}/\mathbb{Z}^{g}$ , we define a scalar

k(M,a,b):=(a\hskip 0.85358pt^{t}\delta-b\hskip 1.13809pt^{t}\gamma)\hskip 1.42262pt\cdot\hskip 0.85358pt^{t}(b\hskip 1.42262pt^{t}\alpha-a\hskip 0.85358pt^{t}\beta+(\alpha\hskip 0.85358pt^{t}\beta)_{0})-a\hskip 0.85358pt^{t}b\vspace{0.5mm}

and two vectors

(2.1)

M.a:=a\hskip 0.85358pt^{t}\delta-b\hskip 1.13809pt^{t}\gamma+\frac{1}{2}(\gamma\hskip 1.42262pt^{t}\delta)_{0},\quad M.b:=b\hskip 1.42262pt^{t}\alpha-a\hskip 0.85358pt^{t}\beta+\frac{1}{2}(\alpha\hskip 0.85358pt^{t}\beta)_{0},\vspace{0.5mm}

where $(\alpha\hskip 0.85358pt^{t}\beta)_{0},(\gamma\hskip 1.42262pt^{t}\delta)_{0}$ denote the row vector of the diagonal elements of $(\alpha\hskip 0.85358pt^{t}\beta),(\gamma\hskip 1.42262pt^{t}\delta)$ . Then, the matrix $M$ acts on the theta function in the following way:

Theorem 2.9 (Theta transformation formula).

For all $a,b\in\frac{1}{2}\mathbb{Z}^{g}/\mathbb{Z}^{g}$ , we have

\theta\bigl{[}\begin{smallmatrix}M.a\\ M.b\end{smallmatrix}\bigr{]}(M.z,M.\varOmega)=c\cdot e^{\pi ik(M,a,b)}\theta\bigl{[}\begin{smallmatrix}a\\ b\end{smallmatrix}\bigr{]}(z,\varOmega),\vspace{0.7mm}

where $c\in\mathbb{C}^{\times}$ is a constant which does not depend $a$ or $b$ .

Proof.

This follows from [32, Theorem 8.6.1]. ∎

Let $n\geq 1$ be an integer and we define

	$\displaystyle\varGamma_{n}$	$\displaystyle:=\bigl{\{}M=\begin{pmatrix}\alpha&\beta\\[-0.85358pt] \gamma&\delta\end{pmatrix}\in{\rm Sp}_{2g}(\mathbb{Z})\,\mid\,\alpha\equiv\delta\equiv{\mathbf{1}}_{g},\ \beta\equiv\gamma\equiv{\mathbf{0}}_{g}\pmod{n}\bigr{\}},$
	$\displaystyle\varGamma_{n,2n}$	$\displaystyle:=\bigl{\{}M=\begin{pmatrix}\alpha&\beta\\[-0.85358pt] \gamma&\delta\end{pmatrix}\in\varGamma_{n}\,\mid\,{\rm diag}(\hskip 0.85358pt^{t}\alpha\gamma)\equiv{\rm diag}(\hskip 0.85358pt^{t}\beta\hskip 0.85358pt\delta)\equiv 0\pmod{2n}\},$

which are both subgroups of ${\rm Sp}_{2g}(\mathbb{Z})$ .

Corollary 2.10.

For all characteristics $a,b\in\frac{1}{2}\mathbb{Z}^{g}/\mathbb{Z}^{g}$ , we have

	$\displaystyle\frac{\theta\bigl{[}\begin{smallmatrix}a\\ b\end{smallmatrix}\bigr{]}(0,M.\varOmega)^{4}}{\theta\bigl{[}\begin{smallmatrix}0\\ 0\end{smallmatrix}\bigr{]}(0,M.\varOmega)^{4}}$	$\displaystyle=\frac{\theta\bigl{[}\begin{smallmatrix}a\\ b\end{smallmatrix}\bigr{]}(0,\varOmega)^{4}}{\theta\bigl{[}\begin{smallmatrix}0\\ 0\end{smallmatrix}\bigr{]}(0,\varOmega)^{4}}\quad{\rm if}\ M\in\varGamma_{2},$
	$\displaystyle\frac{\theta\bigl{[}\begin{smallmatrix}a\\ b\end{smallmatrix}\bigr{]}(0,M.\varOmega)^{2}}{\theta\bigl{[}\begin{smallmatrix}0\\ 0\end{smallmatrix}\bigr{]}(0,M.\varOmega)^{2}}$	$\displaystyle=\frac{\theta\bigl{[}\begin{smallmatrix}a\\ b\end{smallmatrix}\bigr{]}(0,\varOmega)^{2}}{\theta\bigl{[}\begin{smallmatrix}0\\ 0\end{smallmatrix}\bigr{]}(0,\varOmega)^{2}}\quad{\rm if}\ M\in\varGamma_{2,4},$
	$\displaystyle\frac{\theta\bigl{[}\begin{smallmatrix}a\\ b\end{smallmatrix}\bigr{]}(0,M.\varOmega)}{\theta\bigl{[}\begin{smallmatrix}0\\ 0\end{smallmatrix}\bigr{]}(0,M.\varOmega)}$	$\displaystyle=\frac{\theta\bigl{[}\begin{smallmatrix}a\\ b\end{smallmatrix}\bigr{]}(0,\varOmega)}{\theta\bigl{[}\begin{smallmatrix}0\\ 0\end{smallmatrix}\bigr{]}(0,\varOmega)}\quad\ \,{\rm if}\ M\in\varGamma_{4,8}.$

Proof.

It follows from Theorem 2.9 that

\frac{\theta\bigl{[}\begin{smallmatrix}a\\ b\end{smallmatrix}\bigr{]}(0,M.\varOmega)}{\theta\bigl{[}\begin{smallmatrix}0\\ 0\end{smallmatrix}\bigr{]}(0,M.\varOmega)}=\frac{\theta\bigl{[}\begin{smallmatrix}a\\ b\end{smallmatrix}\bigr{]}(0,\varOmega)}{\theta\bigl{[}\begin{smallmatrix}0\\ 0\end{smallmatrix}\bigr{]}(0,\varOmega)}\exp\bigl{(}\pi ia\hskip 0.85358pt^{t}\beta\hskip 0.85358pt\delta\hskip 1.42262pt^{t}a-\pi ib\hskip 0.85358pt^{t}\gamma\hskip 0.85358pt\alpha\hskip 1.42262pt^{t}b-2\pi ia(\hskip 0.56905pt^{t}\alpha-\mathbf{1}_{g})\hskip 0.56905pt^{t}b\bigr{)}.

Hence, we obtain this assertion by simple computations. ∎

2.3. Theta functions for abelian threefolds

In this subsection, we focus only on the case that $A$ is a principally polarized abelian threefold, and we discuss theta functions associated to $A$ . Thanks to [32, Corollary 11.8.2], such $A$ is classified into one of the following four types:

\begin{split}\left\{\begin{array}[]{l}{\rm Type\ 1\!:\ }\,A\cong{\rm Jac}(C)\ \text{where}\ C\ \text{is\ a\ smooth\ plane\ quartic},\\[1.42262pt] {\rm Type\ 2\!:\ }\,A\cong{\rm Jac}(C)\ \text{where}\ C\ \text{is\ a\ hyperelliptic\ curve\ of\ genus\ }3,\\[1.42262pt] {\rm Type\ 3\!:\ }\,A\cong E\times{\rm Jac}(C)\ \text{where}\ E\ \text{and}\ C\text{\ is\ a genus-1 and genus-2 curve},\\[1.42262pt] {\rm Type\ 4\!:\ }\,A\cong E_{1}\times E_{2}\times E_{3}\ \text{where}\ E_{1},E_{2}\text{\ and\ }E_{3}\text{\ are\ elliptic\ curves}.\end{array}\right.\vspace{0.5mm}\end{split}

First of all, we introduce the following notations for the sake of simplicity.

Definition 2.11.

For a period matrix $\varOmega\in\mathcal{H}_{3}$ of $A$ , we write

\vartheta_{i}(z,\varOmega):=\theta\bigl{[}\begin{smallmatrix}a_{1}&a_{2}&a_{3}\\ b_{1}&b_{2}&b_{3}\end{smallmatrix}\bigr{]}(z,\varOmega),\quad i:=2b_{1}+4b_{2}+8b_{3}+16a_{1}+32a_{2}+64a_{3},\vspace{0.8mm}

where $a_{1},a_{2},a_{3},b_{1},b_{2}$ , and $b_{3}$ belong to $\{0,1/2\}$ . Note that $i$ runs over $\{0,\ldots,63\}$ . For each $i$ , the value $\vartheta_{i}(0,\varOmega)$ is written as $\vartheta_{i}$ , and called the $i$ -th theta constant.

We consider a principally polarized abelian threefold $A\cong\mathbb{C}^{3}/(\mathbb{Z}^{3}+\varOmega\mathbb{Z}^{3})$ , then the projective value

[\vartheta_{0}^{2}:\vartheta_{1}^{2}:\cdots:\vartheta_{63}^{2}]\in\mathbb{P}^{63}(\mathbb{C})\vspace{1mm}

is called a (level 2) theta null-point of $A$ . We remark that there are different theta null-points, even for isomorphic principally polarized abelian threefolds.

Definition 2.12.

We define the sets

I_{\rm even}\coloneqq\left\{\begin{array}[]{l}0,\,1,\,2,\,3,\,4,\,5,\,6,\,7,\,8,\,10,\,12,\,14,\,16,\,17,\,20,\,21,\,24,\,27,\,28,\\ 31,\,32,\,33,\,34,\,35,\,40,\,42,\,45,\,47,\,48,\,49,\,54,\,55,\,56,\,59,\,61,\,62\end{array}\right\}

and $I_{\rm odd}\coloneqq\{0,\ldots,63\}\smallsetminus I_{\rm even}$ . Here, we remark that $\#I_{\rm even}=36$ and $\#I_{\rm odd}=28$ .

It follows from Lemma 2.5 that $\vartheta_{i}(z)$ is even (resp. odd) if and only if $i\in I_{\rm even}$ (resp. $i\in I_{\rm odd}$ ). Therefore, we have that $\vartheta_{i}=0$ for all $i\in I_{\rm odd}$ .

Definition 2.13.

Let $S_{\rm van}$ be the set of vanishing indices $i\in I_{\rm even}$ , that is,

S_{\rm van}\coloneqq\{\hskip 0.56905pti\in I_{\rm even}\mid\vartheta_{i}=0\}.\vspace{0.3mm}

In addition, we denote by $N_{\rm van}$ the cardinality of the set $S_{\rm van}$ .

Lemma 2.14.

The value $N_{\rm van}$ in Definition 2.13 is invariant for isomorphic principally polarized abelian threefolds.

Proof.

This follows from Theorem 2.9 together with Proposition 2.3. ∎

Then, we can identify by $N_{\rm van}$ which of the above 4 types $A$ corresponds to (note that this does not depend on a period matrix $\varOmega$ of $A$ by the above lemma).

Proposition 2.15.

The possible values of $N_{\rm van}$ are $0,1,6,9$ and moreover

\left\{\begin{array}[]{l}N_{\rm van}=0\ \Leftrightarrow\,A\cong{\rm Jac}(C)\ \text{where}\ C\ \text{is\ a\ smooth\ plane\ quartic},\\[1.42262pt] N_{\rm van}=1\ \Leftrightarrow\,A\cong{\rm Jac}(C)\ \text{where}\ C\ \text{is\ a\ hyperelliptic\ curve\ of\ genus\ }3,\\[1.42262pt] N_{\rm van}=6\ \Leftrightarrow\,A\cong E\times{\rm Jac}(C)\ \text{where}\ E\ \text{and}\ C\text{\ is\ a genus-1 and genus-2 curve},\\[1.42262pt] N_{\rm van}=9\ \Leftrightarrow\,A\cong E_{1}\times E_{2}\times E_{3}\ \text{where}\ E_{1},E_{2}\text{\ and\ }E_{3}\text{\ are\ elliptic\ curves}.\end{array}\right.\vspace{-0.3mm}

Proof.

The first assertion follows from [20, Theorem 3.1] (the cases where $N_{\rm van}>9$ correspond to singular abelian threefolds). Also, for the second assertion, the cases where $N_{\rm van}=0,1$ follows from [42, Theorem 2]. Note that these results are proved over $\mathbb{C}$ , but valid over an algebraically closed field of characteristic $\neq 2$ as stated in Pieper’s paper [43, Remark 6.3]. In the following, we show the remaining cases.

Let $A$ be a principally polarized abelian threefold isomorphic to the (principally polarized) product of an elliptic curve $A_{1}$ and an abelian surface $A_{2}$ . Denote $\varOmega_{1},\varOmega_{2}$ , and $\varOmega$ to be the period matrices of $A_{1},A_{2}$ , and $A$ respectively. Theorem 2.3 shows that there exists $M\in{\rm Sp}_{6}(\mathbb{Z})$ such that $M.\varOmega={\rm diag}(\varOmega_{1},\varOmega_{2})$ , hence we may assume that $\varOmega={\rm diag}(\varOmega_{1},\varOmega_{2})$ using Lemma 2.14. Then, it follows from Lemma 2.8 that

\theta\bigl{[}\begin{smallmatrix}a_{1}&a_{2}&a_{3}\\ b_{1}&b_{2}&b_{3}\end{smallmatrix}\bigr{]}(z,\varOmega)=\theta\bigl{[}\begin{smallmatrix}a_{1}\\ b_{1}\end{smallmatrix}\bigr{]}(z_{1},\varOmega_{1})\hskip 0.85358pt\theta\bigl{[}\begin{smallmatrix}a_{2}&a_{3}\\ b_{2}&b_{3}\end{smallmatrix}\bigr{]}(z_{2},\varOmega_{2}),\quad z\coloneqq(z_{1},z_{2}).\vspace{1mm}

As known in [34, Remark 3.5] or [46, $\S 16.4$ ], a simple (resp. not simple) principally polarized abelian surface has no (resp. exactly one) vanishing even theta constant. This shows that $N_{\rm van}=6$ if $A_{2}$ is simple and $N_{\rm van}=9$ otherwise, as desired. ∎

In the rest of this subsection, suppose that $A$ corresponds to a genus-3 curve $C$ . Let us review how to restore a defining equation of $C$ from theta constants of $A$ .

Proposition 2.16.

We assume that $S_{\rm van}=\{61\}$ for a period matrix $\varOmega\in\mathcal{H}_{3}$ of $A$ . Then $A$ is isomorphic to the Jacobian of a genus-3 hyperelliptic curve

C:y^{2}=x(x-1)(x-\lambda_{1})(x-\lambda_{2})(x-\lambda_{3})(x-\lambda_{4})(x-\lambda_{5}),\vspace{-0.2mm}

where

(2.2)

\lambda_{1}\coloneqq\frac{\vartheta_{42}^{2}\vartheta_{2}^{2}}{\vartheta_{5}^{2}\vartheta_{45}^{2}},\ \,\lambda_{2}\coloneqq\frac{\vartheta_{42}^{2}\vartheta_{3}^{2}}{\vartheta_{4}^{2}\vartheta_{45}^{2}},\ \,\lambda_{3}\coloneqq\frac{\vartheta_{27}^{2}\vartheta_{42}^{2}}{\vartheta_{45}^{2}\vartheta_{28}^{2}},\ \,\lambda_{4}\coloneqq\frac{\vartheta_{2}^{2}\vartheta_{49}^{2}}{\vartheta_{54}^{2}\vartheta_{5}^{2}},\ \,\lambda_{5}\coloneqq\frac{\vartheta_{3}^{2}\vartheta_{0}^{2}}{\vartheta_{7}^{2}\vartheta_{4}^{2}}.

Proof.

Apply [51, Theorem 1.1] for $g=3$ , or see [47, Theorem 4]. ∎

On the other hand, even in the case that $C$ is a plane quartic, we can also restore a defining equation of $C$ as follows: Taking a period matrix $\varOmega\in\mathcal{H}_{3}$ of $A={\rm Jac}(C)$ , it follows from Proposition 2.15 that $\vartheta_{i}\neq 0$ holds for all $i\in I_{\rm even}$ . Then, we define the following non-zero values

	$\displaystyle a_{11}$	$\displaystyle\coloneqq\frac{\vartheta_{12}\vartheta_{5}}{\vartheta_{40}\vartheta_{33}},$	$\displaystyle\quad a_{21}$	$\displaystyle\coloneqq\frac{\vartheta_{27}\vartheta_{5}}{\vartheta_{40}\vartheta_{54}},$	$\displaystyle\quad a_{31}$	$\displaystyle\coloneqq-\frac{\vartheta_{27}\vartheta_{12}}{\vartheta_{33}\vartheta_{54}},$
(2.3)		$\displaystyle a_{12}$	$\displaystyle\coloneqq\frac{\vartheta_{21}\vartheta_{28}}{\vartheta_{49}\vartheta_{56}},$	$\displaystyle\quad a_{22}$	$\displaystyle\coloneqq\frac{\vartheta_{2}\vartheta_{28}}{\vartheta_{49}\vartheta_{47}},$	$\displaystyle\quad a_{32}$	$\displaystyle\coloneqq\frac{\vartheta_{2}\vartheta_{21}}{\vartheta_{56}\vartheta_{47}},$
	$\displaystyle a_{13}$	$\displaystyle\coloneqq\frac{\vartheta_{7}\vartheta_{14}}{\vartheta_{35}\vartheta_{42}},$	$\displaystyle\quad a_{23}$	$\displaystyle\coloneqq\frac{\vartheta_{16}\vartheta_{14}}{\vartheta_{35}\vartheta_{61}},$	$\displaystyle\quad a_{33}$	$\displaystyle\coloneqq\frac{\vartheta_{16}\vartheta_{7}}{\vartheta_{42}\vartheta_{61}}$

computed as in [15, p. 15]. Then, there exist $k_{1},k_{2},k_{3},\lambda_{1},\lambda_{2},\lambda_{3}\in\mathbb{C}^{\times}$ such that

(2.5)

\begin{pmatrix}\lambda_{1}a_{11}&\lambda_{2}a_{21}&\lambda_{3}a_{31}\\[1.42262pt] \lambda_{1}a_{12}&\lambda_{2}a_{22}&\lambda_{3}a_{32}\\[1.42262pt] \lambda_{1}a_{13}&\lambda_{2}a_{23}&\lambda_{3}a_{33}\end{pmatrix}\begin{pmatrix}k_{1}\\[1.42262pt] k_{2}\\[1.42262pt] k_{3}\end{pmatrix}=\begin{pmatrix}1/a_{11}&1/a_{21}&1/a_{31}\\[1.42262pt] 1/a_{12}&1/a_{22}&1/a_{32}\\[1.42262pt] 1/a_{13}&1/a_{23}&1/a_{33}\end{pmatrix}\begin{pmatrix}\lambda_{1}\\[1.42262pt] \lambda_{2}\\[1.42262pt] \lambda_{3}\end{pmatrix}=\begin{pmatrix}-1\\[1.42262pt] -1\\[1.42262pt] -1\end{pmatrix}\!.

Theorem 2.17 (Weber’s formula).

In the above situation, there exist three homogeneous linear polynomials $\xi_{1},\xi_{2},\xi_{3}\in\mathbb{C}[x,y,z]$ such that

\left\{\begin{array}[]{l}\xi_{1}+\xi_{2}+\xi_{3}+x+y+z=0,\\[1.42262pt] \displaystyle\frac{\xi_{1}}{a_{i1}}+\frac{\xi_{2}}{a_{i2}}+\frac{\xi_{3}}{a_{i3}}+k_{i}(a_{i1}x+a_{i2}y+a_{i3}z)=0,\quad{\rm for\ all}\ i\in\{1,2,3\}.\end{array}\right.

Then $A$ is isomorphic to the Jacobian of a plane quartic

C:(\xi_{1}x+\xi_{2}y-\xi_{3}z)^{2}-4\xi_{1}\xi_{2}xy=0.

Proof.

See [54, $\S 15$ ], [39, Theorem 3.1], or [15, Proposition 2]. ∎

2.4. Superspecial Richelot isogeny graph

Unlike the previous subsections, let us consider abelian varieties over an algebraically closed field of characteristic $p\geq 3$ in this subsection. An elliptic curve $E$ is called supersingular if $E[p]=\{0\}$ .

Definition 2.18.

An abelian variety $A$ is called superspecial if $A$ is isomorphic to the product of supersingular elliptic curves (without a polarization). A superspecial curve is a curve whose Jacobian is a superspecial abelian variety.

We note that if two curves $C,C^{\prime}$ are not isomorphic to each other, then neither are their Jacobians (as principally polarized abelian varieties), by Torelli’s theorem. Let $\Lambda_{g,p}$ be the list of isomorphism classes (over $\overline{\mathbb{F}}_{p}$ ) of superspecial genus- $g$ curves. The numbers $\#\Lambda_{g,p}$ for $g\leq 3$ are given by [12], [24], and [21], respectively (see [30] for a survey). However, it is not known how many of those curves are hyperelliptic for genus $g=3$ . In the following, we explain how to generate the list $\Lambda_{g,p}$ , by using the isogeny graph of superspecial abelian varieties:

Definition 2.19.

Let $\ell$ be a prime integer. The superspecial isogeny graph $\mathcal{G}_{g}(\ell,p)$ is the directed multigraph with the following properties

•

The vertices are isomorphism classes of all principally polarized superspecial abelian varieties of dimension $g$ over $\overline{\mathbb{F}}_{p}$ .
•

The edges are $(\ell,\ldots,\ell)$ -isogenies defined over $\overline{\mathbb{F}}_{p}$ between two vertices.

Then, the graph $\mathcal{G}_{g}(\ell,p)$ is a $N_{g}(\ell)$ -regular multigraph by Proposition 2.4.

The fact that the Pizer’s graph $\mathcal{G}_{1}(\ell,p)$ is connected is well-known (cf. [44]), but more surprisingly, the same assertion holds for $\mathcal{G}_{g}(\ell,p)$ as a generalization:

Theorem 2.20.

The graph $\mathcal{G}_{g}(\ell,p)$ is connected.

Proof.

See [27, Theorem 34] or [1, Corollary 2.8]. ∎

In the following, we discuss the case where $\ell=2$ , then the graph $\mathcal{G}_{g}(2,p)$ is often called the superspecial Richelot isogeny graph. Then, the list $\Lambda_{2,p}$ can be generated by using $\mathcal{G}_{2}(2,p)$ as in the following pseudocode (cf. [31, Algorithm 5.1]).

1:A prime integer

p>5

and the list

\Lambda_{1,p}

of isom. classes of s.sing. ell. curves

2:The list

\Lambda_{2,p}

of isom. classes of s.sp. genus-2 curves

3:Let

\mathcal{L}

be the list of isom. classes of

E_{1}\times E_{2}

with

E_{1},E_{2}\in\Lambda_{1,p}

, and set

k\leftarrow 1

4:repeat

5: Let

A

be the

k

-th abelian surface of

\mathcal{L}

6: for all abelian surfaces

A^{\prime}

which are

(2,2)

-isogenous to

A

7: if

A^{\prime}

is not isomorphic to any abelian surface of

\mathcal{L}

then

8: Add

A^{\prime}

to the end of

\mathcal{L}

9: end if

10: end for

11: Set

k\leftarrow k+1

12:until

k>\#\mathcal{L}

13:return the list of genus-2 curves whose Jacobian belongs to

\mathcal{L}

Algorithm 2.21 ListUpSspGenus2Curves

In Algorithm 2.21, one can compute the codomains of Richelot isogenies of line 4 using formulae in [22, Proposition 4] and [49, Chapter 8]. In addition, isomorphism tests in line 5 can be done using Igusa invariants of a genus-2 curve (cf. [25]).

Remark 2.22.

The structure of the graph $\mathcal{G}_{2}(2,p)$ was studied by many authors (e.g. [28], [17]). Importantly, it is known (cf. [16, Theorem 7.2]) that its Jacobians’ subgraph $\mathcal{J}_{2}(2,p)$ is also connected. Here $\mathcal{J}_{g}(\ell,p)$ is defined to be

•

The vertices are isomorphism classes of Jacobians of superspecial curves of genus $g$ over $\overline{\mathbb{F}}_{p}$ .
•

The edges are $(\ell,\ldots,\ell)$ -isogenies defined over $\overline{\mathbb{F}}_{p}$ between two vertices.

The above mentioned research may enable us to construct a more efficient algorithm for listing superspecial genus-2 curves rather than Algorithm 2.21.

Similarly to the case of $g=2$ , we want to list superspecial curves of genus $g\geq 3$ . However, no explicit formula is known for the computations of $(2,\ldots,2)$ -isogenies in general, unlike the genus-2 case. In the next section, we will propose an algorithm for computing $(2,2,2)$ -isogenies using theta functions.

3. Computing the Richelot isogeny graph of dimension 3

In this section, we give an explicit algorithm (Algorithm 3.16) for listing superspecial genus-3 curves. For this purpose, we describe the following subroutines in the next three sections:

—

Section 3.1: For a given theta null-point of an abelian threefold $A$ , compute a theta null-point of one $A^{\prime}$ which is $(2,2,2)$ -isogenous to $A$ .
—

Section 3.2: For a given theta null-point of an abelian threefold $A$ , compute theta null-points of all $A^{\prime}$ , which are $(2,2,2)$ -isogenous to $A$ .
—

Section 3.3: For a given theta null-point of $A^{\prime}$ corresponding to a curve of genus 3, restore the defining equation of it.

These computations use the theta constants introduced in Section 2.3, but they are still valid over a field of odd characteristic $p\geq 3$ , thanks to the algebraic theory by Mumford [37]. In Section 3.4, we show that our main algorithm can be performed over $\mathbb{F}_{p^{4}}$ , and provide a proof of Theorem 1.1 finally.

3.1. How to compute one path from the vertex

In this subsection, let $A$ be a complex abelian threefold, and $\varOmega\in\mathcal{H}_{3}$ be a period matrix of $A$ . First, we shall prepare the following fact about a theta null-point of $A$ .

Lemma 3.1.

For a matrix $\varOmega\in\mathcal{H}_{3}$ , the equation

\vartheta_{0}\vartheta_{1}\vartheta_{2}\vartheta_{3}-\vartheta_{4}\vartheta_{5}\vartheta_{6}\vartheta_{7}=\vartheta_{32}\vartheta_{33}\vartheta_{34}\vartheta_{35}\vspace{0.5mm}

holds with the notation in Definition 2.11.

Proof.

Applying Theorem 2.6 for four matrices

	$\displaystyle m_{1}$	$\displaystyle\coloneqq\begin{pmatrix}0&0&0\\ 0&0&0\end{pmatrix},\quad$	$\displaystyle m_{2}$	$\displaystyle\coloneqq\begin{pmatrix}0&0&0\\ 1/2&0&0\end{pmatrix},$
	$\displaystyle m_{3}$	$\displaystyle\coloneqq\begin{pmatrix}0&0&0\\ 0&1/2&0\end{pmatrix},\quad$	$\displaystyle m_{4}$	$\displaystyle\coloneqq\begin{pmatrix}0&0&0\\ 1/2&1/2&0\end{pmatrix},\vspace{0.3mm}$

then we obtain this lemma. ∎

We assume that a theta null-point $[\vartheta_{0}^{2}:\vartheta_{1}^{2}:\cdots:\vartheta_{63}^{2}]\in\mathbb{P}^{63}(\mathbb{C})$ of $A$ is obtained. Squaring both sides of the equation in Lemma 3.1, we have

(3.6)

2\vartheta_{0}\vartheta_{1}\vartheta_{2}\vartheta_{3}\vartheta_{4}\vartheta_{5}\vartheta_{6}\vartheta_{7}=\vartheta_{0}^{2}\vartheta_{1}^{2}\vartheta_{2}^{2}\vartheta_{3}^{2}+\vartheta_{4}^{2}\vartheta_{5}^{2}\vartheta_{6}^{2}\vartheta_{7}^{2}-\vartheta_{32}^{2}\vartheta_{33}^{2}\vartheta_{34}^{2}\vartheta_{35}^{2},\vspace{0.5mm}

which means that the value of $\prod_{j=0}^{7}\vartheta_{j}$ is determined by the given theta null-point. On the other hand, we can take any square root of ${\vartheta_{j}}^{2}$ for all $j\in\{0,\ldots,7\}$ so that the equation (3.6) holds, by the following proposition:

Proposition 3.2.

We suppose that $\vartheta_{0}\neq 0$ and let $k$ be an integer with $1\leq k\leq 6$ . Then, there exists a symplectic matrix $M\in{\rm Sp}_{6}(\mathbb{Z})$ such that

\frac{\vartheta_{j}(0,M.\varOmega)}{\vartheta_{0}(0,M.\varOmega)}=\left\{\begin{array}[]{l}-\vartheta_{j}/\vartheta_{0}\quad{\rm if}\ j=k\ {\rm or}\ 7,\\ \vartheta_{j}/\vartheta_{0}\quad\hskip 7.96677pt{\rm otherwise}\end{array}\right.\vspace{1mm}

for all $j\in\{1,\ldots,7\}$ .

Proof.

It follows from Corollary 2.10 that all symplectic matrix in $\varGamma_{4,8}$ (resp. $\varGamma_{2,4}$ ) leave the values (resp. squares) of $\vartheta_{j}/\vartheta_{0}$ for all $j$ . Let us construct $M\in\varGamma_{2,4}\!\smallsetminus\!\varGamma_{4,8}$ which changes the signs of only $\vartheta_{k}/\vartheta_{0}$ and $\vartheta_{7}/\vartheta_{0}$ . We define

(c_{11},c_{12},c_{13},c_{22},c_{23},c_{33}):=\left\{\begin{array}[]{l}(1,1,1,0,0,0)\quad\text{if }\,k=1,\\[1.42262pt] (0,1,0,1,1,0)\quad\text{if }\,k=2,\\[1.42262pt] (0,1,0,0,0,0)\quad\text{if }\,k=3,\\[1.42262pt] (0,0,1,0,1,1)\quad\text{if }\,k=4,\\[1.42262pt] (0,0,1,0,0,0)\quad\text{if }\,k=5,\\[1.42262pt] (0,0,0,0,1,0)\quad\text{if }\,k=6\end{array}\right.

and $M=\begin{pmatrix}\alpha&\beta\\[-0.85358pt] \gamma&\delta\end{pmatrix}$ where

\alpha\coloneqq\mathbf{1}_{3},\quad\beta\coloneqq\mathbf{0}_{3},\quad\gamma\coloneqq\begin{pmatrix}4c_{11}&2c_{12}&2c_{13}\\ 2c_{12}&4c_{22}&2c_{23}\\ 2c_{13}&2c_{23}&4c_{33}\end{pmatrix},\quad\delta\coloneqq\mathbf{1}_{3}.\vspace{0.5mm}

It is obvious that $\alpha\hskip 0.56905pt^{t}\beta={\mathbf{0}}_{3},\,\gamma\hskip 0.85358pt^{t}\delta=\gamma$ are both symmetric and $\alpha\hskip 0.85358pt^{t}\delta-\beta\hskip 1.42262pt^{t}\gamma={\mathbf{1}}_{3}$ holds, hence $M$ belongs to ${\rm Sp}_{6}(\mathbb{Z})$ by Lemma 2.2. Moreover, one can check that $M$ is the desired matrix since

\frac{\theta\bigl{[}\begin{smallmatrix}0\\ b\end{smallmatrix}\bigr{]}(0,M.\varOmega)}{\theta\bigl{[}\begin{smallmatrix}0\\ 0\end{smallmatrix}\bigr{]}(0,M.\varOmega)}=\frac{\theta\bigl{[}\begin{smallmatrix}0\\ b\end{smallmatrix}\bigr{]}(0,\varOmega)}{\theta\bigl{[}\begin{smallmatrix}0\\ 0\end{smallmatrix}\bigr{]}(0,\varOmega)}\exp(-\pi ib\gamma\hskip 1.42262pt^{t}b).\vspace{0.5mm}

by Theorem 2.9. ∎

Summarizing the above discussions, we can compute $[\vartheta_{0}:\vartheta_{1}:\cdots:\vartheta_{7}]\in\mathbb{P}^{7}(\mathbb{C})$ , from a given theta null-point of $A$ . Note that if there exists an index $j\in\{0,\ldots,7\}$ such that ${\vartheta_{j}}^{2}=0$ , then we can take any square root of $\vartheta_{k}^{2}$ for all $k\in\{0,\ldots,7\}\!\smallsetminus\!\{j\}$ . Now, in the following, we consider the $(2,2,2)$ -isogeny

\displaystyle\begin{split}A=\mathbb{C}^{3}/(\mathbb{Z}^{3}+\varOmega\mathbb{Z}^{3})&\longrightarrow\mathbb{C}^{3}/(\mathbb{Z}^{3}+2\varOmega\mathbb{Z}^{3})=B\\[-1.42262pt] z&\longmapsto 2z\end{split}

whose kernel is $\frac{1}{2}\mathbb{Z}^{3}/\mathbb{Z}^{3}$ , and we explain how to compute a theta null-point of the codomain $B$ . Applying Theorem 2.7 for $z=0$ and $\varOmega\mapsto 2\varOmega$ , we obtain

(3.7)

\theta\bigl{[}\begin{smallmatrix}a\\ b\end{smallmatrix}\bigr{]}(0,2\varOmega)^{2}=\frac{1}{2^{3}}\hskip 0.56905pt\sum_{\beta\in\frac{1}{2}\mathbb{Z}^{3}/\mathbb{Z}^{3}}(-1)^{4\hskip 0.56905pt^{t}a\beta}\hskip 0.85358pt\theta\bigl{[}\begin{smallmatrix}0\\ b\end{smallmatrix}\bigr{]}(0,\varOmega)\theta\bigl{[}\begin{smallmatrix}0\\ b+\beta\end{smallmatrix}\bigr{]}(0,\varOmega)\vspace{0.7mm}

for all characteristics $a,b\in\frac{1}{2}\mathbb{Z}^{3}/\mathbb{Z}^{3}$ . Since the right-hand side of (3.7) is obtained by the values $\vartheta_{0},\ldots,\vartheta_{7}$ , one can compute a theta null-point of $B$ . As a summary of the statements in this subsection, we give an explicit algorithm (Algorithm 3.3) to compute a null-point of the codomain of a $(2,2,2)$ -isogeny:

1:A theta null-point

[\vartheta_{0}^{2}:\vartheta_{1}^{2}:\cdots:\vartheta_{63}^{2}]\in\mathbb{P}^{63}

of an abelian threefold

A

2:A theta null-point of

B

, one of abelian threefolds

(2,2,2)

-isogenous to

A

3:if any of

\vartheta_{0}^{2},\ldots,\vartheta_{7}^{2}

is non-zero then

4: Choose a square root of

(\vartheta_{k}/\vartheta_{0})^{2}

for all

k\in\{1,\ldots,6\}

5: Compute

\vartheta_{7}/\vartheta_{0}

from the equation (3.6)

6:else

7: Find an index

j\in\{0,\ldots,7\}

such that

{\vartheta_{j}}^{2}\neq 0

8: Choose a square root of

(\vartheta_{k}/\vartheta_{j})^{2}

for all

k\in\{0,\ldots,7\}\!\smallsetminus\!\{j\}

9:end if

10:for

i\hskip 0.85358pt\in\{0,\ldots,63\}

11: Write

i={a_{3}a_{2}a_{1}b_{3}b_{2}b_{1}}_{(2)}

as a binary and initialize

{\vartheta^{\prime}}_{\!\!i}^{2}\leftarrow 0

12: for

j\in\{0,\ldots,7\}

13: Write

j={\beta_{3}\beta_{2}\beta_{1}}_{(2)}

and set

k\leftarrow i+j\pmod{8}

14: Let

{\vartheta^{\prime}}_{\!\!i}^{2}\leftarrow{\vartheta^{\prime}}_{\!\!i}^{2}+(-1)^{a_{1}\beta_{1}+a_{2}\beta_{2}+a_{3}\beta_{3}}\cdot\vartheta_{j}\vartheta_{k}

15: end for

16:end for

17:return

[{\vartheta^{\prime}}_{\!\!0}^{2}:{\vartheta^{\prime}}_{\!\!1}^{2}:\cdots:{\vartheta^{\prime}}_{\!\!63}^{2}]\in\mathbb{P}^{63}

Algorithm 3.3 ComputeOneRichelotIsogeny

We remark that there exists an index $j\in\{0,\ldots,7\}$ such that ${\vartheta_{j}}^{2}\neq 0$ in Step 5.
Indeed if $\vartheta_{0}=\vartheta_{1}=\cdots=\vartheta_{7}=0$ , then it follows from (3.7) that $\theta\bigl{[}\begin{smallmatrix}a\\ b\end{smallmatrix}\bigr{]}(0,2\varOmega)^{2}=0$ for all characteristics $a,b\in\frac{1}{2}\mathbb{Z}^{3}/\mathbb{Z}^{3}$ , which contradicts Proposition 2.15.

Remark 3.4.

It suffices to do the loops in lines 8–14 only for $i\in I_{\rm even}$ , since we have that ${\vartheta^{\prime}}_{\!\!i}^{2}=0$ for all $i\in I_{\rm odd}$ , in fact.

3.2. How to compute all paths from the vertex

We continue to assume that a theta null-point of a complex abelian threefold $A$ with the period matrix $\varOmega\in\mathcal{H}_{3}$ is given, and let $P,Q,R\in A$ as follows:

Then, the $(2,2,2)$ -isogeny computed in Algorithm 3.3 has the kernel $G=\langle P,\hskip 0.85358ptQ,R\rangle$ , which is determined by a given theta null-point of $A$ . To compute different $(2,2,2)$ -isogenies, we have to act an appropriate matrix $M\in{\rm Sp}_{6}(\mathbb{Z})$ on the theta null-point. Specifically, if we write $M=\begin{pmatrix}\alpha&\beta\\[-0.85358pt] \gamma&\delta\end{pmatrix}$ and let $M.P,M.Q,M.R\in A$ corresponding to

(\alpha_{1j}/2,\alpha_{2j}/2,\alpha_{3j}/2)+\varOmega(\gamma_{1j}/2,\gamma_{2j}/2,\gamma_{3j}/2),\quad{\rm where}\ \alpha=(\alpha_{ij}),\ \gamma=(\gamma_{ij})\vspace{0.5mm}

for $j\in\{1,2,3\}$ respectively, then we can compute the $(2,2,2)$ -isogeny whose kernel is given as $M.G\coloneqq\langle M.P,M.Q,M.R\rangle$ after acting $M$ on the theta null-point. Here, to state a necessary and sufficient condition that $M.G=G$ , we define as follows:

Definition 3.5.

For an integer $n\geq 1$ , the set

\bigl{\{}M=\begin{pmatrix}\alpha&\beta\\[-0.85358pt] \gamma&\delta\end{pmatrix}\in{\rm Sp}_{6}(\mathbb{Z})\mid\gamma\equiv\mathbf{0}_{3}\!\pmod{n}\bigr{\}}\vspace{0.3mm}

is a subgroup of ${\rm Sp}_{6}(\mathbb{Z})$ , and we denote it by $\varGamma_{0}(n)$ .

For a symplectic matrix $M\in{\rm Sp}_{6}(\mathbb{Z})$ , it is well-known (cf. [7, $\S 2.3.2$ ]) that $G$ is invariant (i.e. $M.G=G$ ) under the action by $M$ if and only if $M\in\varGamma_{0}(2)$ . Hence, we need to act an element of ${\rm Sp}_{6}(\mathbb{Z})/\varGamma_{0}(2)$ on the theta null-point of $A$ in order to obtain a different $(2,2,2)$ -isogeny from $A$ .

Remark 3.6.

There exist exactly $N_{3}(2)=135$ different $(2,2,2)$ -isogenies from $A$ by Proposition 2.4, and therefore the order of ${\rm Sp}_{6}(\mathbb{Z})/\varGamma_{0}(2)$ is equal to $135$ . Indeed, Tsuyumine [52, $\S{\rm V}.21$ ] described all the actions by $135$ elements of ${\rm Sp}_{6}(\mathbb{Z})/\varGamma_{0}(2)$ .

For the reader’s convinience, we give a concrete algorithm for generating the list consisting of all different elements of ${\rm Sp}_{6}(\mathbb{Z})/\varGamma_{0}(2)$ as follows:

1:None

2:The list consisting of all different elements of

{\rm Sp}_{6}(\mathbb{Z})/\varGamma_{0}(2)

3:Initialize

\mathcal{M}\leftarrow\emptyset

4:repeat

5: Generate a symplectic matrix

M\in{\rm Sp}_{6}(\mathbb{Z})

randomly

6: if

MN^{-1}\notin\varGamma_{0}(2)

for all

N\in\mathcal{M}

then

7: Add

M

to the list

\mathcal{M}

8: end if

9:until

\#\mathcal{M}=135

10:return

\mathcal{M}

Algorithm 3.7 CollectSymplecticMatrices

Remark 3.8.

This list $\mathcal{M}$ does not depend on the characteristic $p$ of a field or an abelian threefold $A$ . Hence, we only need to run Algorithm 3.7 once, and the result can be used many times thereafter.

In the following, let $\mathcal{M}$ be the output of the function CollectSymplecticMatrices. Then, we can compute theta null-points of $135$ abelian threefolds $(2,2,2)$ -isogenous to $A$ by using Theorem 2.9 as in the following algorithm:

1:A theta null-point

[\vartheta_{0}^{2}:\vartheta_{1}^{2}:\cdots:\vartheta_{63}^{2}]\in\mathbb{P}^{63}

of an abelian threefold

A

2:A list of theta null-points of all abelian threefolds

(2,2,2)

-isogenous to

A

3:Initialize

\mathcal{B}\leftarrow\emptyset

4:for all

M\in\mathcal{M}

5: Write

M=\begin{pmatrix}\alpha&\beta\\[-0.85358pt] \gamma&\delta\end{pmatrix}

where

\alpha,\beta,\gamma

and

\delta

are all

(3\times 3)

-matrices

6: for

i\in I_{\rm even}

7: Let

{\vartheta^{\prime}}_{\!\!i}^{2}

be the value determined from the action in Theorem 2.9

8: end for

9: Add ComputeOneRichelotIsogeny

([{\vartheta^{\prime}}_{\!\!i}^{2}]_{i})

to the list

\mathcal{B}

10:end for

11:return

\mathcal{B}

Algorithm 3.9 ComputeAllRichelotIsogenies

3.3. How to restore a defining equation of a genus-3 curve

In Section 2.3, we introduced some relations between a genus-3 curve $C$ and a theta null-point of the Jacobian of $C$ . However, these are not enough to recover $C$ from a given theta null-point. Specifically, there are the following problems as it stands:

—

If $N_{\rm van}=1$ and $S_{\rm van}\neq\{61\}$ , how should we restore a defining equation of the corresponding genus-3 curve?
—

If $N_{\rm van}=0$ , we have to take some square roots of $\vartheta_{k}^{2}$ to use Theorem 2.17. How should we determine them?

In this subsection, we explain in detail how to do this.

Definition 3.10.

For each $i\in I_{\rm even}$ , we write $\vartheta_{i}=\theta\bigl{[}\begin{smallmatrix}a_{1}&a_{2}&a_{3}\\ b_{1}&b_{2}&b_{3}\end{smallmatrix}\bigr{]}(0,\varOmega)$ and define

P_{i}\coloneqq\begin{pmatrix}\mathbf{1}_{3}&\beta_{i}\\[-0.85358pt] \gamma_{i}&\mathbf{1}_{3}\end{pmatrix},\vspace{-0.8mm}

where

•

If $i\notin\{27,31,45,47,54,55,59,61,62\}$ , then

$\beta_{i}\coloneqq{\rm diag}(b_{1},b_{2},b_{3}),\quad\gamma_{i}\coloneqq{\rm diag}(a_{1},a_{2},a_{3}).\vspace{0.6mm}$

•

Otherwise, we define as follows:

{\small\mbox{$\beta_{27}\coloneqq\begin{pmatrix}1&1&0\\ 1&1&0\\ 0&0&0\end{pmatrix}$}},\quad\gamma_{27}\coloneqq{\small\begin{pmatrix}1&-1&0\\ -1&1&0\\ 0&0&0\end{pmatrix}}.

\beta_{31}\coloneqq{\small\begin{pmatrix}1&1&1\\ 1&1&1\\ 1&1&1\end{pmatrix}},\quad\gamma_{31}\coloneqq{\small\begin{pmatrix}1&-1&0\\ -1&1&0\\ 0&0&0\end{pmatrix}}.

\beta_{45}\coloneqq{\small\begin{pmatrix}1&0&1\\ 0&0&0\\ 1&0&1\end{pmatrix}},\quad\gamma_{45}\coloneqq{\small\begin{pmatrix}1&0&-1\\ 0&0&0\\ -1&0&1\end{pmatrix}}.

\beta_{47}\coloneqq{\small\begin{pmatrix}1&1&1\\ 1&1&1\\ 1&1&1\end{pmatrix}},\quad\gamma_{47}\coloneqq{\small\begin{pmatrix}1&0&-1\\ 0&0&0\\ -1&0&1\end{pmatrix}}.

\beta_{54}\coloneqq{\small\begin{pmatrix}0&0&0\\ 0&1&1\\ 0&1&1\end{pmatrix}},\quad\gamma_{54}\coloneqq{\small\begin{pmatrix}0&0&0\\ 0&1&-1\\ 0&-1&1\end{pmatrix}}.

\beta_{55}\coloneqq{\small\begin{pmatrix}1&1&1\\ 1&1&1\\ 1&1&1\end{pmatrix}},\quad\gamma_{55}\coloneqq{\small\begin{pmatrix}0&0&0\\ 0&1&-1\\ 0&-1&1\end{pmatrix}}.

{\small\beta_{59}\coloneqq\begin{pmatrix}1&-1&0\\ -1&1&0\\ 0&0&0\end{pmatrix}},\quad\gamma_{59}\coloneqq{\small\begin{pmatrix}1&1&1\\ 1&1&1\\ 1&1&1\end{pmatrix}}.

\beta_{61}\coloneqq{\small\begin{pmatrix}1&0&-1\\ 0&0&0\\ -1&0&1\end{pmatrix}},\quad\gamma_{61}\coloneqq{\small\begin{pmatrix}1&1&1\\ 1&1&1\\ 1&1&1\end{pmatrix}}.

\beta_{62}\coloneqq{\small\begin{pmatrix}0&0&0\\ 0&1&-1\\ 0&-1&1\end{pmatrix}},\quad\gamma_{62}\coloneqq{\small\begin{pmatrix}1&1&1\\ 1&1&1\\ 1&1&1\end{pmatrix}}.\vspace{0.3mm}

Lemma 3.11.

For each $i\in I_{\rm even}$ , the matrix $P_{i}$ defined as above belongs to ${\rm Sp}_{6}(\mathbb{Z})$ . In addition, if we suppose that $\vartheta_{0}=0$ , then we obtain $\vartheta_{i}(0,P_{i}\hskip 0.85358pt.\hskip 0.28453pt\varOmega)=0$ .

Proof.

One can check that $\beta_{i},\gamma_{i}$ are both symmetric matrices and $\beta_{i}\gamma_{i}=\gamma_{i}\beta_{i}=\mathbf{0}_{3}$ , which means that $P_{i}\in{\rm Sp}_{6}(\mathbb{Z})$ by Lemma 2.2. In addition for $a=b=0$ , we have

	$\displaystyle P_{i}\hskip 0.85358pt.\hskip 0.56905pta$	$\displaystyle=a\hskip 0.85358pt^{t}\delta_{i}-b\hskip 1.13809pt^{t}\gamma_{i}+\frac{1}{2}(\gamma_{i}\hskip 0.28453pt^{t}\delta_{i})_{0}=\frac{1}{2}(\gamma_{i}\hskip 0.28453pt^{t}\delta_{i})_{0}=(a_{1},a_{2},a_{3}),$
	$\displaystyle P_{i}\hskip 0.85358pt.\hskip 0.56905ptb$	$\displaystyle=b\hskip 1.42262pt^{t}\alpha_{i}-a\hskip 0.85358pt^{t}\beta_{i}+\frac{1}{2}(\alpha_{i}\hskip 0.28453pt^{t}\beta_{i})_{0}=\frac{1}{2}(\alpha_{i}\hskip 0.28453pt^{t}\beta_{i})_{0}=(b_{1},b_{2},b_{3})$

by the equation (2.1). Then, this lemma follows from Theorem 2.9. ∎

In the following, let $A$ be a principally polarized abelian threefold isomorphic to the Jacobian of a genus-3 hyperelliptic curve $C$ and suppose that a theta null-point of $A$ is given. By Proposition 2.15, there uniquely exists $i\in I_{\rm even}$ such that $\vartheta_{i}=0$ . If $i=61$ , we can apply Proposition 2.16 for restoring a defining equation of $C$ , but otherwise not. Here, let $M:=P_{61}{P_{i}}^{-1}\in{\rm Sp}_{6}(\mathbb{Z})$ and $[{\vartheta^{\prime}}_{\!\!i}^{2}]_{i}$ be the theta null-point after acting $M$ on given $[\vartheta_{i}^{2}]_{i}$ , and we have ${\vartheta^{\prime}}_{\!\!61}=0$ by Lemma 3.11. Then one can apply Proposition 2.16 to $[{\vartheta^{\prime}}_{\!\!i}^{2}]_{i}$ , which enables us to recover $C$ as follows:

1:A theta null-point

[\vartheta_{i}^{2}]_{i}\in\mathbb{P}^{63}

with

N_{\rm van}=1

of a p.p.a.v.

A

\dim=3

2:A hyperelliptic curve

C

of genus

3

such that

A\cong{\rm Jac}(C)

3:Let

i\in I_{\rm even}

such that

\vartheta_{i}^{2}=0

and define

M:=P_{61}{P_{i}}^{-1}

4:Write

M=\begin{pmatrix}\alpha&\beta\\[-0.85358pt] \gamma&\delta\end{pmatrix}

where

\alpha,\beta,\gamma

and

\delta

are all

(3\times 3)

-matrices

5:for

i\in I_{\rm even}

6: Let

{\vartheta^{\prime}}_{\!\!i}^{2}

be the value determined from the action in Theorem 2.9

7:end for

8:Compute

\lambda_{1},\ldots,\lambda_{5}

from the theta null-point

[{\vartheta^{\prime}}_{\!\!i}^{2}]_{i}\in\mathbb{P}^{63}

by the equation (2.2)

9:return the curve

y^{2}=x(x-1)\prod_{k=1}^{5}(x-\lambda_{k})

Algorithm 3.12 RestoreHyperellipticCurve

Next, we suppose that a theta null-points of $A={\rm Jac}(C)$ is given, where $C$ is a smooth plane quartic. In the following, we explain how to recover $C$ according to the method in [35, $\S 6.4$ ]. It follows from Theorem 2.6 that

(3.8)

\left\{\begin{array}[]{l}\vartheta_{0}\vartheta_{16}\vartheta_{45}\vartheta_{61}=\vartheta_{5}\vartheta_{21}\vartheta_{40}\vartheta_{56}-\vartheta_{12}\vartheta_{28}\vartheta_{33}\vartheta_{49},\\ \vartheta_{12}\vartheta_{21}\vartheta_{33}\vartheta_{56}=\vartheta_{5}\vartheta_{28}\vartheta_{40}\vartheta_{49}-\vartheta_{2}\vartheta_{27}\vartheta_{47}\vartheta_{54},\\ \vartheta_{3}\vartheta_{20}\vartheta_{32}\vartheta_{55}=\vartheta_{2}\vartheta_{21}\vartheta_{33}\vartheta_{54}+\vartheta_{12}\vartheta_{27}\vartheta_{47}\vartheta_{56},\end{array}\right.

(3.9)

\left\{\begin{array}[]{l}\vartheta_{7}\vartheta_{14}\vartheta_{35}\vartheta_{42}=\vartheta_{5}\vartheta_{12}\vartheta_{33}\vartheta_{40}-\vartheta_{21}\vartheta_{28}\vartheta_{49}\vartheta_{56},\\ \vartheta_{14}\vartheta_{16}\vartheta_{35}\vartheta_{61}=\vartheta_{2}\vartheta_{28}\vartheta_{47}\vartheta_{49}-\vartheta_{5}\vartheta_{27}\vartheta_{40}\vartheta_{54},\\ \vartheta_{7}\vartheta_{16}\vartheta_{42}\vartheta_{61}=\vartheta_{2}\vartheta_{21}\vartheta_{47}\vartheta_{56}-\vartheta_{12}\vartheta_{27}\vartheta_{33}\vartheta_{54}\end{array}\right.

for appropriate inputs $m_{1},m_{2},m_{3},m_{4}$ . Squaring both sides of (3.8), we obtain

(3.10)

\displaystyle\begin{split}2\vartheta_{5}\vartheta_{12}\vartheta_{21}\vartheta_{28}\vartheta_{33}\vartheta_{40}\vartheta_{49}\vartheta_{56}&=\vartheta_{5}^{2}\vartheta_{21}^{2}\vartheta_{40}^{2}\vartheta_{56}^{2}+\vartheta_{12}^{2}\vartheta_{28}^{2}\vartheta_{33}^{2}\vartheta_{49}^{2}-\vartheta_{0}^{2}\vartheta_{16}^{2}\vartheta_{45}^{2}\vartheta_{61}^{2},\\ 2\vartheta_{2}\vartheta_{5}\vartheta_{27}\vartheta_{28}\vartheta_{40}\vartheta_{47}\vartheta_{49}\vartheta_{54}&=\vartheta_{5}^{2}\vartheta_{28}^{2}\vartheta_{40}^{2}\vartheta_{49}^{2}+\vartheta_{2}^{2}\vartheta_{27}^{2}\vartheta_{47}^{2}\vartheta_{54}^{2}-\vartheta_{12}^{2}\vartheta_{21}^{2}\vartheta_{33}^{2}\vartheta_{56}^{2},\\ 2\vartheta_{2}\vartheta_{12}\vartheta_{21}\vartheta_{27}\vartheta_{33}\vartheta_{47}\vartheta_{54}\vartheta_{56}&=\vartheta_{3}^{2}\vartheta_{20}^{2}\vartheta_{32}^{2}\vartheta_{55}^{2}-\vartheta_{2}^{2}\vartheta_{21}^{2}\vartheta_{33}^{2}\vartheta_{54}^{2}-\vartheta_{12}^{2}\vartheta_{27}^{2}\vartheta_{47}^{2}\vartheta_{56}^{2}.\end{split}

By using these equations, we can restore a defining equation of a plane quartic $C$ , as follows:

1:A theta null-point

[\vartheta_{i}^{2}]_{i}\in\mathbb{P}^{63}

with

N_{\rm van}=0

of a p.p.a.v.

A

\dim=3

2:A plane quartic

C

of genus

3

such that

A\cong{\rm Jac}(C)

3:Compute

{a_{11}}^{\!\!\!2},\,{a_{12}}^{\!\!\!2},\,{a_{13}}^{\!\!\!2}

by the equation (2.3) and choose their square roots

4:Compute

\vartheta_{5}\vartheta_{12}\vartheta_{33}\vartheta_{40},\,\vartheta_{5}\vartheta_{27}\vartheta_{40}\vartheta_{54},\,\vartheta_{12}\vartheta_{27}\vartheta_{33}\vartheta_{54}

from values

a_{11},a_{21},a_{31}

5:Compute

\vartheta_{21}\vartheta_{28}\vartheta_{49}\vartheta_{56},\,\vartheta_{2}\vartheta_{28}\vartheta_{47}\vartheta_{49},\,\vartheta_{2}\vartheta_{12}\vartheta_{33}\vartheta_{56}

from the equation (3.10)

6:Compute

\vartheta_{7}\vartheta_{14}\vartheta_{35}\vartheta_{42},\,\vartheta_{14}\vartheta_{16}\vartheta_{35}\vartheta_{61},\,\vartheta_{7}\vartheta_{16}\vartheta_{42}\vartheta_{61}

from the equation (3.9)

7:Compute

a_{12},a_{13},a_{22},a_{23},a_{32},

and

a_{33}

from the equation (2.3)

8:Compute

k_{1},k_{2},k_{3}

from the equation (2.5)

9:Compute homogenous linear polynomials

\xi_{1},\xi_{2},\xi_{3}

from Theorem 2.17

10:return the curve

(\xi_{1}x+\xi_{2}y-\xi_{3}z)^{2}-4\xi_{1}\xi_{2}xy=0

Algorithm 3.13 RestorePlaneQuartic

3.4. Algorithm for listing superspecial genus-3 curves

Let $p>2$ be a prime integer, and all principally polarized abelian varieties considered in this subsection are defined over a field of positive characteristic $p$ . We emphasize that, even in this situation, all the algorithms until the previous subsection still work, thanks to the Mumford’s theory [37] (see also [11, $\S 2$ ]). First, we prepare the following theorem:

Theorem 3.14.

Let $\hskip 0.56905ptC$ be a superspecial genus- $g$ curve in odd characteristic $p>2$ . Then, the theta null-point of its Jacobian is $\mathbb{F}_{p^{2}}$ -rational.

Proof.

First, recall from [45, Corollary 2.11.4] that a theta null-point of a principally polarized abelian variety $A$ of dimension $g$ is $\mathbb{F}_{p^{2}}$ -rational if and only if there exists a symplectic basis $(e_{1},\ldots,e_{g},{e^{\prime}}_{1},\ldots,{e^{\prime}}_{g})\in A[2]$ such that

(i)

all $e_{i},{e^{\prime}}_{i}$ are $\mathbb{F}_{p^{2}}$ -rational, and
(ii)

$e_{T,2}(e_{i},e_{i})=e_{T,2}({e^{\prime}}_{i},{e^{\prime}}_{i})=1$ where $e_{T,2}$ denotes the $2$ -Tate pairing.

Here, it is well-known (cf. [29, Theorem 2.6]) that a superspecial curve $C$ descends to a curve $C^{\prime}$ over $\mathbb{F}_{p^{2}}$ where the $p^{2}$ -power Frobenius map $\pi_{p^{2}}$ is equal to $\pm p$ . Then, it follows from Theorem 2.9 that it suffices to show that the theta null-point of ${\rm Jac}(C^{\prime})$ is $\mathbb{F}_{p^{2}}$ -rational, since $\sqrt{-1}$ belongs to $\mathbb{F}_{p^{2}}$ .

By Ekedahl’s result [14, Proposition 1.2], there is an isomorphism ${\rm Jac}(C^{\prime})\cong E^{g}$ over $\mathbb{F}_{p^{2}}$ as a non-polarized abelian variety where $E$ is a supersingular elliptic curve. Since all the $2$ -torsion points of $E$ are $\mathbb{F}_{p^{2}}$ -rational, all $e_{i},{e^{\prime}}_{i}$ are $\mathbb{F}_{p^{2}}$ -rational for any symplectic basis $(e_{1},\ldots,e_{g},{e^{\prime}}_{1},\ldots,{e^{\prime}}_{g})\in{\rm Jac}(C^{\prime})[2]$ . Moreover, if we write $2f_{i}=e_{i}$ with $f_{i}\in{\rm Jac}(C^{\prime})[4]$ for each $i\in\{1,\ldots,g\}$ , we have that

	$\displaystyle e_{T,2}(e_{i},e_{i})$	$\displaystyle=e_{W,2}(e_{i},\pi_{p^{2}}(f_{i})-f_{i})$
		$\displaystyle=e_{W,2}(e_{i},(\pm p-1)f_{i})=e_{W,2}(e_{i},e_{i})^{(\pm p-1)/2}=1$

where $e_{W,2}$ denotes the $2$ -Weil pairing. In the same way, we obtain $e_{T,2}({e^{\prime}}_{i},{e^{\prime}}_{i})=1$ . Since ${\rm Jac}(C^{\prime})$ satisfies the two previous conditions, and hence its theta null-point is $\mathbb{F}_{p^{2}}$ -rational. This completes the proof of the theorem. ∎

Corollary 3.15.

The theta null-point $[\vartheta_{0}^{2}:\vartheta_{1}^{2}:\cdots:\vartheta_{63}^{2}]\in\mathbb{P}^{63}$ of any superspecial principally polarized abelian threefold is $\mathbb{F}_{p^{2}}$ -rational.

Proof.

Recall from the beginning of Section 2.3 that every superspecial principally polarized abelian threefold is isomorphic to the Jacobian of a superspecial curve or a product of them. In the former case, the theta null-point is $\mathbb{F}_{p^{2}}$ -rational by using Theorem 3.14. In the latter case as well, the assertion follows from Lemma 2.8. ∎

Now, we can list all superspecial genus-3 curves in characteristic $p>7$ as in the following Algorithm 3.16 (as inputs $\Lambda_{1,p}$ and $\Lambda_{2,p}$ ).

1:A prime integer

p>7

and the two lists

\Lambda_{1,p}

and

\Lambda_{2,p}

2:The list

\Lambda_{3,p}

of isom. classes of s.sp. genus-3 curves

3:Initialize

\mathcal{L}_{1}\leftarrow\emptyset,\,\mathcal{L}_{2}\leftarrow\emptyset

, and

\mathcal{S}\leftarrow\emptyset

4:Let

\mathcal{L}_{4}

be the list of isom. classes of

E_{1}\times E_{2}\times E_{3}

with

E_{1},E_{2},E_{3}\in\Lambda_{1,p}

5:Let

\mathcal{L}_{3}

be the list of isom. classes of

E\times{\rm Jac}(C)

with

E\in\Lambda_{1,p}

and

C\in\Lambda_{2,p}

6:Let

\mathcal{S}

be the list of theta null-points of threefolds in

\mathcal{L}_{3}\sqcup\mathcal{L}_{4}

, and set

k\leftarrow 1

7:repeat

8: Let

[\vartheta^{2}_{i}]_{i}

be the

k

-th theta null-point of

\mathcal{S}

9: for all

[{\vartheta^{\prime}}_{\!\!i}^{2}]_{i}\in\hskip 2.27621pt

ComputeAllRichelotIsogenies(

[\vartheta^{2}_{i}]_{i}

) do

10: Compute the number

N_{\rm van}

of indices

i\in I_{\rm even}

such that

{\vartheta^{\prime}}_{\!\!i}^{2}=0

11: if

N_{\rm van}=0

then

12: Let

inv^{\prime}

be the Dixmier-Ohno invariant of

{\sf RestorePlaneQuartic}([{\vartheta^{\prime}}_{\!\!i}^{2}]_{i})

13: if

inv^{\prime}\notin\mathcal{L}_{1}

then

14: Add

inv^{\prime}

to the list

\mathcal{L}_{1}

and

[{\vartheta^{\prime}}_{\!\!i}^{2}]_{i}

to the end of

\mathcal{S}

15: end if

16: else if

N_{\rm van}=1

then

17: Let

inv^{\prime}

be the Shioda invariant of

{\sf RestoreHyperellipticCurve}([{\vartheta^{\prime}}_{\!\!i}^{2}]_{i})

18: if

inv^{\prime}\notin\mathcal{L}_{2}

then

19: Add

inv^{\prime}

to the list

\mathcal{L}_{2}

and

[{\vartheta^{\prime}}_{\!\!i}^{2}]_{i}

to the end of

\mathcal{S}

20: end if

21: end if

22: end for

23: Set

k\leftarrow k+1

24:until

k\hskip 0.56905pt>\#\mathcal{S}

25:return the list of genus-3 curves whose invariants belong to

\mathcal{L}_{1}\sqcup\mathcal{L}_{2}

Algorithm 3.16 ListUpSspGenus3Curves

In the following, we provide some notes regarding the above Algorithm 3.16.

—

In line 4, we need compute theta null-points of all superspecial principally polarized abelian threefolds of Types 3 and 4. We explain how to get them: The theta null-point of an elliptic curve $y^{2}=x(x-1)(x-t)$ can be given as $[1:\sqrt{t}:\sqrt{1-t}:0]\in\mathbb{P}^{3}$ using Thomae’s formula (cf. [38, Theorem 8.1]). Also, applying formulae in [19, $\S 7.5$ ] or [8, $\S 5.1$ ], one can compute the theta null-point of the Jacobian of a genus-2 curve. Then, Lemma 2.8 enables us to compute the theta null-point of their product.

Remark 3.17.

To generate the list $\Lambda_{2,p}$ , we can also use theta functions (without going into detail), instead of Algorithm 2.21. If we use such a method, then we get theta null-points of all superspecial abelian surfaces in process, and we do not need to use formulas in [19, $\S 7.5$ ] or [8, $\S 5.1$ ].

—

In lines 7–20, we compute all the $135$ edges starting from a vertex in turn. If the codomain corresponds to an abelian threefold of Type 1, we compute the Dixmier-Ohno invariants (cf. [13], [40]) of the underlying plane quartic. If the codomain corresponds to a Type 2 one, then we compute the Shioda invariants (cf. [48]) of the underlying hyperelliptic curve of genus 3. These invariants are given by polynomials of degree bounded by constants, thus these can be computed in constant time. We note that there is no need to do anything for the codomain of Type 3 or 4 since they have already been obtained in lines 2–3.

Remark 3.18.

If smooth plane quartics $C$ and $C^{\prime}$ over an algebraically closed field have different Dixmier-Ohno invariants, then they are not isomorphic to each other. It is proven that the converse holds when the characteristic is $0$ , and it is suspected that this similarly holds in the case of characteristic $p>7$ , see [33] for details. This is the main reason why we assume that $p>7$ in Algorithm 3.16.

—

In line 22, the condition $k\hskip 0.56905pt>\#\mathcal{S}$ can be replaced by $\#\mathcal{L}_{1}+\#\mathcal{L}_{2}=\#\Lambda_{3,p}$ , where $\#\Lambda_{3,p}$ denotes the number of superspecial genus-3 curves, which can be obtained from Brock [4, Theorem 3.10]. The algorithm terminates faster with the modified condition; however, in our computational experiment, we ran Algorithm 3.16 with the original condition for reasons explained later.

Finally let us show Theorem 1.1, which asserts the complexity of Algorithm 3.16. (even if the condition in line 22 is replaced, the complexity remains the same).

Proof of Theorem 1.1.

It follows from Corollary 3.15 that all the theta null-points appearing in Algorithm 3.16 are $\mathbb{F}_{p^{2}}$ -rational. In lines 7 and 10, it is necessary to take their square roots to obtain theta constants, which belong to $\mathbb{F}_{p^{4}}$ . Then, one can see that all the computations in Algorithm 3.16 can be done over $\mathbb{F}_{p^{4}}$ .

It follows from

\#\Lambda_{1,p}=O(p),\quad\#\Lambda_{2,p}=O(p^{3})\quad{\rm and}\quad\#\Lambda_{3,p}=O(p^{6})

that the complexity of line 2 and line 3 is given by $\widetilde{O}(p^{3})$ and $\widetilde{O}(p^{4})$ , respectively. Next, lines 6–20 are repeated $O(p^{6})$ times since ultimately $\#\mathcal{S}$ equals the number of isomorphism classes of superspecial principally polarized abelian threefolds. For each loop, lines 8–19 are executed exactly $135$ times, and hence all operations can be performed in polynomial time of $\log{p}$ . Therefore, Algorithm 3.16 can be done with $\widetilde{O}(p^{6})$ operations over $\mathbb{F}_{p^{4}}$ . ∎

4. Computational results

In this section, we give some computational results obtained by executing main algorithm. We implemented the algorithms with Magma V2.26-10 and ran it on a machine with an AMD EPYC $7742$ CPU and $2$ TB of RAM.

4.1. Enumeration of superspecial hyperelliptic genus-3 curves

In this subsection, we enumerate superspecial hyperelliptic genus-3 curves in characteristics $p$ with $11\leq p<100$ . For this, we only need to modify Algorithm 3.16 to output the cardinalities of lists $\mathcal{L}_{1},\mathcal{L}_{2},\mathcal{L}_{3}$ , and $\mathcal{L}_{4}$ . We note that $\#\mathcal{L}_{1}$ (resp. $\#\mathcal{L}_{2}$ ) is equal to the number of isomorphism classes of non-hyperelliptic (resp. hyperelliptic) curves of genus 3.

Remark 4.1.

For the cardinalities of $\mathcal{L}_{3}$ and $\mathcal{L}_{4}$ , we have

\#\mathcal{L}_{3}=\#\Lambda_{1,p}\cdot\#\Lambda_{2,p},\quad\#\mathcal{L}_{4}=\binom{\#\Lambda_{1,p}+2}{3}.\vspace{0.3mm}

Here, the numbers $\#\Lambda_{1,p}$ and $\#\Lambda_{2,p}$ are obtained by [12] and [24].

The following table (Table 1) shows the cardinalities of four lists $\mathcal{L}_{1},\mathcal{L}_{2},\mathcal{L}_{3},\mathcal{L}_{4}$ , and the time it took. As mentioned earlier, the time can be shortened by changing the condition $k\hskip 0.56905pt>\#\mathcal{S}$ in line 22 to $\#\mathcal{L}_{1}+\#\mathcal{L}_{2}=\#\Lambda_{3,p}$ . However, for the purpose of verifying that our result are consistent with Brock’s result [4, Theorem 3.10], we executed it without changing the condition in the following experiment:

char.	$\#\mathcal{L}_{1}$	$\#\mathcal{L}_{2}$	$\#\mathcal{L}_{3}$	$\#\mathcal{L}_{4}$	total	compl	Time (sec)
$p=11$	10	1	4	4	19	7	9.660
$p=13$	18	1	3	1	23	5	7.790
$p=17$	54	2	10	4	70	24	19.320
$p=19$	87	4	14	4	109	48	29.860
$p=23$	213	9	30	10	262	70	66.070
$p=29$	681	10	54	10	755	249	189.090
$p=31$	950	27	60	10	1047	249	258.710
$p=37$	2448	35	93	10	2586	591	615.930
$p=41$	4292	54	160	20	4526	1754	1070.190
$p=43$	5567	82	180	20	5849	2685	1385.360
$p=47$	9138	125	285	35	9583	3277	2295.400
$p=53$	18032	153	390	35	18610	6663	4484.100
$p=59$	33204	299	624	56	34183	11964	8719.060
$p=61$	40259	262	565	35	41121	13015	10670.860
$p=67$	69132	451	870	56	70509	25780	19646.600
$p=71$	96717	647	1190	84	98638	30861	29135.890
$p=73$	113778	582	1098	56	115514	35823	34888.620
$p=79$	180273	942	1596	84	182895	51592	59625.370
$p=83$	240755	1136	2080	120	244091	70445	83296.450
$p=89$	362720	1402	2528	120	366770	122626	134065.830
$p=97$	602062	2002	3200	120	607384	208415	230401.380

Table 1. The cardinalities of lists

\mathcal{L}_{1},\mathcal{L}_{2},\mathcal{L}_{3}

and

\mathcal{L}_{4}

Notably, the final results obtained show that $\#\mathcal{L}_{1}+\#\mathcal{L}_{2}=\#\Lambda_{3,p}$ , and therefore our result is consistent with Brock’s result.

Here, the column total (resp. compl) indicates the number of times lines 6–21 are performed until the condition $k\hskip 0.56905pt>\#\mathcal{S}$ (resp. $\#\mathcal{L}_{1}+\#\mathcal{L}_{2}=\#\Lambda_{3,p}$ ) is satisfied. Note that ${\tt total}$ is equal to the sum of $\#\mathcal{L}_{1},\#\mathcal{L}_{2},\#\mathcal{L}_{3}$ , and $\#\mathcal{L}_{4}$ . As can be seen in Table 1, the ratio ${\tt compl}/{\tt total}$ is approximately equal to $1/3$ for each $p$ . Hence, if the termination condition is changed to $\#\mathcal{L}_{1}+\#\mathcal{L}_{2}=\#\Lambda_{3,p}$ , and then the time required is considered to be only $1/3$ of the time written in Table 1. For example, listing up all superspecial genus-3 curves in characteristic $p=97$ takes about a day.

4.2. The existence of a superspecial hyperellitptic genus-3 curve

In this subsection, we check that there exists at least one superspecial hyperelliptic curve of genus $3$ in characteristic $p$ with $7\leq p<10000$ .

Remark 4.2.

Moriya-Kudo’s method [36] is more efficient for finding such curves, but their method can only generate curves whose automorphism groups contain the Klein $4$ -group. Even if their approach cannot find such a curve for some $p$ , there is a possibility that our approach will find them.

First, the sufficient condition for the characteristic $p$ for such a curve to exist is given by the following proposition:

Theorem 4.3 ([4, Theorem 3.15] or [53, Theorem 2]).

(1)

The curve $y^{2}=x^{7}-1$ is superspecial if and only if $p\equiv 6\pmod{7}$ .
(2)

The curve $y^{2}=x^{7}-x$ is superspecial if and only if $p\equiv 3\pmod{4}$ .
(3)

The curve $y^{2}=x^{8}-1$ is superspecial if and only if $p\equiv 7\pmod{8}$ .

By the above theorem, there exists a superspecial hyperelliptic curve of genus $3$ if $p\equiv 6\pmod{7}$ or $p\equiv 3\pmod{4}$ . To investigate the other cases, we implemented the probabilistic algorithm (Algorithm 4.4) which outputs such a curve:

1:A prime integer

p\geq 7

2:A superspecial hyperelliptic curve of genus

3

in characteristic

p

3:Generate a supersingular elliptic curve

E

and

\mathcal{M}\leftarrow{\sf CollectSymplecticMatrices}()

4:Compute a theta null-point

[{\vartheta^{\prime}}_{\!\!i}^{2}]_{i}

E^{3}

by using Lemma 2.8

5:repeat

6: Set

[\vartheta_{i}^{2}]_{i}\leftarrow[{\vartheta^{\prime}}_{\!\!i}^{2}]_{i}

and choose a symplectic matrix

M\in\mathcal{M}

randomly

7: Write

M=\begin{pmatrix}\alpha&\beta\\[-0.85358pt] \gamma&\delta\end{pmatrix}

, where

\alpha,\beta,\gamma

, and

\delta

are

(3\times 3)

-matrices

8: for

i\in I_{\rm even}

9: Let

{\vartheta^{\prime}}_{\!\!i}^{2}

be the value determined from the action in Theorem 2.9

10: end for

11: Compute the number

N_{\rm van}

of indices

i\in I_{\rm even}

such that

{\vartheta^{\prime}}_{\!\!i}^{2}=0

12:until

N_{\rm van}=1

13:return

{\sf RestoreHyperellipticCurve}([{\vartheta^{\prime}}_{\!\!i}^{2}]_{i})

Algorithm 4.4 GenerateSingleSSpHypGenus3Curve

Then, one can obtain a superspecial hyperelliptic genus-3 curve as follows:

•

If $p\equiv 6\pmod{7}$ , then output the curve $y^{2}=x^{7}-1$ .
•

If $p\equiv 3\pmod{4}$ , then output the curve $y^{2}=x^{7}-x$ .
•

Otherwise, execute the Algorithm 4.4.

We remark that Algorithm 4.4 may return different outputs depending on randomness in lines $1$ and $4$ .

Executing the above procedure, we have checked that a superspecial hyperelliptic genus-3 curve exists for all characteristic $7\leq p<10000$ . As Algorithm 4.4 performs a random walk on $\mathcal{G}_{3}(2,p)$ , the time to output such a curve varies, but we will list them below for reference:

•

For each $7\leq p<10000$ , the above procedure ends within 10,000 seconds. Moreover for almost $p$ , it took less than 3,600 seconds.
•

It took 9,952 seconds for $p=8893$ , which is the maximal time for $p<10000$ .
•

It took very short time for some $p$ (e.g. about $30$ seconds for $p=9209$ ).

4.3. Drawing the graph $\mathcal{G}_{3}(2,p)$ for small $p$

By slightly modifying main algorithm, one can draw the graph $\mathcal{G}_{3}(2,p)$ for small $p$ as follows:

Refer to caption — Figure 1. The graph $\mathcal{G}_{3}(2,11)$

In the above figures, we write abelian threefolds of Types $1$ , $2$ , $3$ , and $4$ are written as blue, green, yellow, and red vertices respectively. Moreover, to improve visibility, we draw all the graphs as undirected graphs.

5. Concluding remarks

We proposed an algorithm (Algorithm 3.16) for computing the Richelot isogeny graph of superspecial abelian threefolds in characteristic $p>7$ . As applications of this algorithm, we succeeded in listing up superspecial genus-3 curves for $p<100$ (Section 4.1), and finding one superspecial hyperelliptic curve of genus 3 for all $p$ with $7\leq p<10000$ (Section 4.2). Our implementation with the Magma algebraic system [3] for listing up superspecial genus-3 curves is available at

https://github.com/Ryo-Ohashi/sspg3list.

The complexity of our algorithm is estimated as $\widetilde{O}(p^{6})$ operations in the field $\mathbb{F}_{p^{4}}$ . However, in our computational experiments, all the operations were performed over the field $\mathbb{F}_{p^{2}}$ in practice, and hence it may be possible to assert something stronger from Theorem 1.1.

As other future works, it would be interesting to list up superspecial curves of genus $3$ in characteristic $p>100$ . However, since enumerating in $p=97$ took about a day, enumerating even in $p=199$ can be expected to take more than two months. Hence, improving the algorithm in Section 4.1 is an important task. This requires an efficient search to obtain new nodes on the graph $\mathcal{G}_{3}(2,p)$ , and then we need to investigate the structure of $\mathcal{G}_{3}(2,p)$ in detail.

On the other hand, we can expect from the results in Table 1 that

Expectation 5.1.

The number of isomorphism classes of superspecial hyperelliptic curves of genus $3$ in characteristic $p$ is $O(p^{4})$ .

Recall that the order of magnitude of all superspecial genus-3 curves is $O(p^{6})$ , then the proportion of hyperelliptic ones among them can be estimated to be about $1/p^{2}$ under this expectation. A theoretical proof of it is also one of our future works.

Acknowledgements

The authors are grateful to Damien Robert and Everett Howe for their helpful comments on earlier version of this paper. This research was conducted under a contract of “Research and development on new generation cryptography for secure wireless communication services” among “Research and Development for Expansion of Radio Wave Resources (JPJ000254)”, which was supported by the Ministry of Internal Affairs and Communications, Japan. This work was also supported by JSPS Grant-in-Aid for Young Scientists 20K14301 and 23K12949, and WISE program (MEXT) at Kyushu University.

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Ryo Ohashi: Graduate School of Information Science and Technology, The University of Tokyo — 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan.

E-mail address: [email protected].

Hiroshi Onuki: Graduate School of Information Science and Technology, The University of Tokyo — 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan.

E-mail address: [email protected].

Momonari Kudo: Fukuoka Institute of Technology — 3-30-1 Wajiro-higashi, Higashi-ku, Fukuoka, 811-0295, Japan.

E-mail address: [email protected].

Ryo Yoshizumi: Joint Graduate Program of Mathematics for Innovation, Kyushu University — 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan.

E-mail address: [email protected].

Koji Nuida: Institute of Mathematics for Industry, Kyushu university — 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan / National Institute of Advanced Industrial Science and Technology (AIST) — 2-3-26 Aomi, Koto-ku, Tokyo, 135-0064, Japan.

E-mail address: [email protected].

Listing superspecial curves of genus three by using Richelot isogeny graph

Abstract.

1. Introduction

Theorem 1.1.

Theorem 1.2.

2. Preliminaries

2.1. Abelian varieties and their isogenies

Definition 2.1.

Lemma 2.2 ([32, Lemma 8.2.1]).

Proposition 2.3 ([32, Proposition 8.1.3]).

Proposition 2.4.

Proof.

2.2. Theta functions

Lemma 2.5.

Proof.

Theorem 2.6 (Riemann’s theta formula).

Proof.

Theorem 2.7 (Duplication formula).

Proof.

Lemma 2.8 ([10, Lemma F.3.1]).

Theorem 2.9 (Theta transformation formula).

Proof.

Corollary 2.10.

Proof.

2.3. Theta functions for abelian threefolds

Definition 2.11.

Definition 2.12.

Definition 2.13.

Lemma 2.14.

Proof.

Proposition 2.15.

Proof.

Proposition 2.16.

Proof.

Theorem 2.17 (Weber’s formula).

Proof.

2.4. Superspecial Richelot isogeny graph

Definition 2.18.

Definition 2.19.

Theorem 2.20.

Proof.

Remark 2.22.

3. Computing the Richelot isogeny graph of dimension 3

3.1. How to compute one path from the vertex

Lemma 3.1.

Proof.

Proposition 3.2.

Proof.

Remark 3.4.

3.2. How to compute all paths from the vertex

Definition 3.5.

Remark 3.6.

Remark 3.8.

3.3. How to restore a defining equation of a genus-3 curve

Definition 3.10.

Lemma 3.11.

Proof.

3.4. Algorithm for listing superspecial genus-3 curves

Theorem 3.14.

Proof.

Corollary 3.15.

Proof.

Remark 3.17.

Remark 3.18.

Proof of Theorem 1.1.

4. Computational results

4.1. Enumeration of superspecial hyperelliptic genus-3 curves

Remark 4.1.

4.2. The existence of a superspecial hyperellitptic genus-3 curve

Remark 4.2.

Theorem 4.3 ([4, Theorem 3.15] or [53, Theorem 2]).

4.3. Drawing the graph 𝒢3​(2,p)\mathcal{G}_{3}(2,p) for small pp

5. Concluding remarks

Expectation 5.1.

Acknowledgements

References

4.3. Drawing the graph $\mathcal{G}_{3}(2,p)$ for small $p$