arXiv subjects

Ryo Ohashi

Publications and source records attributed to Ryo Ohashi.

14 recordsLinked to original sources

arXiv2026

On superspecial hyperelliptic curves of Rosenhain forms

Any genus-$g$ hyperelliptic curve $C$ defined over an algebraically closed field of characteristic $p \geq 3$ can be written in a Rosenhain form as $y^2 = x(x-1)\prod_{i=1}^{2g-1}(x-\lambda_i)$. In this paper, we first show that, if $C$ is superspecial, then each of $\lambda_i,1-\lambda_i$, and $\lambda_i-\lambda_j$ is a square in $\mathbb{F}_{p^2}$. As an application, we propose a new algorithm for enumerating superspecial hyperelliptic curves in small characteristic. By implementing our algorithm, we successfully computed the number of isomorphism classes of such curves of genera $4$ and $5$ in all characteristics $p \leq 41$, and of genus $6$ in all characteristics $p \leq 31$.

math.AG↗

arXiv2026

Superspecial plane quintics with large automorphism groups

In this paper, we study plane quintic curves whose automorphism groups have order greater than 10, as well as those with cyclic automorphism groups of order 8 and 10. The latter two cases are represented as one-parameter families, where their superspeciality can be explicitly described in terms of a truncation of certain Gaussian hypergeometric series. Applying this characterization, we determine the exact number of isomorphism classes of superspecial plane quintic curves with automorphism groups $\cong \mathbb{Z}/10\mathbb{Z}$. We also provide an efficient algorithm to enumerate such curves with automorphism groups $\cong \mathbb{Z}/8\mathbb{Z}$, and provide the computational results for the range $13 < p < 10000$.

math.AG↗

arXiv2026

Generalized Howe curves of genus 4, 5, and 6 with completely decomposable Jacobians

Superspecial curves are important objects in number theory and algebraic geometry, and the existence in genus $g \geq 4$ remains an open problem for all but finitely many characteristics $p > 0$. As a computational approach to this problem, Kudo-Harashita-Howe (2020) showed that a superspecial curve of genus 4 exists in each characteristic $p$ with $7 < p < 20000$. Their method restricted attention to a specific class of curves, known as Howe curves, for which superspeciality is reduced to those of curves of genus at most 2. In this paper, we focus on a more specific class of curves, namely Howe curves whose Jacobians decompose into a product of four elliptic curves. By restricting our attention to such curves, the superspeciality reduces to the supersingularity of elliptic curves, which enables us to construct a superspecial curve of genus 4 more efficiently than Kudo-Harashita-Howe's method. As our first main result, we confirmed by computer the existence of such superspecial curves of genus 4 in characteristics $p$ with $20000 < p < 10^6$. Using a similar approach, we also propose constructions of superspecial curves of genera 5 and 6 from only supersingular elliptic curves. Furthermore, computational experiments establish the existence of superspecial curves of genus 5 (resp. genus 6) in characteristics $p$ with $13 < p < 10^5$ (resp. $7 < p < 10^5$).

math.AG↗

arXiv2026

On superspecial hyperelliptic curves of genus 5 whose automorphism groups contain $(\mathbb{Z}/2\mathbb{Z})^3$

While the numbers of superspecial curves of genus at most 3 are well understood, and several computational approaches have been developed to count superspecial curves of genus 4 with large automorphism groups, much less is known in higher genera. In this paper, we construct a feasible algorithm to enumerate superspecial hyperelliptic curves of genus 5 whose automorphism groups contain $(\mathbb{Z}/2\mathbb{Z})^3$. We implement and executing our algorithm in Magma, we succeeded in enumerating such superspecial curves in every characteristic $11 < p < 1000$.

math.AG↗

arXiv2025

Generalized Poisson Matrix Factorization for Overdispersed Count Data

Non-negative matrix factorization (NMF) is widely used as a feature extraction technique for matrices with non-negative entries, such as image data, purchase histories, and other types of count data. In NMF, a non-negative matrix is decomposed into the product of two non-negative matrices, and the approximation accuracy is evaluated by a loss function. If the Kullback-Leibler divergence is chosen as the loss function, the estimation coincides with maximum likelihood under the assumption that the data entries are distributed according to a Poisson distribution. To address overdispersion, negative binomial matrix factorization has recently been proposed as an extension of the Poisson-based model. However, the negative binomial distribution often generates an excessive number of zeros, which limits its expressive capacity. In this study, we propose a non-negative matrix factorization based on the generalized Poisson distribution, which can flexibly accommodate overdispersion, and we introduce a maximum likelihood approach for parameter estimation. This methodology provides a more versatile framework than existing models, thereby extending the applicability of NMF to a broader class of count data.

stat.CO↗

arXiv2025

Enumeration Algorithm for Genus-4 Superspecial Hyperelliptic Curves with Automorphism Group $Q_8$

In this paper, we propose an algorithm to enumerate genus-4 superspecial hyperelliptic curves whose automorphism groups isomorphic to the quaternion group. By implementing this algorithm with Magma, we successfully obtain the number of isomorphism classes of such curves in every characteristic $7 \leq p < 10000$. Interestingly, the experimental results lead us to the conjecture that there exist exactly $[p/48]$ isomorphism classes of such curves if $p \equiv 1,7 \pmod{8}$, whereas such curves exist if $p \equiv 3,5 \pmod{8}$

math.AG↗

arXiv2024

Superspecial genus-$4$ double covers of elliptic curves

In this paper we study genus-$4$ curves obtained as double covers of elliptic curves. Firstly we shall give explicit defining equations of such curves with explicit criterion for whether it is nonsingular, and show the irreducibility of the long polynomial determining whether the genus-4 curve is nonsingular or not, in any characteristic $\ne 2,3$. Secondly, as an application, we enumerate superspecial genus-$4$ double covers of elliptic curves in small characteristic.

math.AG↗

arXiv2024

Listing superspecial curves of genus three using Richelot isogeny graphs

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 explicit algorithm for computing the Richelot isogeny graph of superspecial principally polarized abelian varieties of dimension 3 using theta functions. In particular, one can determine whether a given vertex in the graph corresponds to the Jacobian of a genus-3 curve or not, and restore the defining equation of such a genus-3 curve from its theta constants. Our algorithm enables efficient enumeration of superspecial genus-3 curves, as all operations can be performed in $\mathbb{F}_{p^2}$. By implementing the algorithm in Magma, we successfully counted the number of hyperelliptic curves among them for all primes $11 \leq p < 100$.

math.AG↗

arXiv2023

Computing superspecial hyperelliptic curves of genus 4 with automorphism group properly containing the Klein 4-group

In algebraic geometry, enumerating or finding superspecial curves in positive characteristic $p$ is important both in theory and in computation. In this paper, we propose feasible algorithms to enumerate or find superspecial hyperelliptic curves of genus $4$ with automorphism group properly containing the Klein $4$-group. Executing the algorithms on Magma, we succeeded in enumerating such superspecial curves for every $p$ with $19 \leq p < 500$, and in finding a single one for every $p$ with $19 \leq p < 7000$.

math.AG↗

arXiv2023

On the Rosenhain forms of superspecial curves of genus two

In this paper, we examine superspecial genus-2 curves $C: y^2 = x(x-1)(x-\lambda)(x-\mu)(x-\nu)$ in odd characteristic $p$. As a main result, we show that the difference between any two elements in $\{0,1,\lambda,\mu,\nu\}$ is a square in $\mathbb{F}_{p^2}$. Moreover, we show that $C$ is maximal or minimal over $\mathbb{F}_{p^2}$ without taking its $\mathbb{F}_{p^2}$-form (we also give a criterion in terms of $p$ that tells whether $C$ is maximal or minimal). As these applications, we study the maximality of superspecial hyperelliptic curves of genus $3$ and $4$ whose automorphism groups contain $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$.

math.AG↗

arXiv2022

Differential forms on the curves associated to Appell-Lauricella hypergeometric series and the Cartier operator on them

Archinard studied the curve $C$ over $\mathbb{C}$ associated to an Appell-Lauricella hypergeometric series and differential forms on its desingularization. In this paper, firstly as a generalization of Archinard's results, we describe a partial desingularization of $C$ over a field $K$ under a mild condition on its characteristic and the space of global sections of its dualizing sheaf, especially we give an explicit basis of it. Secondly, when the characteristic is positive, we show that the Cartier operator on the space can be defined and describe it in terms of Appell-Lauricella hypergeometric series.

math.AG↗

arXiv2022

On the maximality of hyperelliptic Howe curves of genus 3

In this paper, we study a Howe curve $C$ in positive characteristic $p \geq 3$ which is of genus 3 and is hyperelliptic. We will show that if $C$ is superspecial, then its standard form is maximal or minimal over $\mathbb{F}_{p^2}$ without taking its $\mathbb{F}_{p^2}$-form.

math.AG↗

arXiv2022

On the maximality of genus-3 nonhyperelliptic curves of Ciani type

In this paper, we study a Ciani curve $C: x^4 + y^4 + z^4 + rx^2y^2 + sy^2z^2 + tz^2x^2 = 0$ in positive characteristic $p \geq 3$. We will show that if $C$ is superspecial, then its standard form is maximal or minimal over $\mathbb{F}_{p^2}$ without taking its $\mathbb{F}_{p^2}$-form.

math.AG↗

arXiv2021

The $a$-numbers of non-hyperelliptic curves of genus 3 with cyclic automorphism group of order 6

In this paper, we study non-hyperelliptic curves of genus $3$ with cyclic automorphism group of order $6$. Over an algebraically closed field $K$ of characteristic $\neq 2,3$, such curves are written as plane quartics $C_r: x^3 z + y^4 + r y^2 z^2 + z^4 = 0$ with one parameter $r$. As the first main theorem, we show that $r\neq 0,\pm 2$ and give a necessary and sufficient condition with respect to $r$ and $r'$ such that $C_r \cong C_{r'}$. By describing the Hasse-Witt matrix of $C_r$ in terms of a certain Gauss' hypergeometric series, we obtain the second main theorem, where we determine the possible $a$-number of $C_r$, and give the exact number of isomorphism classes over $K$ of such curves attaining the possible maximal $a$-number.

math.AG↗