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Shushi Harashita

Publications and source records attributed to Shushi Harashita.

At least 19 recordsLinked to original sources
arXiv2026

Automorphism groups of hyperelliptic curves of $2$-rank zero

In this paper, we determine the reduced automorphism groups of hyperelliptic curves of a small genus in characteristic $2$, when they are of $2$-rank $0$. Such a curve is an Artin-Schreier curve defined in the form $y^2-y=f(x)$ for a polynomial $f(x)$. After we clarify semidirect-product structures of the automorphism groups for an arbitrary genus, we derive the detailed group structures for the reduced automorphism groups of the curves of a small genus, through computations using the computational algebra system Magma. With these experiments, we formulate two conjectures, which are analogues for our curves of the Oort conjecture on automorphism groups of generic principally polarized supersingular abelian varieties.

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arXiv2026

The Lang-Trotter conjecture on average for genus-$2$ curves with $S_3$ reduced automorphism group

For an elliptic curve $E$ over $\mathbb{Q}$ without complex multiplication, Lang and Trotter conjectured that the number of primes $p 0$ is a constant depending only on $E$. While it remains an open question, an average estimation related to the Lang-Trotter conjecture was established by Fouvry and Murty. This result is called the Lang-Trotter conjecture on average. We extend the Lang-Trotter conjecture to curves of genus $2$ and obtain a similar result to the Lang-Trotter conjecture on average for the family of curves $C_λ:y^2=x(x-1)(x-λ)(x-(λ-1)/λ)(x-1/ (1-λ))$. These curves are characterized as curves of genus $2$ with reduced automorphism group containing symmetric group $S_3$.

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arXiv2025

The multiplicity-one theorem for the superspeciality of curves of genus two

Igusa proved in 1958 that the polynomial determining the supersingularity of elliptic curve in Legendre form is separable. In this paper, we get an analogous result for curves of genus $2$ in Rosenhain form. More precisely we show that the ideal determining the superspeciality of the curve has multiplicity one at every superspecial point. Igusa used a Picard-Fucks differential operator annihilating a Gauß hypergeometric series. We shall use Lauricella system (of type D) of hypergeometric differential equations in three variables.

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arXiv2025

The cycle class of the supersingular locus of principally polarized abelian varieties

We prove a formula for the cycle class of the supersingular locus in the Chow ring with rational coefficients of the moduli space of principally polarized abelian varieties in characteristic $p$. This formula determines this class as a monomial in the Chern classes of the Hodge bundle up to a factor that is a polynomial in $p$. This factor is known for $g\leq 3$. We determine the factor for $g=4$.

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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.

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arXiv2024

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.

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arXiv2023

Representation of non-special curves of genus 5 as plane sextic curves and its application to finding curves with many rational points

In algebraic geometry, it is important to provide effective parametrizations for families of curves, both in theory and in practice. In this paper, we present such an effective parametrization for the moduli of genus-$5$ curves that are neither hyperelliptic nor trigonal. Subsequently, we construct an algorithm for a complete enumeration of non-special genus-$5$ curves having more rational points than a specified bound, where ``non-special curve'' means that the curve is non-hyperelliptic and non-trigonal with mild singularities of the associated sextic model that we propose. As a practical application, we implement this algorithm using the computer algebra system MAGMA, specifically for curves over the prime field of characteristic $3$.

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arXiv2022

Genus-five hyperelliptic or trigonal curves with many rational points in characteristic three

The number $N_9(5)$, the maximal number of $\mathbb{F}_9$-rational points on curves over $\mathbb{F}_9$ of genus $5$ is unknown, but it is known that $32 \le N_9(5)\le 35$. In this paper, we enumerate hyperelliptic curves and trigonal curves over $\mathbb{F}_3$ which have many $\mathbb{F}_9$-rational points (and $\mathbb{F}_3$-rational points), especially the maximal number of $\mathbb{F}_9$-rational points of those curves is $30$. Kudo-Harashita studied the nonhyperelliptic and nontrigonal case,where they found a new example of curves (over $\mathbb{F}_3$) of genus five which attains $32$ and proved that there is no example attaining more than $32$, among sextic plane curves with mild singularities. We conclude from the main results in this paper that we need to search sextic models (i.e., nonhyperelliptic and nontrigonal) with bad singularities, in order to find a genus-five curve over $\mathbb{F}_3$ with at least $33$ $\mathbb{F}_9$-rational points.

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arXiv2022

Computing the space of differential forms of a plane curve and its Cartier-Manin matrix

In this paper, we propose a feasible algorithm to give an explicit basis of the space of regular differential forms on the nonsingular projective model of any given plane algebraic curve. The algorithm is demonstrated for concrete examples, with our implementation over the computer algebra system Magma. As an application, we also describe the Cartier-Manin matrix of the nonsingular projective curve with respect to the basis computed by the algorithm.

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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.

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arXiv2020

Asymptotic formula of the number of Newton polygons

In this paper, we enumerate Newton polygons asymptotically. The number of Newton polygons is computable by a simple recurrence equation, but unexpectedly the asymptotic formula of its logarithm contains growing oscillatory terms. As the terms come from non-trivial zeros of the Riemann zeta function, an estimation of the amplitude of the oscillating part is equivalent to the Riemann hypothesis.

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arXiv2020

Algorithm to enumerate superspecial Howe curves of genus $4$

A Howe curve is a curve of genus $4$ obtained as the fiber product over $\mathbf{P}^1$ of two elliptic curves. Any Howe curve is canonical. This paper provides an efficient algorithm to find superspecial Howe curves and that to enumerate their isomorphism classes. We discuss not only an algorithm to test the superspeciality but also an algorithm to test isomorphisms for Howe curves. Our algorithms are much more efficient than conventional ones proposed by the authors so far for general canonical curves. We show the existence of a superspecial Howe curve in characteristic $7<p\le 331$ and enumerate the isomorphism classes of superspecial Howe curves in characteristic $p\le 53$, by executing our algorithms over the computer algebra system Magma.

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arXiv2019

Algorithmic study of superspecial hyperelliptic curves over finite fields

This paper presents algorithmic approaches to study superspecial hyperelliptic curves. The algorithms proposed in this paper are: an algorithm to enumerate superspecial hyperelliptic curves of genus $g$ over finite fields $\mathbb{F}_q$, and an algorithm to compute the automorphism group of a (not necessarily superspecial) hyperelliptic curve over finite fields. The first algorithm works for any $(g,q)$ such that $q$ and $2g+2$ are coprime and $q>2g+1$. As an application, we enumerate superspecial hyperelliptic curves of genus $g=4$ over $\mathbb{F}_{p}$ for $11 \leq p \leq 23$ and over $\mathbb{F}_{p^2}$ for $11 \leq p \leq 19$ with our implementation on a computer algebra system Magma. Moreover, we found maximal hyperelliptic curves and minimal hyperelliptic curves over $\mathbb{F}_{p^2}$ from among enumerated superspecial ones. The second algorithm computes an automorphism as a concrete element in (a quotient of) a linear group in the general linear group of degree $2$.

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arXiv2018

Superspecial trigonal curves of genus $5$

This paper provides an algorithm enumerating superspecial trigonal curves of genus $5$ over finite fields. Executing the algorithm over a computer algebra system Magma, we enumerate them over finite fields $\mathbb{F}_{p^a}$ for any natural number $a$ if $p \leq 7$ and for odd $a$ if $p \leq 13$.

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arXiv2018

On specializations of minimal $p$-divisible groups

In this paper, for any pair $(ζ, ξ)$ of Newton polygons with $ζ\prec ξ$, we construct a concrete specialization from the minimal $p$-divisible group of $ξ$ to the minimal $p$-divisible group of $ζ$ by a beautiful induction. This in particular gives the affirmative answer to the unpolarized analogue of a question by Oort on the boundaries of central streams, and gives another proof of the dimension formula of the central leaves in the unpolarized case.

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arXiv2017

Automorphism groups of superspecial curves of genus $4$ over $\mathbb{F}_{11}$

In this paper, we explicitly determine the automorphism group of every nonhyperelliptic superspecial curve of genus $4$ over $\mathbb{F}_{11}$. Our algorithm determining automorphism groups works for any nonhyperelliptic curves of genus $4$ over finite fields. With this computation, we show the compatibility between the enumeration of superspecial curves of genus $4$ over $\mathbb{F}_{11}$ obtained computationally by the first and second authors in 2017 and an enumeration by Galois cohomology theory.

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arXiv2017

Enumerating superspecial curves of genus $4$ over prime fields

In this paper we enumerate nonhyperelliptic superspecial curves of genus $4$ over prime fields of characteristic $p\le 11$. Our algorithm works for nonhyperelliptic curves over an arbitrary finite field in characteristic $p \ge 5$. We execute the algorithm for prime fields of $p\le 11$ with our implementation on a computer algebra system Magma. Thanks to the fact that the cardinality of $\mathbb{F}_{p^a}$-isomorphism classes of superspecial curves over $\mathbb{F}_{p^a}$ of a fixed genus depends only on the parity of $a$, this paper contributes to the odd-degree case for genus $4$, whereas our previous paper contributes to the even-degree case.

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