arXiv subjects

Hiroshi Onuki

Publications and source records attributed to Hiroshi Onuki.

4 recordsLinked to original sources

arXiv2026

Geometric construction of modular polynomials with level structures

The classical modular polynomial for $j$-invariants describes the relation between two elliptic curves connected by isogenies. This polynomial has been applied to various algorithms in computational number theory, such as point counting on elliptic curves. In addition, computing the modular polynomial itself is also an important problem, and various algorithms to compute it have been proposed. On the other hand, modular polynomials for other invariants of higher level structures have also been studied. For example, the modular polynomials for the Legendre $λ$-invariant and the Weber functions are well-known. In this paper, we give another approach to construct modular polynomials of higher level purely algebraically. In particular, we show the existence of modular polynomials for invariants directly related to models of elliptic curves, such as the coefficients of Montgomery and Hessian curves. We also show that these modular polynomials have integer coefficients and are symmetric and irreducible in certain cases, and give an algorithm to compute them, which is based on the deformation method by Kunzweiler and Robert.

math.NT↗

arXiv2025

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↗

arXiv2024

Primality proving using elliptic curves with complex multiplication by imaginary quadratic fields of class number three

In 2015, Abatzoglou, Silverberg, Sutherland, and Wong presented a framework for primality proving algorithms for special sequences of integers using an elliptic curve with complex multiplication. They applied their framework to obtain algorithms for elliptic curves with complex multiplication by imaginary quadratic field of class numbers one and two, but, they were not able to obtain primality proving algorithms in cases of higher class number. In this paper, we present a method to apply their framework to imaginary quadratic fields of class number three. In particular, our method provides a more efficient primality proving algorithm for special sequences of integers than the existing algorithms by using an imaginary quadratic field of class number three in which 2 splits. As an application, we give two special sequences of integers derived from $\Q(\sqrt{-23})$ and $\Q(\sqrt{-31})$, which are all the imaginary quadratic fields of class number three in which 2 splits. Finally, we give a computational result for the primality of these sequences.

math.NT↗

arXiv2020

On oriented supersingular elliptic curves

We revisit theoretical background on OSIDH, that is an isogeny-based key-exchange protocol proposed by Colò and Kohel at NutMiC 2019. We give a proof of a fundamental theorem for OSIDH. The theorem was stated by Colò and Kohel without proof. Furthermore, we consider parameters of OSIDH, give a sufficient condition on the parameters that the protocol works, and estimate the size of the parameters for a certain security level.

math.NT↗