arXiv subjects

Hiroki Aoki

Publications and source records attributed to Hiroki Aoki.

7 recordsLinked to original sources

arXiv2026

Modules of Jacobi forms of degree two of small levels

The purpose of this paper is to describe explicitly the modules of (Siegel-)Jacobi forms of degree two of index one of any scalar valued weight with respect to some congruence subgroups of small levels $N\leq 4$. Such a structure for the full Siegel modular group as a module over scalar valued Siegel modular forms of even weight has been explicitly given by T.~Ibukiyama. There we used an explicit structure theorem of rings of scalar valued Siegel modular forms by Igusa and that of vector valued Siegel modular forms of weight $\det^k\, \Sym^2$ by T. Satoh and T. Ibukiyama. On the other hand, for levels $N=2$, $3$, $4$, ring structures of scalar valued case have been also known by H. Aoki and T. Ibukiyama and $\det^k \Sym^2$-valued case by H. Aoki. In this paper, by merging these results, we give the same sort of simple structure theorems on modules of Jacobi forms of degree of two of index one for level $2$, $3$ and $4$.

math.NT↗

arXiv2025

Existence of primes in the interval $ [15x,16x] $ -- An entirely elementary proof --

In this paper, we give a short and entirely elementary proof of the proposition ``For any positive integer $ N $, there exists a real number $ L $ such that for any real number $ x \geqq L $, there are at least $ N $ primes in the interval $ [kx, (k+1)x] $'' for $ k \leqq 15 $. Our proof is based on the idea of the proof by Erd\"{o}s for $ k=1 $ and its improvement by Hitotsumatsu and by Sainose for $ k=2 $. In the case of $ k=3 $ and $ k=4 $, the method is very similar to the case of $ k=2 $, however, in the case of $ k \geqq 5 $, we need new idea to complete the proof.

math.NT↗

arXiv2025

A characterization of root systems from the viewpoint of denominator formulae

Root systems are sets with remarkable symmetries and therefore they appear in many situations in mathematics. Among others, denominator formulae of root systems are very beautiful and mysterious equations which have several meanings from a variety of disciplines in mathematics. In this paper, we show a converse statement of this phenomena. Namely, for a given finite subset $ S $ of a Euclidean vector space $ V $, define an equation $ F $ in the group ring $ {\mathbb{Z}}[V] $ featuring the product part of denominator formulae. Then, a geometric condition for the support of $ F $ characterizes $ S $ being a set of positive roots of a finite/affine root system, recovering the denominator formula. This gives a novel characterization of the sets of positive roots of reduced finite/affine root systems.

math.RA↗

arXiv2024

Formal series of Jacobi forms

We prove for general paramodular level that formal series of scalar Jacobi forms with an involution condition necessarily converge and are therefore the Fourier-Jacobi expansions at the standard 1-cusp of paramodular Fricke eigenforms.

math.NT↗

arXiv2024

On holomorphicity of Hartogs series satisfying algebraic relations

We consider a formal power series in one variable whose coefficients are holomorphic functions in a given multidimensional complex domain. Assume the following two conditions on the series. (C1) The restriction of the series at each point of a dense subset of the domain converges in an open disk of a fixed radius. (C2) The series is algebraic over the ring of holomophic functions on the direct product space of the domain and the disk. The main theorem of the present note is that the series defines a holomorphic function on the direct product space. We also give an example where the condition (C2) is essentially necessary.

math.CV↗

arXiv2019

Modular forms from the Weierstrass functions

We construct holomorphic elliptic modular forms of weight 2 and weight 1, by special values of Weierstrass p-functions, and by differences of special values of Weierstrass zeta-functions, respectively. Also we calculated the values of these forms at some cusps.

math.NT↗

arXiv2006

Generalizations of Ho-Lee's binomial interest rate model I: from one- to multi-factor

In this paper a multi-factor generalization of Ho-Lee model is proposed. In sharp contrast to the classical Ho-Lee, this generalization allows for those movements other than parallel shifts, while it still is described by a recombining tree, and is stationary to be compatible with principal component analysis. Based on the model, generalizations of duration-based hedging are proposed. A continuous-time limit of the model is also discussed.

math.PR↗