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Junichi Shigezumi

Publications and source records attributed to Junichi Shigezumi.

13 recordsLinked to original sources
arXiv2014

A detailed note on the zeros of Eisenstein series for $Γ_0^* (5)$ and $Γ_0^* (7)$

The present paper provides the details omitted from the more concise study "On the zeros of Eisenstein series for $Γ_0^* (5)$ and $Γ_0^* (7)$." We locate almost all of the zeros of the Eisenstein series associated with the Fricke groups of level 5 and 7 in their fundamental domains by applying and extending the method of F. K. C. Rankin and H. P. F. Swinnerton-Dyer (1970). We also use the arguments of some terms of the Eisenstein series in order to improve existing error bounds.

math.NT↗
arXiv2014

On the zeros of Eisenstein series for $Γ_0^* (5)$ and $Γ_0^* (7)$

We locate almost all the zeros of the Eisenstein series associated with the Fricke groups of level 5 and 7 in their fundamental domains by applying and extending the method of F. K. C. Rankin and H. P. F. Swinnerton-Dyer (1970). We also use the arguments of some terms of the Eisenstein series in order to improve existing error bounds.

math.NT↗
arXiv2012

Maximal $m$-distance sets containing the representation of the Johnson graph $J(n, m)$

We classify the maximal $m$-distance sets in $\mathbb{R}^{n-1}$ which contain the representation of the Johnson graph $J(n, m)$ for $m = 2, 3$. Furthermore, we determine the necessary and sufficient condition for $n$ and $m$ such that the representation of the Johnson graph $J(n, m)$ is not maximal as an $m$-distance set. Also, we classify the maximal two-distance sets in $\mathbb{R}^{n-1}$ which contain the representation of $J(n - 1, 2)$.

math.CO↗
arXiv2009

Spherical designs from norm-3 shell of integral lattices

A set of vectors all of which have a constant (non-zero) norm value in an Euclidean lattice is called a shell of the lattice. Venkov classified strongly perfect lattices of minimum 3 (Réseaux et "designs" sphérique, 2001), whose minimal shell is a spherical 5-design. This note considers the classification of integral lattices whose shells of norm 3 are 5-designs.

math.CO↗
arXiv2008

On 3-lattices and spherical designs

An integral lattice which is generated by some vectors of norm $q$ is called $q$-lattice. Classification of 3-lattices of dimension at most four is given by Mimura (On 3-lattice, 2006). As a expansion, we give a classification of 3-lattices of dimension at most seven. In addition, we consider the spherical designs from its shells.

math.CO↗
arXiv2008

A note on zeros of Eisenstein series for genus zero Fuchsian groups

Let $Γ\subseteq \text{SL}_2(\mathbb{R})$ be a genus zero Fuchsian group of the first kind having $\infty$ as a cusp, and let $E_{2 k}^Γ$ be the holomorphic Eisenstein series associated with $Γ$ for the $\infty$ cusp that does not vanish at $\infty$ but vanishes at all the other cusps. In the paper "On zeros of Eisenstein series for genus zero Fuchsian groups", under assumptions on $Γ$, and on a certain fundamental domain $\mathcal{F}$, H. Hahn proved that all but at most $c(Γ, \mathcal{F})$ (a constant) of the zeros of $E_{2 k}^Γ$ lie on a certain subset of $\{z \in \mathfrak{H} : j_Γ(z) \in \mathbb{R}\}$. In this note, we consider a small generalization of Hahn's result on the domain locating the zeros of $E_{2 k}^Γ$. We can prove most of the zeros of $E_{2 k}^Γ$ in $\mathcal{F}$ lie on its lower arcs under the same assumption.

math.NT↗
arXiv2008

A construction of spherical designs from finite graphs with the theory of crystal lattice

We want to introduce a construction of spherical designs from finite graphs with the theory of crystal lattice. We start from a finite graph, and we consider standard realization of the crystal lattices as the maximal Abelian covering of the graph. Then, we take the set of vectors which form the crystal lattice. If every vector has the same norm, then we can consider a finite set on Euclidean sphere, and then we get a spherical design. In this paper, we observe the results by numerical calculations. We tried constructing vectors from various finite graphs, strongly regular graphs, distance regular graphs, and so on. We also introduce some facts and conjectures.

math.CO↗
arXiv2006

On the zeros of Eisenstein series for $Γ_0^* (2)$ and $Γ_0^* (3)$

We locate all of the zeros of the Eisenstein series associated with the Fricke groups $Γ_0^{*}(2)$ and $Γ_0^{*}(3)$ in their fundamental domains by applying and expanding the method of F. K. C. Rankin and H. P. F. Swinnerton-Dyer (``{\it On the zeros of Eisenstein series}'', 1970).

math.NT↗