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Koji Tasaka

Publications and source records attributed to Koji Tasaka.

At least 19 recordsLinked to original sources
arXiv2026

Symmetric multiple Eisenstein series

In this paper, we introduce the symmetric multiple Eisenstein series, a variant of the multiple Eisenstein series. As a fundamental result, we show that they satisfy the linear shuffle relation. As a case study, we investigate the vector space spanned by symmetric double Eisenstein series of weight $k$. When $k$ is even, it coincides with the space spanned by modular forms of weight $k$ and the derivative of the Eisenstein series of weight $k-2$. For $k$ odd, we prove that its dimension equals $\lfloor k/3\rfloor$. We further provide an explicit correspondence between the linear shuffle relation and the Fay-shuffle relation satisfied by elliptic double zeta values, which may be of independent interest. In connection with modular forms, we prove that every modular form can be expressed as a linear combination of symmetric triple Eisenstein series. This will serve as a first step toward understanding modular phenomena for symmeric multiple zeta values observed by Kaneko and Zagier.

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arXiv2025

Spherical designs for finite quaternionic unit groups and their applications to modular forms

For a finite subset $X$ of the $d$-dimensional unit sphere, the harmonic strength $T(X)$ of $X$ is the set of $\ell\in \mathbb{N}$ such that $\sum_{x\in X} P(x)=0$ for all harmonic polynomials $P$ of homogeneous degree $\ell$. We will study three exceptional finite groups of unit quaternions, called the binary tetrahedral group $2T$ of order 24, the octahedral group $2O$ of order 48, and the icosahedral group $2I$ of order 120, which can be viewed as a subset of the 3-dimensional unit sphere. For these three groups, we determine the harmonic strength and show the minimality and the uniqueness as spherical designs. In particular, the group $2O$ is unique as a minimal subset $X$ of the 3-dimensional unit sphere with $T(X)=\{22,14,10,6,4,2 \}\cup \mathbb{O}^+$, where $\mathbb{O}^+$ denotes the set of all positive odd integers. This result provides the first characterization of $2O$ from the spherical design viewpoint. For $G\in \{2T,2O,2I\}$, we consider the lattice $\mathcal{O}_{G}$ generated by $G$ over $R_G$ on which the group $G$ acts on by multiplication, where $R_{2T}=\mathbb{Z},\ R_{2O}=\mathbb{Z}[\sqrt{2}],\ R_{2I}=\mathbb{Z}[(1+\sqrt{5})/2]$ are the ring of integers. We introduce the spherical theta function $θ_{G,P}(z)$ attached to the lattice $\mathcal{O}_G$ and a harmonic polynomial $P$ of degree $\ell$ and prove that they are modular forms. By applying our results on the characterization of $G$ as a spherical design, we determine the cases in which the $\mathbb{C}$-vector space spanned by all $θ_{G,P}(z)$ of harmonic polynomials $P$ of homogeneous degree $\ell$ has dimension zero--without relying on the theory of modular forms.

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arXiv2023

Spherical designs and modular forms of the $D_4$ lattice

In this paper, we study shells of the $D_4$ lattice with a {slight generalization} of spherical $t$-designs due to Delsarte-Goethals-Seidel, namely, the spherical design of harmonic index $T$ (spherical $T$-design for short) introduced by Delsarte-Seidel. We first observe that{, for any positive integer $m$,} the $2m$-shell of $D_4$ is an antipodal spherical $\{10,4,2\}$-design on the three dimensional sphere. We then prove that the $2$-shell, which is the $D_4$ root system, is a tight $\{10,4,2\}$-design, using the linear programming method. The uniqueness of the $D_4$ root system as an antipodal spherical $\{10,4,2\}$-design with 24 points is shown. We give two applications of the uniqueness: a decomposition of the shells of the $D_4$ lattice in terms of orthogonal transformations of the $D_4$ root system, and the uniqueness of the $D_4$ lattice as an even integral lattice of level 2 in the four dimensional Euclidean space. We also reveal a connection between the harmonic strength of the shells of the $D_4$ lattice and non-vanishing of the Fourier coefficients of a certain newform of level 2. Motivated by this, congruence relations for the Fourier coefficients are discussed.

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arXiv2023

Catalogue of modular relations for double zeta values

This note is a compilation of related research on modular relations for multiple zeta values. Roughly speaking, modular relations are (homogeneous) linear relations of multiple zeta values of fixed weight whose coefficients are `originated' from modular forms on (a subgroup of) the full modular group. Several kinds of such relations, in particular, for double zeta values have been found up to now. It is our purpose to review these results and also succinctly outline potential avenues for future research projects.

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arXiv2023

The harmonic strength of shells of lattices and weighted theta series

This is the write-up of a talk given in RIMS conference ``Analytic and arithmetic aspects of automorphic representations", where I outlined two kinds of different results related to the D4 lattice, obtained in a joint work with Hirao and Nozaki. In general, from the design theoretical viewpoints, we are interested in determining the harmonic strength of finite subsets on the unit sphere. When considering the shells of a lattice, this problem shifts to the non-vanishing problem of the Fourier coefficients of elliptic cusp forms. Various instances, mostly from root lattices, are listed in this note.

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arXiv2022

Supercongruences of multiple harmonic $q$-sums and generalized finite/symmetric multiple zeta values

The Kaneko--Zagier conjecture describes a correspondence between finite multiple zeta values and symmetric multiple zeta values. Its refined version has been established by Jarossay, Rosen and Ono--Seki--Yamamoto. In this paper, we explicate these conjectures through studies of multiple harmonic $q$-sums. We show that the (generalized) finite/symmetric multiple zeta value are obtained by taking an algebraic/analytic limit of multiple harmonic $q$-sums. As applications, new proofs of reversal, duality and cyclic sum formulas for the generalized finite/symmetric multiple zeta values are given.

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arXiv2021

Finite and symmetric colored multiple zeta values and multiple harmonic q-series at roots of unity

The Kaneko-Zagier conjecture states that finite and symmetric multiple zeta values satisfy the same relations. In the previous work with H.~Bachmann and Y.~Takeyama, we proved that the finite and symmetric multiple zeta value are obtained as an `algebraic' and `analytic' limit at $q\rightarrow 1$ of a certain truncated multiple harmonic $q$-series, and studied its relations in order to give partial evidence of the Kaneko-Zagier conjecture. In this paper, we start with truncated multiple harmonic $q$-series of level $N$, which is a $q$-analogue of the truncated colored multiple zeta value. We introduce our finite and symmetric colored multiple zeta values as an algebraic and analytic limit of the truncated multiple harmonic $q$-series of level $N$ and discuss a higher level (or a cyclotomic) analogue of the Kaneko-Zagier conjecture.

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arXiv2019

On Ecalle's and Brown's polar solutions to the double shuffle equations modulo products

Two explicit sets of solutions to the double shuffle equations modulo products were introduced by Ecalle and Brown respectively. We place the two solutions into the same algebraic framework and compare them. We find that they agree up to and including depth four but differ in depth five by an explicit solution to the linearized double shuffle equations with an exotic pole structure.

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arXiv2019

Hecke eigenform and double Eisenstein series

Cusp forms for the full modular group can be written as linear combination of double Eisenstein series introduced by Gangl, Kaneko and Zagier. We give an explicit formula for decomposing a Hecke eigenform into double Eisenstein series.

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arXiv2018

Cyclotomic analogues of finite multiple zeta values

We introduce the notion of finite multiple harmonic q-series at a primitive root of unity and show that these specialize to the finite multiple zeta value (FMZV) and the symmetrized multiple zeta value (SMZV) through an algebraic and analytic operation, respectively. Further, we obtain families of linear relations among these series which induce linear relations among FMZVs and SMZVs of the same form. This gives evidence towards a conjecture of Kaneko and Zagier relating FMZVs and SMZVs. Motivated by the above results, we define cyclotomic analogues of FMZVs, which conjecturally generate a vector space of the same dimension as that spanned by the finite multiple harmonic q-series at a primitive root of unity of sufficiently large degree.

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arXiv2018

Special values of finite multiple harmonic q-series at roots of unity

We study special values of finite multiple harmonic q-series at roots of unity. These objects were recently introduced by the authors and it was shown that they have connections to finite and symmetric multiple zeta values and the Kaneko-Zagier conjecture. In this note we give new explicit evaluations for finite multiple harmonic q-series at roots of unity and prove Ohno-Zagier-type relations for them.

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arXiv2017

Evaluation of Tornheim's type of double series

We examine values of certain Tornheim's type of double series with odd weight. As a result, an affirmative answer to a conjecture about the parity theorem for the zeta function of the root system of the exceptional Lie algebra $G_2$, proposed by Komori, Matsumoto and Tsumura, is given.

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