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Kyoji Saito

Publications and source records attributed to Kyoji Saito.

At least 19 recordsLinked to original sources
arXiv2025

Winding quotients for virtual period maps of rank 1

We illustrate a rank 1 model of virtual period maps and their associated winding quotient, where the winding quotient is a new phenomenon appeared in a recent study of virtual period maps and it requires a reformulation of the classical Jacobi inversion problem for the period maps due to the appearance of exponents which are imaginary numbers. We answer to the new inversion problem by introducing the q-multiplicatively periodic function, whose pull-back to the winding covering space is the Weierstrass p-function up to a correction by Eisenstein series E2. The function appears also in the study of mathematical physics as the propagator on elliptic curves.

math.CV↗
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↗
arXiv2023

Second homotopy classes associated with non-cancellative monoids

We construct second homotopy classes associated with twins of non-cancellative tuples of a monoid, where the monoid is defined by the semi-positive fundamental relations of the fundamental group of a CW-complex. As an application, we reconstruct the second homotopy classes for the complement of generic lines arrangement studied by Akio Hattori. We aim to apply the theory for the complement of elliptic discriminant loci in a forthcoming work.

math.AT↗
arXiv2020

Primitive Forms without Higher Residue Structure and Integrable Hierarchies (I)

We introduce primitive forms with or without higher residue structure and explore their connection with the flat structures with or without a metric and integrable hierarchies of KdV type. Just as the classical case of primitive forms with metric arXiv:1311.1659, the primitive forms without metrics are constructed as the positive part of the Birkhoff decomposition of formal oscillatory integrals with respect to the descendent variable. The oscilating integrals of a primitive form without metric give rise to a hierarchy of commuting PDE of the KdV type as in the case of primitive forms with metric. This shall be studied in (II).

nlin.SI↗
arXiv2020

A view on elliptic integrals from primitive forms (Period integrals of type $\mathrm{A_2, B_2}$ and $\mathrm{G_2}$

Elliptic integrals, since Euler's finding of addition theorem 1751, has been studied extensively from various view points. Present paper gives a view point from primitive integrals of types $\mathrm{A_2}, \mathrm{B_2}$ and $\mathrm{G_2}$ for the three families of elliptic curves of Weierstrass, Jacobi-Legendre and Hesse, respectively. We solve Jacobi inversion problem for the period maps in the sense explained in the introduction (see [Siegel] Chap.1,13) by introducing certain generalized Eisenstein series of types $\mathrm{A_2}, \mathrm{B_2}$ and $\mathrm{\mathrm{G_2}}$, which generate the ring of invariant functions on the period domain for the congruence subgroups $Γ_1(N)$ ($N=1,2$ and $3$). In particular, Eisenstein series of type $\mathrm{B_2}$ includes the case of weight two, and Eisenstein series of type $\mathrm{G_2}$ includes the cases of weight one and two, which seem to be of new feature. The goal of the paper is a partial answer to the discriminant conjecture, which claims an existence of certain cusp form of weight 1 with character of topological origin, giving a power root of the discriminant form (Aspects Math., E36,p.\ 265-320.\ 2004). See \S12 Concluding Remarks for more about back grounds of the present paper.

math.AG↗
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↗
arXiv2016

Coherence of direct images of the De Rham complex

We show the coherence of the direct images of the De Rham complex relative to a flat holomorphic map with suitable boundary conditions. For this purpose, a notion of bi-dg-algbera called the Koszul-De Rham algbera is dveloped.

math.AG↗
arXiv2016

The Zero Locus of the $F$-triangle

We are interested in the zero locus of a Chapoton's $F$-triangle as a polynomial in two real variables $x$ and $y$. An expectation is that (1) the $F$-triangle of rank $l$ as a polynomial in $x$ for each fixed $y\in[0,1]$, has exactly $l$ distinct real roots in $[0,1]$, and (2) $i$-th root $x_i(y)$ ($1 \le i \le l$) as a function on $y \in [0,1]$ is monotone decreasing. In order to understand these phenomena, we slightly generalized the concept of $F$-triangles and study the problem on the space of such generalized triangles. We analyze the case of low rank in details and show that the above expectation is true. We formulate inductive conjectures and questions for further rank cases. This study gives a new insight on the zero loci of $f^+$- and $f$-polynomials.

math.CO↗
arXiv2016

Zero loci of skew-growth functions for dual Artin monoids

We show that the skew-growth function of a dual Artin monoid of finite type P has exactly rank(P) =: l simple real zeros on the interval (0, 1]. The proofs for types A_l and B_l are based on an unexpected fact that the skew-growth functions, up to a trivial factor, are expressed by Jacobi polynomials due to a Rodrigues type formula in the theory of orthogonal polynomials. The skew-growth functions for type D_l also satisfy Rodrigues type formulae, but the relation with Jacobi polynomials is not straightforward, and the proof is intricate. We show that the smallest root converges to zero as the rank l of all the above types tend to infinity.

math.CO↗
arXiv2014

Mirror symmetry for exceptional unimodular singularities

In this paper, we prove the mirror symmetry conjecture between the Saito-Givental theory of exceptional unimodular singularities on Landau-Ginzburg B-side and the Fan-Jarvis-Ruan-Witten theory of their mirror partners on Landau-Ginzburg A-side. On the B-side, we compute the genus-zero generating function from a perturbative formula of primitive forms introduced by the first three authors recently. This computation matches the orbifold-Grothendieck-Riemann-Roch and WDVV calculations in FJRW theory on the A-side. The coincidence of the full data at all genera is established by reconstruction techniques. Our result establishes the first examples of LG-LG mirror symmetry of all genera for weighted homogeneous polynomials of central charge greater than one (i.e. which contain negative degree deformation parameters).

math.AG↗
arXiv2014

Primitive forms via polyvector fields

We develop a complex differential geometric approach to the theory of higher residues and primitive forms from the viewpoint of Kodaira-Spencer gauge theory, unifying the semi-infinite period maps for Calabi-Yau models and Landau-Ginzburg models. We give an explicit perturbative construction of primitive forms with respect to opposite filtrations and primitive elements. This leads to a concrete algorithm to compute the Taylor expansions of primitive forms as well as the description of their moduli space for all weighted homogenous cases. As an example, we present unknown perturbative expressions for the primitive form of E_12 singularity and illustrate its application to Landau-Ginzburg mirror symmetry with FJRW-theory.

math.AG↗
arXiv2013

The skew-growth function on the monoid of square matrices

We develop an elementary theory of divisibility on the monoid $M(n,R)^\times$ consisting of all square matrices of size $n\ge 1$ of non-zero determinants with coefficients in a principal ideal domain $R$. In particular, we show that any finite subset of the monoid has the least left common multiple up to a right unit factor. When $R$ is residue finite, we consider a signed generating series, called the skew growth function, of least common multiples of finite right equivalence classes of irreducible elements. As an elementary application of the divisibility theory, we show that the skew-growth function decomposes into Euler products.

math.GR↗
arXiv2012

Inversion formula for the growth function of a cancellative monoid

We consider any cancellative monoid $M$ equipped with a discrete degree map $deg:M\to R_{\ge0}$ and associated generating function $P(t)=\sum_{m\in M}t^{deg(m)}$, called the growth function of $M$. We also introduce, using some towers of minimal common multiple sets in $M$, another signed generating function $N(t)$, called the skew-growth function of $M$. We show that these functions satisfy the inversion formula $P(t)N(t)=1$. In case the monoid is the set of positive integers with ordinary product structure and the degree map is logarithm function, using the coordinate change $t=exp(-s)$, the inversion formula turns out to be the Euler product formula for the Riemann's zeta function.

math.CO↗
arXiv2012

Limit elements in the configuration algebra for a cancellative monoid

We introduce two spaces $Ω(Γ,G)$ and $Ω(P_{Γ,G})$ of pre-partition functions and of opposite series, respectively, which are associated with a Cayley graph $(Γ,G)$ of a cancellative monoid $Γ$ with a finite generating system $G$ and with its growth function $P_{Γ,G}(t)$. Under mild assumptions on $(Γ,G)$, we introduce a fibration $π_Ω:Ω(Γ,G)\to Ω(P_{Γ,G})$ equivariant with a $\Z_{\ge0}$-action, which is transitive if it is of finite order. Then, the sum of pre-partition functions in a fiber is a linear combination of residues of the proportion of two growth functions $P_{Γ,G}(t)$ and $P_{Γ,G}\mathcal{M}(t)$ attached to $(Γ,G)$ at the places of poles on the circle of the convergent radius.

math.GR↗
arXiv2012

Opposite power series

Let $γ_n$ ($n\in \mathbb{Z}_{\ge0}$) be a sequence of complex numbers, which is tame: $0<\exists u\le γ_{n-1}/γ_n \le \exists v<\infty$ for all $n>0$. We show a resonance between the singularities of the function of the power series $P(t):=\sum_{n=0}^\infty γ_n t^n$ on its boundary of the disc of convergence and the oscillation behavior of the sequences $γ_{n-k}/γ_n$ ($n\in \mathbb{Z}_{>>0}$) for $k>0$. The resonance is proven by introducing the space of opposite power series, which is the compact subspace of the space of all formal power series in the opposite variable $s=1/t$ and is defined as the accumulating set of the sequence $X_n(s):=\sum_{k=0}^n\frac{γ_{n-k}}{γ_n}t^k$ ($n\in \mathbb{Z}_{\ge0}$). We analyze in details an example of the growth series $P(t)$ for the modular group $PSL(2,Z)$ due to Machi.

math.CA↗
arXiv2012

Coxeter elements for vanishing cycles of type $A_{1/2\infty}$ and $D_{1/2\infty}$

We introduce two entire functions $f_{A_{1/2\infty}}$and $f_{D_{1/2\infty}}$ in two variables. Both of them have only two critical values 0 and 1, and the associated maps $\C^2 to \C$ define topologically locally trivial fibrations over $\C\setminus{0,1\}$. All critical points are ordinary double points, and the associated vanishing cycles span the middle homology group of the general fiber, whose intersection diagram forms bi-partitely decomposed quivers of type $A_{1/2\infty}$ and $D_{1/2\infty}$, respectively. Coxeter elements of type $A_{1/2\infty}$ and $D_{1/2\infty}$, acting on the middle homology group, are introduced as the product of the monodromies around 0 and 1. We describe the spectra of the Coxeter elements by embedding the middle homology group into a Hilbert space. The spectra turn out to be strongly continuous on the interval $(-1/2,1/2)$ except at 0 for type $\DD$.

math.AG↗
arXiv2010

Growth partition functions for cancellative infinite monoids

We introduce the {\it growth partition function} $Z_{Γ,G}(t)$ associate with any cancellative infinite monoid $Γ$ with a finite generator system $G$. It is a power series in $t$ whose coefficients lie in integral Lie-like space $\mathcal{L}_{\Z}(Γ,G)$ in the configuration algebra associated with the Cayley graph $(Γ,G)$. We determine them for homogeneous monoids admitting left greatest common divisor and right common multiple. Then, for braid monoids and Artin monoids of finite type, using that formula, we explicitly determine their limit partition functions $ω_{Γ,G}$.

math.GR↗