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Satoshi Wakatsuki

Publications and source records attributed to Satoshi Wakatsuki.

At least 19 recordsLinked to original sources
arXiv2026

On the absolute convergence of the spectral side of the twisted trace formula

We establish the absolute convergence of the spectral side of the twisted trace formula for reductive algebraic groups. More precisely, we extend the absolute convergence theorem of the spectral side due to Finis--Lapid--M\"uller to the twisted setting. This paper provides a detailed proof of the absolute convergence theorem, whose proof is known to experts but does not seem to have appeared in the literature. The resulting formulas are motivated by applications to limit multiplicity problems and to Weyl laws for self-dual automorphic representations of $\mathrm{GL}_n$.

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arXiv2026

The Eichler--Selberg trace formula for Hilbert cusp forms, the class numbers of quartic CM fields, and their distributions

Motivated by Su's construction of Cohen-type Eisenstein series of half-integral weight over totally real number fields \cite{Su16}, we introduce a generalization of Hurwitz class numbers to totally real number fields. Using these generalized Hurwitz class numbers, we establish an Eichler--Selberg trace formula for the space of holomorphic Hilbert cusp forms over real quadratic fields of narrow class number one. While the classical Hurwitz class numbers are defined in terms of class numbers of imaginary quadratic fields, the generalized Hurwitz class numbers appearing in our Eichler--Selberg trace formula are defined in terms of class numbers of quartic CM fields. For applications of this Eichler--Selberg trace formula, we study the distribution of the generalized Hurwitz class numbers, prove class number relations, and carry out numerical computations of traces of Hecke operators for $\mathbb{Q}(\sqrt{5})$ and $\mathbb{Q}(\sqrt{29})$.

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arXiv2025

Central limit theorem for Hecke eigenvalues

In this paper, we obtain the central limit theorem of Hecke eigenvalues in very general setting of split simple algebraic groups over $\mathbb{Q}$, using irreducible characters of compact Lie groups.

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arXiv2024

Zeta functions and nonvanishing theorems for toric periods on $\mathrm{GL}_2$

Let $F$ be a number field and $D$ a quaternion algebra over $F$. Take a cuspidal automorphic representation $π$ of $D_\mathbb{A}^\times$ with trivial central cahracter. We study the zeta functions with period integrals on $π$ for the perhomogeneous vector space $(D^\times\times D^\times\times\mathrm{GL}_2, D\oplus D)$. We show their meromorphic continuation and functional equation, determine the location and orders of possible poles and compute the residue. Arguing along the theory of Saito and computing unramified local factors, the explicit formula of the zeta functions is obtained. Counting the order of possible poles of this explicit formula, we show that if $L(1/2, π)\neq0$, there are infinitely many quadratic extension $E$ of $F$ which embeds in $D$, such that $π$ has nonvanishing toric period with respect to $E$.

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arXiv2024

Explicit mean value theorems for toric periods and automorphic $L$-functions

Let $F$ be a number field and $D$ a quaternion algebra over $F$. Take a cuspidal automorphic representation $π$ of $D_{\mathbb{A}}^\times$ with trivial central character and a cusp form $ϕ$ in $π$. Using the prehomogeneous zeta function, we find an explicit mean value of the toric periods of $ϕ$ with respect to quadratic algebras over $F$. The result can also be written as a mean value formula for the central values of automorphic $L$-functions twisted by quadratic characters.

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arXiv2024

Asymptotic behavior for twisted traces of self-dual and conjugate self-dual representations of $\mathrm{GL}_n$

In this paper, we study the asymptotic behavior of the sum of twisted traces of self-dual or conjugate self-dual discrete automorphic representations of $\mathrm{GL}_n$ for the level aspect of principal congruence subgroups under some conditions. Our asymptotic formula is derived from the Arthur twisted trace formula, and it is regarded as a twisted version of limit multiplicity formula on Lie groups. We determine the main terms for the asymptotic behavior under different conditions, and also obtain explicit forms of their Fourier transforms, which correspond to endoscopic lifts from classical groups. Its main application is the self-dual (resp. conjugate self-dual) globalization of local self-dual (resp. conjugate self-dual) representations of $\mathrm{GL}_n$. We further derive an automorphic density theorem for conjugate self-dual representations of $\mathrm{GL}_n$.

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arXiv2023

Non-vanishing theorems for prime twists of some modular $L$-functions

In this paper, we give some non-vanishing results on the central values of prime twists of modular $L$-functions by imaginary quadratic fields for specific elliptic modular forms. In particular, we show that the central values of prime twists of $L$-functions of some elliptic modular forms are always non-vanishing whenever the root number is positive.

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arXiv2022

Distribution of toric periods of modular forms on definite quaternion algebras

Let $D$ be a definite quaternion algebra over $\mathbb{Q}$ and $\mathcal{O}$ an Eichler order in $D$ of square-free level. We study distribution of the toric periods of algebraic modular forms of level $\mathcal{O}$. We focus on two problems: non-vanishing and sign changes. Firstly, under certain conditions on $\mathcal{O}$, we prove the non-vanishing of the toric periods for positive proportion of imaginary quadratic fields. This improves the known lower bounds toward Goldfeld's conjecture in some cases and provides evidence for similar non-vanishing conjectures for central values of twisted automorphic $L$-functions. Secondly, we show that the sequence of toric periods has infinitely many sign changes. This proves the sign changes of the Fourier coefficients $\{a(n)\}_n$ of weight 3/2 modular forms, where $n$ ranges over fundamental discriminants. In the final section, we present numerical experiments in some cases and formulate several conjectures based on them.

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arXiv2022

Equidistribution theorems for holomorphic Siegel cusp forms of general degree: the level aspect

We prove equidistribution theorems for a family of holomorphic Siegel cusp forms of general degree in the level aspect. Our main contribution is to estimate unipotent contributions for general degree in the geometric side of Arthur's invariant trace formula in terms of Shintani zeta functions. Several applications including the vertical Sato-Tate theorem and low-lying zeros for standard $L$-functions of holomorphic Siegel cusp forms are discussed. We also show that the "non-genuine forms" which come from non-trivial endoscopic contributions by Langlands functoriality classified by Arthur are negligible.

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arXiv2020

Asymptotics for Hecke eigenvalues of automorphic forms on compact arithmetic quotients

In this paper, we describe the asymptotic distribution of Hecke eigenvalues in the Laplace eigenvalue aspect for certain families of Hecke-Maass forms on compact arithmetic quotients. Instead of relying on the trace formula, which was the primary tool in preceding studies on the subject, we use Fourier integral operator methods. This allows us to treat not only spherical, but also non-spherical Hecke-Maass forms with corresponding remainder estimates. Our asymptotic formulas are available for arbitrary simple and connected algebraic groups over number fields with cocompact arithmetic subgroups.

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arXiv2020

The Shintani double zeta functions

In this paper, we give an explicit formula of the Shintani double zeta functions with any ramification in the most general setting of adeles over an arbitrary number field. Three applications of the explicit formula are given. First, we obtain a functional equation satisfied by the Shintani double zeta functions in addition to Shintani's functional equations. Second, we establish the holomorphicity of a certain Dirichlet series generalizing a result by Ibukiyama and Saito. This Dirichlet series occurs in the study of unipotent contributions of the geometric side of the Arthur-Selberg trace formula of the symplectic group. Third, we prove an asymptotic formula of the weighted average of the central values of quadratic Dirichlet $L$-functions.

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arXiv2019

Mass formulas and Eisenstein congruences in higher rank

We use mass formulas to construct minimal parabolic Eisenstein congruences for algebraic modular forms on reductive groups compact at infinity, and study when these yield congruences between cusp forms and Eisenstein series on the quasi-split inner form. This extends recent work of the first author on weight 2 Eisenstein congruences for GL(2) to higher rank. Two issues in higher rank are that the transfer to the quasi-split form is not always cuspidal and sometimes the congruences come from lower rank (e.g., are "endoscopic"). We show our construction yields Eisenstein congruences with non-endoscopic cuspidal automorphic forms on quasi-split unitary groups by using certain unitary groups over division algebras. On the other hand, when using unitary groups over fields, or other groups of Lie type, these Eisenstein congruences typically appear to be endoscopic. This suggests a new way to see higher weight Eisenstein congruences for GL(2), and leads to various conjectures about GL(2) Eisenstein congruences. In supplementary sections, we also generalize previous weight 2 Eisenstein congruences for Hilbert modular forms, and prove some special congruence mod p results between cusp forms on U(p).

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arXiv2018

Subconvex bounds for Hecke-Maass forms on compact arithmetic quotients of semisimple Lie groups

Let $H$ be a semisimple algebraic group, $K$ a maximal compact subgroup of $G:=H(\mathbb{R})$, and $Γ\subset H(\mathbb{Q})$ a congruence arithmetic subgroup. In this paper, we generalize existing subconvex bounds for Hecke-Maass forms on the locally symmetric space $Γ\backslash G/K$ to corresponding bounds on the arithmetic quotient $Γ\backslash G$ for cocompact lattices using the spectral function of an elliptic operator. The bounds obtained extend known subconvex bounds for automorphic forms to non-trivial $K$-types, yielding subconvex bounds for new classes of automorphic representations, and constitute subconvex bounds for eigenfunctions on compact manifolds with both positive and negative sectional curvature. We also obtain new subconvex bounds for holomorphic modular forms in the weight aspect.

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arXiv2018

Equidistribution theorems for holomorphic Siegel modular forms for $GSp_4$; Hecke fields and $n$-level density

This paper is a continuation of the author's previous wotk. We supplement four results on a family of holomorphic Siegel cusp forms for $GSp_4/\mathbb{Q}$. First, we improve the result on Hecke fields. Namely, we prove that the degree of Hecke fields is unbounded on the subspace of genuine forms which do not come from functorial lift of smaller subgroups of $GSp_4$ under a conjecture in local-global compatibility and Arthur's classification for $GSp_4$. Second, we prove simultaneous vertical Sato-Tate theorem. Namely, we prove simultaneous equidistribution of Hecke eigenvalues at finitely many primes. Third, we compute the $n$-level density of degree 4 spinor $L$-functions, and thus we can distinguish the symmetry type depending on the root numbers. This is conditional on certain conjecture on root numbers. Fourth, we consider equidistribution of paramodular forms. In this case, we can prove a result on root numbers. Main tools are the equidistribution theorem in our previous work and Shin-Templier's work.

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arXiv2016

The dimensions of spaces of Siegel cusp forms of general degree

In this paper, we give a dimension formula for spaces of Siegel cusp forms of general degree with respect to neat arithmetic subgroups. The formula was conjectured before by several researchers. The dimensions are expressed by special values of Shintani zeta functions for spaces of symmetric matrices at non-positive integers. This formula was given by Shintani for only a small part of the geometric side of the trace formula. To be precise, it is the contribution of unipotent elements corresponding to the partitions $(2^j,1^{2n-2j})$, where $n$ denotes the degree and $0\leq j \leq n$. Hence, our work is to show that all the other contributions vanish. In addition, one finds that Shintani's formula means the dimension itself. Combining our formula and an explicit formula of the Shintani zeta functions, which was discovered by Ibukiyama and Saito, we can derive an explicit dimension formula for the principal congruence subgroups of level greater than two. In this explicit dimension formula, the dimensions are described by degree $n$, weight $k$, level $N$, and the Bernoulli numbers $B_m$.

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arXiv2016

An equidistribution theorem for holomorphic Siegel modular forms for $GSp_4$

We prove an equidistribution theorem for a family of holomorphic Siegel cusp forms for $GSp_4/\mathbb{Q}$ in various aspects. A main tool is Arthur's invariant trace formula. While Shin and Shin-Templier used Euler-Poincaré functions at infinity in the formula, we use a pseudo-coefficient of a holomorphic discrete series to extract holomorphic Siegel cusp forms. Then the non-semisimple contributions arise from the geometric side, and this provides new second main terms $A, B_1$ in the main theorem which have not been studied and a mysterious second term $B_2$ also appears in the second main term coming from the semisimple elements. Furthermore our explicit study enables us to treat more general aspects in the weight. We also give several applications including the vertical Sato-Tate theorem, the unboundedness of Hecke fields and low-lying zeros for degree 4 spinor $L$-functions and degree 5 standard $L$-functions of holomorphic Siegel cusp forms.

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