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Masao Tsuzuki

Publications and source records attributed to Masao Tsuzuki.

13 recordsLinked to original sources
arXiv2025

Counting Lattices with Local Hecke Series

We count the maximal lattices over $p$-adic fields and the rational number field. For this, we use the theory of Hecke series for a reductive group over nonarchimedean local fields, which was developed by Andrianov and Hina-Sugano. By treating the Euler factors of the counting Dirichlet series for lattices, we obtain zeta functions of classical groups, which were earlier studied with $p$-adic cone integrals. When our counting series equals the existing zeta functions of groups, we recover the known results in a simple way. Further we obtain some new zeta functions for non-split even orthogonal and odd orthogonal groups.

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arXiv2025

The Ranking-Selberg integral on ${\bf GSp(2)}$ for square free levels

We explicitly compute the Rankin-Selberg type integral introduced by Piatetski-Shapiro over adeles for vector-valued Siegel cusp forms of square-free levels $Γ_0(N)$. On the way, for particular test functions in the Bessel models of irreducible admissible representations, exact evaluations of the local zeta-integrals are given.

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arXiv2023

Functional equations and gamma factors of local zeta functions for the metaplectic cover of SL_2

We introduce a local zeta-function for an irreducible admissible supercuspidal representation $π$ of the metaplectic double cover of $\SL_2$ over a non-archimedean local field of characteristic zero. We prove a functional equation of the local zeta-functions showing that the gamma factor is given by a Mellin type transform of the Bessel function of $π$. We obtain an expression of the gamma factor, which shows its entireness on $\C$. Moreover, we show that, through the local theta-correspondence, the local zeta-function on the covering group is essentially identified with the local zeta-integral for spherical functions on ${\rm PGL}_2\cong {\rm SO}_3$ associated with the prehomogenous vector space of binary symmetric matrices.

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arXiv2021

Quantitative non-vanishing of central values of certain $L$-functions on ${\rm GL}(2)\times {\rm GL}(3)$

Let $ϕ$ be an even Hecke-Maass cusp form on ${\rm SL}_2(\mathbb{Z})$ whose $L$-function does not vanish at the center of the functional equation. In this article, we obtain an exact formula of the average of triple products of $ϕ$, $f$ and $\bar f$, where $f$ runs over an orthonormal basis $H_k$ of Hecke eigen elliptic cusp forms on ${\rm SL}_2(\mathbb{Z})$ of a fixed weight $k\geq 4$. As an application, we prove a quantitative non-vanishing results on the central values for the family of degree $6$ $L$-functions $L(s,ϕ\times {\rm Ad}\,f)$ with $f$ in the union of $H_k$ $({\rm K} \leq k < 2{\rm K})$ as ${\rm K}\rightarrow \infty$.

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arXiv2021

Hecke-Maass cusp forms on PGL_n of large levels with non-vanishing L-values

We introduce a new trace formula of Kuznetsov type involving the central standard L-values and the Whittaker periods of cuspidal automorphic representations of PGL_n(Q) which are spherical at the archimedean place. As an application, we show a simultaneous non-vanishing of standard L-values at n-1 points on the critical strip for infinitely many Hecke-Maass cuspidal newforms.

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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

Spectral average of central values of automorphic L-functions for holomorphic cusp forms on SO_0(m,2) II

Given a maximal even-integral lattice $\cL$ of signature $(m+, 2-)$ with an odd $m\geq 3$, we consider the holomorphic cusp forms $F$ of weight $l$ on the bounded symmetric domain of type IV of dimension $m$ with respect to the discriminant subgroup of the orthogonal group $O(\cL)$ defined by $\cL$. Under a non-negativity assumption on the central $L$-values, we prove an equidistribution result of Satake parameters in an ensemble constructed from the central values of standard $L$-functions and the square of the Whittaker-Bessel periods.

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arXiv2018

Optimal estimates for an average of Hurwitz class numbers

In this paper, we give an optimal estimate of an average of Hurwitz class numbers. As an application, we give an equidistribution result of the family $\{\frac{t}{2q^{ν/2}} \ | \ ν\in \mathbb{N}, t \in \mathbb{Z}, |t|<2q^{ν/2}\}$ with $q$ prime, weighted by Hurwitz class numbers. This equidistribution produces many asymptotic relations among Hurwitz class numbers. Our proof relies on the resolvent trace formula of Hecke operators on elliptic cusp forms of weight $k\ge 2$.

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arXiv2018

An explicit trace formula of Jacquet-Zagier type for Hilbert modular forms

We give an exact formula of the average of adjoint $L$-functions of holomorphic Hilbert cusp forms with a fixed weight and a square-free level, which is a generalization of Zagier's formula known for the case of elliptic cusp forms on ${\rm SL}_2(\mathbb{Z})$. As an application, we prove that the Satake parameters of Hilbert cusp forms with a fixed weight and with growing square-free levels are equidistributed in an ensemble constructed by values of the adjoint $L$-functions.

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arXiv2015

Relative trace formulas and subconvexity estimates of L-functions for Hilbert modular forms

We elaborate an explicit version of the relative trace formula on $\PGL(2)$ over a totally real number field for the toral periods of Hilbert cusp forms along the diagonal split torus. As an application, we prove (i) a spectral equidistribution result in the level aspect for Satake parameters of holomorphic Hilbert cusp forms weighted by central $L$-values, and (ii) a bound of quadratic base change $L$-functions for Hilbert cusp forms with a subconvex exponent in the weight aspect.

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arXiv2014

Existence of Hilbert cusp forms with non-vanishing $L$-values

We give a derivative version of the relative trace formula on PGL(2) studied in our previous work, and obtain a formula of an average of central values (derivatives) of automorphic $L$-functions for Hilbert cusp forms. As an application, we prove existence of Hilbert cusp forms with non-vanishing central values (derivatives) such that the absolute degrees of their Hecke fields are sufficiently large.

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