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Shingo Sugiyama

Publications and source records attributed to Shingo Sugiyama.

13 recordsLinked to original sources
arXiv2024

Integrality of Hecke eigenvalues and the growth of Hecke fields

We prove that Hecke eigenvalues for any Hilbert and Siegel modular forms are algebraic integers. Our method does not rely on cohomologicality nor Galois representations. We apply the integrality of Hecke eigenvalues for Hilbert modular forms of non-parallel weight to the estimation of the growth of Hecke fields of Hilbert cusp forms with non-vanishing central $L$-values. As a further application, we give the growth of the fields of rationality of cuspidal automorphic representations of ${\rm GL}_{2d}(\mathbb{A}_\mathbb{Q})$ for a prime number $d$ with non-vanishing central $L$-values. We also apply the integrality of Hecke eigenvalues for holomorphic Siegel cusp forms of general degree in order to give the growth of the Hecke fields of those forms.

math.NT↗
arXiv2023

Lattice sums of $I$-Bessel functions, theta functions, linear codes and heat equations

We extend a certain type of identities on sums of $I$-Bessel functions on lattices, previously given by G. Chinta, J. Jorgenson, A. Karlsson and M. Neuhauser. Moreover we prove that, with continuum limit, the transformation formulas of theta functions such as the Dedekind eta function can be given by $I$-Bessel lattice sum identities with characters. We consider analogues of theta functions of lattices coming from linear codes and show that sums of $I$-Bessel functions defined by linear codes can be expressed by complete weight enumerators. We also prove that $I$-Bessel lattice sums appear as solutions of heat equations on general lattices. As a further application, we obtain an explicit solution of the heat equation on $\mathbb{Z}^n$ whose initial condition is given by a linear code.

math-ph↗
arXiv2022

Weighted one-level density of low-lying zeros of Dirichlet $L$-functions

In this paper, we compute the one-level density of low-lying zeros of Dirichlet $L$-functions in a family weighted by special values of Dirichlet $L$-functions at a fixed $s \in [1/2, 1)$. We verify both Fazzari's conjecture and the first author's conjecture on the weighted one-level density for our family of $L$-functions.

math.NT↗
arXiv2022

A remark on the existence of equivariant functions

Let $\Gamma$ be a Fuchsian group in ${\rm SL}_2(\mathbb{R})$. In this note, we discuss the existence of $\rho$-equivariant functions for a two-dimensional representation $\rho$ of $\Gamma$. This assertion was first stated by Saber and Sebbar in 2020, and this note partially fills a gap of their statement by proving the assertion for a certain class of Fuchsian groups such as conjugates of subgroups of ${\rm SL}_2(\mathbb{Z})$.

math.NT↗
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 ${\rm K}\rightarrow \infty$.

math.NT↗
arXiv2021

Low-lying zeros of symmetric power $L$-functions weighted by symmetric square $L$-values

For a totally real number field $F$ and its adèle ring $\mathbb{A}_F$, let $π$ vary in the set of irreducible cuspidal automorphic representations of ${\rm PGL}_2(\mathbb{A}_F)$ corresponding to primitive Hilbert modular forms of a fixed weight. Then, we determine the symmetry type of the one-level density of low-lying zeros of the symmetric power $L$-functions $L(s,{\rm Sym}^r(π))$ weighted by special values of symmetric square $L$-functions $L(\frac{z+1}{2},{\rm Sym}^2(π))$ at $z \in [0, 1]$ in the level aspect. If $0 < z \le 1$, our weighted density in the level aspect has the same symmetry type as Ricotta and Royer's density of low-lying zeros of symmetric power $L$-functions for $F=\mathbb{Q}$ with harmonic weight. Hence our result is regarded as a $z$-interpolation of Ricotta and Royer's result. If $z=0$, density of low-lying zeros weighted by central values is a different type only when $r=2$, and it does not appear in random matrix theory as Katz and Sarnak predicted. Moreover, we propose a conjecture on weighted density of low-lying zeros of $L$-functions by special $L$-values. In the latter part, Appendices A, B and C are dedicated to the comparison among several generalizations of Zagier's parameterized trace formula. We prove that the explicit Jacquet-Zagier type trace formula (the ST trace formula) by Tsuzuki and the author recovers all of Zagier's, Takase's and Mizumoto's formulas by specializing several data. Such comparison is not so straightforward and includes non-trivial analytic evaluations.

math.NT↗
arXiv2020

The limit theorem with respect to the matrices on non-backtracking paths of a graph

We give a limit theorem with respect to the matrices related to non-backtracking paths of a regular graph. The limit obtained closely resembles the $k$th moments of the arcsine law. Furthermore, we obtain the asymptotics of the averages of the $p^m$th Fourier coefficients of the cusp forms related to the Ramanujan graphs defined by A. Lubotzky, R. Phillips and P. Sarnak.

math.CO↗
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 $k\ge 2$.

math.NT↗
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.

math.NT↗
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.

math.NT↗
arXiv2014

Asymptotic behaviors of means of central values of automorphic $L$-functions for GL(2)

Let $\mathbb{A}$ be the adele ring of a totally real algebraic number field $F$. We push forward an explicit computation of a relative trace formula for periods of automorphic forms along a split torus in $GL(2)$ from a square free level case done by Masao Tsuzuki, to an arbitrary level case. By using a relative trace formula, we study central values of automorphic $L$-functions for cuspidal automorphic representations of $GL(2, \mathbb{A})$ corresponding to Maass forms with arbitrary level.

math.NT↗
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.

math.NT↗