Updated on 2022/07/02

写真a

 
NARITA, Hiroaki
 
Affiliation
Faculty of Science and Engineering, School of Fundamental Science and Engineering
Job title
Professor

Concurrent Post

  • Faculty of Science and Engineering   Graduate School of Fundamental Science and Engineering

Research Institute

  • 2020
    -
    2022

    理工学術院総合研究所   兼任研究員

Education

  •  
    -
    2000

    The University of Tokyo  

  •  
    -
    2000

    The University of Tokyo   Graduate School, Division of Mathematical Sciences  

  •  
    -
    1995

    Waseda University   School of Science and Engineering  

  •  
    -
    1995

    Waseda University   Faculty of Science and Engineering  

Degree

  • 博士(数理科学)

Research Experience

  • 2018
    -
    Now

    Waseda University   Faculty of Science and Engineering

  • 2008
    -
    2017

    Kumamoto University   Graduate School of Science and Technology

  • 2008
    -
     

    Associate Professor, ,Graduate School of Science and Technology(Science group),Kumamoto University

  • 2006
    -
    2008

    大阪市立大学 数学研究所 COE研究所員

  • 2006
    -
    2008

    COE Researcher, Advanced Mathematical Institute,Osaka City University

  • 2005
    -
    2006

    マックスプランク数学研究所 客員研究員

  • 2005
    -
    2006

    Guest Researcher,Max-Planck-Institute for Mathematics

  • 2002
    -
    2005

    日本学術振興会 特別研究員

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Research Areas

  • Algebra

Research Interests

  • 保型形式論

  • 整数論

  • automorphic forms

  • Number theory

Papers

  • Jacquet-Langlands-Shimizu correspondence for theta lifts to GSp(2) and its inner forms II: an explicit formula for Bessel periods and the non-vanishing of theta lifts

    Hiro-aki Narita

    Journal of the mathematical society of Japan   73 ( 1 ) 125 - 159  2021  [Refereed]

  • Modular degrees of elliptic curves and some quotients of L-values

    Kousuke Sugimoto, Hiro-aki Narita

    Tokyo Journal of mathematics   43 ( 2 ) 279 - 293  2020  [Refereed]

    Authorship:Last author

  • An explicit construction of non-tempered cusp forms on O(1,8n+1)

    Yingkun Li, Hiro-aki Narita, Ameya Pitale

    Annales math. Quebec   44 ( 2 ) 349 - 384  2020  [Refereed]

    Authorship:Lead author

  • Jacquet-Langlands-Shimizu correspondence for theta lifts to &ITGSp&IT(2) and its inner forms I: An explicit functorial correspondence

    Hiro-aki Narita, Ralf Schmidt

    JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN   69 ( 4 ) 1443 - 1474  2017.10  [Refereed]

     View Summary

    As was first essentially pointed out by Tomoyoshi Ibukiyama, Hecke eigenforms on the indefinite symplectic group GSp(1,1) or the definite symplectic group GSp*(2) over Q right invariant by a (global) maximal open compact subgroup are conjectured to have the same spinor L-functions as those of paramodular new forms of some specified level on the symplectic group GSp(2) (or GSp(4)). This can be viewed as a generalization of the Jacquet-Langlands-Shimizu correspondence to the case of GSp(2) and its inner forms GSp(1,1) and GSp*(2).& para;& para;In this paper we provide evidence of the conjecture on this explicit functorial correspondence with theta lifts: a theta lift from GL(2) x B-x to GSp(1,1) or GSp*(2) and a theta lift from GL(2) x GL(2) (or GO(2,2)) to GSp(2). Here B denotes a definite quaternion algebra over Q. Our explicit functorial correspondence given by these theta lifts are proved to be compatible with archimedean and non-archimedean local Jacquet-Langlands correspondences. Regarding the non-archimedean local theory we need some explicit functorial correspondence for spherical representations of the inner form and non-supercuspidal representations of GSp(2), which is studied in the appendix by Ralf Schmidt.

    DOI

  • LIFTING TO GL(2) OVER A DIVISION QUATERNION ALGEBRA, AND AN EXPLICIT CONSTRUCTION OF CAP REPRESENTATION

    Masanori Muto, Hiro-Aki Narita, Ameya Pitale

    NAGOYA MATHEMATICAL JOURNAL   222 ( 1 ) 137 - 185  2016.06  [Refereed]

     View Summary

    The aim of this paper is to carry out an explicit construction of CAP representations of GL(2) over a division quaternion algebra with discriminant two. We first construct cusp forms on such a group explicitly by lifting from Maass cusp forms for the congruence subgroup Gamma(0)(2). We show that this lifting is nonzero and Hecke-equivariant. This allows us to determine each local component of a cuspidal representation generated by such a lifting. We then show that our cuspidal representations provide examples of CAP (cuspidal representation associated to a parabolic subgroup) representations, and, in fact, counterexamples to the Ramanujan conjecture.

    DOI

  • Fourier expansion of Arakawa lifting II: Relation with central L-values

    Atsushi Murase, Hiro-aki Narita

    INTERNATIONAL JOURNAL OF MATHEMATICS   27 ( 1 )  2016.01  [Refereed]

     View Summary

    This is a continuation of our previous paper [Fourier expansion of Arakawa lifting I: An explicit formula and examples of non-vanishing lifts, Israel J. Math. 187 (2012) 317-369]. The aim of the paper here is to study the Fourier coefficients of Arakawa lifts in relation with central values of automorphic L-functions. In the previous paper we provide an explicit formula for the Fourier coefficients in terms of toral integrals of automorphic forms on multiplicative groups of quaternion algebras. In this paper, after studying explicit relations between the toral integrals and the central L-values, we explicitly determine the constant of proportionality relating the square norm of a Fourier coefficient of an Arakawa lift with the central L-value. We can relate the square norm with the central value of some L-function of convolution type attached to the lift and a Hecke character. We also discuss the existence of strictly positive central values of the L-functions in our concern.

    DOI

  • Bessel Periods of Theta Lifts to GSp(1,1) and Central Values of Some L-Functions of Convolution Type

    Hiro-aki Narita

    AUTOMORPHIC FORMS: RESEARCH IN NUMBER THEORY FROM OMAN   115   179 - 191  2014  [Refereed]

    DOI

  • IRREDUCIBILITY CRITERIA FOR LOCAL AND GLOBAL REPRESENTATIONS

    Hiro-Aki Narita, Ameya Pitale, Ralf Schmidt

    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY   141 ( 1 ) 55 - 63  2013.01  [Refereed]

     View Summary

    It is proved that certain types of modular cusp forms generate irreducible automorphic representations of the underlying algebraic group. Analogous Archimedean and non-Archimedean local statements are also given.

    DOI

  • SOME VECTOR-VALUED SINGULAR AUTOMORPHIC FORMS ON U(2,2) AND THEIR RESTRICTION TO Sp(1,1)

    Atsuo Yamauchi, Hiro-Aki Narita

    INTERNATIONAL JOURNAL OF MATHEMATICS   23 ( 10 )  2012.10  [Refereed]

     View Summary

    In this paper we provide a construction of theta series on the real symplectic group of signature (1, 1) or the 4-dimensional hyperbolic space. We obtain these by considering the restriction of some vector-valued singular theta series on the unitary group of signature (2, 2) to this indefinite symplectic group. Our (vector-valued) theta series are proved to have algebraic Fourier coefficients, and lead to a new explicit construction of automorphic forms generating quaternionic discrete series representations and automorphic functions on the hyperbolic space.

    DOI

  • Fourier expansion of Arakawa lifting I: An explicit formula and examples of non-vanishing lifts

    Atsushi Murase, Hiroaki Narita

    Israel Journal of Mathematics   187 ( 1 ) 317 - 369  2012.01  [Refereed]

     View Summary

    Given an elliptic cusp form f and an automorphic form f′ on a definite quaternion algebra over ℚ, there is a theta lifting from (f, f′) to an automorphic form L(f, f′) on the quaternion unitary group GSp(1, 1) generating quaternionic discrete series at the Archimedean place. The aim of this paper is to provide an explicit formula for Fourier coefficients of L(f, f′) in terms of periods of f and f′ with respect to a unitary character χ of an imaginary quadratic field. As an application, we show the existence of (f, f′) with L(f, f′) ≠ 0.

    DOI

  • Theta lifting from elliptic cusp forms to automorphic forms on Sp(1, q)

    Hiro-aki Narita

    MATHEMATISCHE ZEITSCHRIFT   259 ( 3 ) 591 - 615  2008.07  [Refereed]

     View Summary

    Tsuneo Arakawa formulated a theta lifting from elliptic cusp forms to automorphic forms on Sp(1,q) in his unpublished note, which was inspired by "Kudla lifting", i.e. a theta lifting from elliptic modular forms to holomorphic automorphic forms on SU(1,q). We prove that the images of Arakawa's theta lifting belong to the space of bounded automorphic forms generating quaternionic discrete series, which are non-holomorphic forms. In the appendix we provide the construction of Eisenstein series and Poincare series generating such discrete series.

    DOI

  • Commutation relations of Hecke operators for Arakawa lifting

    Atsushi Murase, Hiro-Aki Narita

    TOHOKU MATHEMATICAL JOURNAL   60 ( 2 ) 227 - 251  2008.06  [Refereed]

     View Summary

    T. Arakawa. in his Unpublished note, constructed and Studied it theta lifting front elliptic cusp forms to automorphic forms on the quaternion unitary group Of Signature (1. q), The second named author proved that such a lifting provides bounded (or cuspidal) automorphic forms generating quaternionic discrete series. In this paper. restricting ourselves to the case of q = 1. we reformulate Arakawa's theta lifting as it theta correspondence in the adelic setting and determine a commutation relation of Hecke operators satisfied by the lifting. As in application, we show that the theta lift of an elliptic Hecke eigenform is also it Hecke eigenform On the quaternion unitary group. We furthermore Study the spinor L-function attached to the theta lift.

    DOI

  • Fourier-Jacobi expansion of automorphic forms on Sp-(1, q) generating quaternionic discrete series

    Hiro-aki Narita

    JOURNAL OF FUNCTIONAL ANALYSIS   239 ( 2 ) 638 - 682  2006.10  [Refereed]

     View Summary

    The aim of this paper is to develop the notion of the Fourier expansion of automotphic forms on Sp(1, q) generating quaternionic discrete series, which are non-holomorphic forms. There is such an expansion given by Tsuneo Arakawa, assuming the boundedness of the forms and the integrability of the discrete series. We study these automorphic forms without such assumptions. When q > 1 we prove the "Koecher principle" for such automorphic forms, whose validity is known for holomorphic automorphic forms except elliptic modular forms. (c) 2006 Elsevier Inc. All rights reserved.

    DOI

  • Fourier expansion of holomorphic modular forms on classical lie groups of tube type along the minimal parabolic subgroup

    H Narita

    ABHANDLUNGEN AUS DEM MATHEMATISCHEN SEMINAR DER UNIVERSITAT HAMBURG   74   253 - 279  2004  [Refereed]

     View Summary

    For holomorphic modular forms,on tube domains, there are two types of known Fourier expansions, i.e. the classical Fourier expansion and the Fourier-Jacobi expansion. Either of them is along a maximal parabolic subgroup. In this paper, we discuss Fourier expansion of holomorphic modular forms on tube domains of classical type along the minimal parabolic subgroup. We also relate our Fourier expansion to the two known ones in terms of Fourier coefficients and theta series appearing in these expansions.

    DOI

  • Fourier expansion of holomorphic Siegel modular forms of genus n along the minimal parabolic subgroup

    NARITA Hiroaki

    Journal of the Mathematical Sciences, the University of Tokyo   10  2003  [Refereed]

  • Fourier expansion of holomorphic Siegel modular forms with respect to the minimal parabolic subgroup

    H Narita

    MATHEMATISCHE ZEITSCHRIFT   231 ( 3 ) 557 - 588  1999.07  [Refereed]

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Research Projects

  • 実双曲空間上の実解析保型形式のリフティングによる多様な構成と多方面分野への応用

    科学研究費補助金(基盤研究C)

    Project Year :

    2016
    -
    2018
     

    成田 宏秋

  • 保型形式の整数論、具体的構成の観点からの研究領域の拡張

    科学研究費補助金(基盤研究C)

    Project Year :

    2012
    -
    2014
     

    成田 宏秋

  • 保型形式の具体的構成とその数論的及び幾何学的応用

    科学研究費補助金(若手研究B)

    Project Year :

    2009
    -
    2011
     

    成田 宏秋

  • 四元数離散系列表現を生成する保型形式の解析的及び数論的研究

    科学研究費補助金(若手研究B)

    Project Year :

    2006
    -
    2008
     

    成田 宏秋

  • Research on analysis and arithmetic of automorphic forms generating quaternionic discrete series

    Grant-in-Aid for Scientific Research

    Project Year :

    2006
    -
    2008
     

Presentations

  • EXplicit constructions of non-tempered cusp forms on orthogonal groups of low split ranks

    成田 宏秋  [Invited]

    RIMS共同研究(公開型) 保型形式の解析的・数論的研究 

    Presentation date: 2018.01

  • Explicit constructions of non-tempered cusp forms on orthogonal groups of low split ranks

    成田 宏秋  [Invited]

    The third Japanese-German Number Theory Workshop (at MPIM) 

    Presentation date: 2017.11

  • Fourier expansion of Siegel modular forms along the minimal parabolic subgroup

    成田 宏秋  [Invited]

    第19回整数論オータムワークショップ (於 長野県白馬村 白馬ハイマウントホテル) 

    Presentation date: 2016.11

  • Lifting to an inner form of GL(4) and counterexamples of the Ramanujan conjecture

     [Invited]

    Presentation date: 2014.11

  • Lifting from Maass cusp forms for Γ_0(2) to cusp forms on GL(2) over a division quaternion algebra

     [Invited]

    Presentation date: 2014.01

  • Bessel periods of theta lifts to GSp(1,1) and central values of some L-functions of convolution type

     [Invited]

    International conference on automorphic forms and number theory (at Oman) 

    Presentation date: 2012.02

  • Jacquet-Langlands-Shimizu correspondence for two theta lifts to GSp(2) and GSp(1,1)

     [Invited]

    Presentation date: 2012.01

  • Fourier coefficients of Arakawa lifting and central values of some Rankin-Selberg L-functions

    成田 宏秋  [Invited]

    第55回代数学シンポジウム (於 北海道大学) 

    Presentation date: 2010.08

  • 四元数ユニタリー群Sp(1,q)上の実解析的保型形式について

    成田 宏秋  [Invited]

    第53回代数学シンポジウム (於 いわて県民情報交流センター) 

    Presentation date: 2008.08

  • Generalized Whittaker functions on Sp(1,q) for quaternionic discrete series and their application to automorphic forms

    NARITA Hiroaki  [Invited]

    The northern workshop on representation theory of LIe groups and Lie algebras 

    Presentation date: 2007.03

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Specific Research

  • 実双曲空間上のテータリフトの内積公式とその応用

    2020   Ameya Pitale

     View Summary

    オクラホマ大学のAmeya Pitale氏との共同研究により、以前より研究していた複素上半平面上のMaassカスプ形式からのテータリフトについて、その内積公式を明示的に求める計算を行ってきた。この公式はEuler積表示を持ち、不分岐有限素点の成分は既に一般的に計算されているが分岐有限素点と無限素点の成分は自力で計算する必要がある。今年度の研究で我々のテータ―リフトの内積公式の無限素点の明示形を求めることができた。有限分岐素点の明示形も導出のための基本的な材料が揃った段階で現時点で「時間が十分あればできる」という段階に至っている。

  • 実双曲空間上の保型形式の逆定理と周期の研究

    2019  

     View Summary

    有理数体上定符号四元数環係数の2次一般線形群上のカスプ形式のリフティングによる構成について、判別式が2の場合で得られていたものを一般の素数判別式の場合に一般化することができた。また非消滅の例も判別式2の場合以外にいくつか更に与えることができた。周期の研究についてはリフティングのWeil表現による定式化を進めそのPetersson内積などの周期に関連する研究を更に進めることで次年度も継続することとした。逆定理のアデール化についても研究したがBruhat細胞分解の観点からの保型性の特徴づけに部分的解決は得られたもののまだ不十分という現時点での結果となった。本研究はOklahoma大学のAmeya Pitale氏との共同研究に基づく。

  • 実双曲空間上の実解析的保型形式のリフティングによる構成

    2018  

     View Summary

    これまで行ってきた実双曲空間上の保型形式、特にカスプ形式の具体的構成について、既に与えた具体的構成を広い枠組みで捉える一般論の構築に成功した。より詳細には、これまで複素上半平面上の実解析的Maassカスプ形式からのリフティングによる構成を与えてきたが、これは有限素点で「非緩増加」という性質を満たし、所謂Ramanujan予想の反例条件を満たす。このリフティングの定性的側面として「特殊Bessel模型を持つ」というのがある。本研究期間において、一般の直交群上の保型形式が特殊Bessel模型を持ち且つ「Maass関係式」が成り立つ有限素点が存在すれば、そこで非緩増加であるという一般論を与えた。

 

Syllabus

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