Updated on 2024/04/18

写真a

 
HIRONAKA, Yumiko
 
Affiliation
Faculty of Education and Integrated Arts and Sciences
Job title
Professor Emeritus
Degree
理学博士 ( 筑波大学 )
Doctor of Science

Research Experience

  • 1998
    -
     

    Waseda University, School of Education, Professor

  • 1992
    -
    1998

    Shinshu University, Faculty of Science, Associate Professor

  • 1982
    -
    1992

    Shinshu University, Faculty of Science, Assistant Professor

Professional Memberships

  •  
     
     

    日本数学会

Research Areas

  • Algebra

Research Interests

  • Number Theory

 

Papers

  • Harmonic analysis on the space of p-adic unitary hermitian matrices, mainly for dyadic case

    Yumiko Hironaka

    Tokyo Journal of Mathematics   40 ( 2 ) 517 - 564  2017.12

     View Summary

    We are interested in harmonic analysis on p-adic homogeneous spaces based on spherical functions. In the present paper, we investigate the space X of unitary hermitian matrices of size m over a p-adic field k mainly for dyadic case, and give the unified description with our previous papers for non-dyadic case. The space becomes complicated for dyadic case, and the set of integral elements in X has plural Cartan orbits. We introduce a typical spherical function ω(x
    z) on X, study its functional equations, which depend on m and the ramification index e of 2 in k, and give its explicit formula, where Hall-Littlewood polynomials of type Cn appear as a main term with different specialization according as the parity m = 2n or 2n + 1, but independent of e. By spherical transform, we show the Schwartz space S(K\\X) is a free Hecke algebra H(G, K)-module of rank 2n, and give parametrization of all the spherical functions on X and the explicit Plancherel formula on S(K\\X). The Plancherel measure does not depend on e, but the normalization of G-invariant measure on X depends.

    DOI

    Scopus

    1
    Citation
    (Scopus)
  • Harmonic analysis on the space of p-adic unitary hermitian matrices, mainly for dyadic case

    Hironaka, Yumiko

    Tokyo Journal of Mathematics   40 ( 2 ) 517 - 564  2017.12

     View Summary

    We are interested in harmonic analysis on p-adic homogeneous spaces based on spherical functions. In the present paper, we investigate the space X of unitary hermitian matrices of size m over a p-adic field k mainly for dyadic case, and give the unified description with our previous papers for non-dyadic case. The space becomes complicated for dyadic case, and the set of integral elements in X has plural Cartan orbits. We introduce a typical spherical function ω(x; z) on X, study its functional equations, which depend on m and the ramification index e of 2 in k, and give its explicit formula, where Hall-Littlewood polynomials of type C n appear as a main term with different specialization according as the parity m = 2n or 2n + 1, but independent of e. By spherical transform, we show the Schwartz space S(K\X) is a free Hecke algebra H(G, K)-module of rank 2 n , and give parametrization of all the spherical functions on X and the explicit Plancherel formula on S(K\X). The Plancherel measure does not depend on e, but the normalization of G-invariant measure on X depends.

    DOI

    Scopus

    1
    Citation
    (Scopus)
  • Zeta functions of finite groups by enumerating subgroups

    Yumiko Hironaka

    Communications in Algebra   45   3365 - 3376  2017  [Refereed]

  • Zeta functions of finite groups by enumerating subgroups

    Yumiko Hironaka

    COMMUNICATIONS IN ALGEBRA   45 ( 8 ) 3365 - 3376  2017  [Refereed]

     View Summary

    For a finite group G, we consider the zeta function zeta G(s) = Sigma(H)vertical bar H vertical bar(-s), where H runs over the subgroups of G. First we give simple examples of abelian p-group G and non-abelian p-group G' of order p(m), m >= 3 for odd p (resp. 2(m), m >= 4) for which zeta G(s) = zeta(G')(s) . Hence we see there are many non-abelian groups whose zeta functions have symmetry and Euler product, like the case of abelian groups. On the other hand, we show that zeta(G)(s) determines the isomorphism class of G within abelian groups, by estimating the number of subgroups of abelian p-groups. Finally we study the problem which abelian p-group is associated with a non-abelian group having the same zeta function.

    DOI

    Scopus

    1
    Citation
    (Scopus)
  • SPHERICAL FUNCTIONS ON THE SPACE OF p-ADIC UNITARY HERMITIAN MATRICES

    Yumiko Hironaka, Yasushi Komori

    INTERNATIONAL JOURNAL OF NUMBER THEORY   10 ( 2 ) 513 - 558  2014.03  [Refereed]

     View Summary

    We investigate the space X of unitary hermitian matrices over p-adic fields through spherical functions. First we consider Cartan decomposition of X, and give precise representatives for fields with odd residual characteristic, i.e. 2 is not an element of p. From Sec. 2.2 till the end of Sec. 4, we assume odd residual characteristic, and give explicit formulas of typical spherical functions on X, where Hall-Littlewood symmetric polynomials of type C-n appear as a main term, parametrization of all the spherical functions. By spherical Fourier transform, we show that the Schwartz space S(K\X) is a free Hecke algebra H(G, K)-module of rank 2(n), where 2n is the size of matrices in X, and give the explicit Plancherel formula on S(K\X).

    DOI

    Scopus

    2
    Citation
    (Scopus)
  • Spherical functions on the space of $p$-adic unitary hermitian matrices II, the case of odd size

    Yumiko Hironaka, Yasushi Komori

    Commentarii Mathematici Universitatis Sacnti Pauli   63 ( 1 ) 47 - 78  2014  [Refereed]

    DOI CiNii

  • Spherical functions on U(2n)/(U(n) x U(n)) and hermitian Siegel series

    Yumiko Hironaka

    GEOMETRY AND ANALYSIS OF AUTOMORPHIC FORMS OF SEVERAL VARIABLES   7   120 - 159  2012  [Refereed]

  • Spherical functions on $p$-adic homogeneous spaces

    Yumiko, Hironaka

    MSJ Memoires   21   50 - 72  2010  [Refereed]

  • Linear independence of local densities of quadratic forms and its application to the theory of Siegel modular forms

    Siegfried Boecherer, Yumiko Hironaka, Fumihiro Sato

    QUADRATIC FORMS - ALGEBRA, ARITHMETIC, AND GEOMETRY   493   51 - +  2009  [Refereed]

     View Summary

    Regarding local densities alpha(p) (S,T) of quadratic forms as functions of T, we study their linear independence when S varies by scaling of hyperbolic planes. Since local densities can be regarded as Fourier transforms of certain Gauss sums, we study their linear independence by reducing it to the corresponding problem for these Gauss sums. We assume throughout that the size m of S is large compared to the size n of T.
    Since local densities appear in Fourier coefficients of genus theta series, our local results allow us to exhibit many linearly independent genus theta series of fixed level. They generate the full space of Siegel Eisenstein series if the level is squarefree, but only a quite small subspace otherwise. We characterize this small subspace intrinsically within the space of Siegel Eisenstein series of fixed level.
    Our work was inspired by an article of Katsurada and Schulze- Pillot [14], where the case of prime level was considered.

  • The Siegel series and spherical functions on O(2n)/(O(n) x O(n))

    Y Hironaka, F Sato

    AUTOMORPHIC FORMS AND ZETA FUNCTIONS     150 - +  2006  [Refereed]

  • Spherical functions on p-adic homogeneous spaces

    Y Hironaka

    Number Theory: Tradition and Modernization   15   81 - 95  2006  [Refereed]

     View Summary

    In closed integral 1, after defining spherical functions on homogeneous spaces, we examine the case of symmetric forms as an enlightening example. In closed integral 2, we introduce a general formula of spherical functions using functional equations under suitable assumption. In closed integral 3, we study a certain mechanism of functional equations of spherical functions.

  • Spherical functions on S-P2 as a spherical homogeneous S-P2 x (S-P1)(2)-space

    Y Hironaka

    JOURNAL OF NUMBER THEORY   112 ( 2 ) 238 - 286  2005.06  [Refereed]

     View Summary

    We investigate spherical functions on Sp(2) as a spherical homogeneous G = Sp(2) x (Sp(1))(2)-space over a p-adic field k, which form a 4-dimensional vector space for each eigenvalue given by Satake parameter. Explicit expressions of spherical functions and Cartan decomposition Of Sp(2) are given. Using spherical transform, we determine Hecke module structure of the Schwartz-Bruhat space S(K\Sp(2)), which is free of rank 4. © 2004 Elsevier Inc. All rights reserved.

    DOI

    Scopus

    6
    Citation
    (Scopus)
  • 数論とは何か?

    日本の科学者   40 ( 3 )  2005.03

  • Functional equations of spherical functions on p-adic homogeneous spaces

    Y Hironaka

    ABHANDLUNGEN AUS DEM MATHEMATISCHEN SEMINAR DER UNIVERSITAT HAMBURG   75   285 - 311  2005  [Refereed]

     View Summary

    Let G be a connected reductive linear algebraic group and X a G-homogeneous affine algebraic variety both defined over a p-adic field k, where we assume a minimal k-parabolic subgroup of G acts with open orbit. We are interested in spherical functions on X = X(k). In the present papaer, we give a unified method to obtain functional equations of spherical functions on X under the condition (AF) in the introduction, and explain functional equations are reduced to those of p-adic local zeta functions of small prehomogeneous vector spaces of limited type.

  • Spherical function on $p$-adic spherical homogeneous spaces

    Prceedings of the Japan-korea Joint Seminar on Number Theory     45 - 58  2004.10

  • Spherical functions on certain spherical homogeneous spaces over $p$-adic fiels

    京都大学数理解析研究所講究録   1338   91 - 106  2003.07

  • $p$進線型空間上の球関数や局所密度 --- 概均質ベクトル空間の理論の応用として---

    第10回 整数論サマースクール「概均質ベクトル空間」     195 - 204  2003.02

  • Spherical functions on some spherical homogeneous spaces

    Proceedings of Japanese-German Seminar Explicit structure of Modular forms and Zeta functions     233 - 239  2002

  • A remark on Kitaoka's power series attached to local densities

    Hironaka Yumiko

    Commentarii Mathematici Universitatis Sancti Pauli   50 ( 2 ) 141 - 146  2001  [Refereed]

    CiNii

  • Classification by Iwahori subgroup and local densities on hermitian forms

    京都大学数理解析研究所講究録   1173   143 - 154  2000.10

  • Local densities of representations of quadratic forms over p-adic integers (the non-dyadic case)

    F Sato, Y Hironaka

    JOURNAL OF NUMBER THEORY   83 ( 1 ) 106 - 136  2000.07  [Refereed]

     View Summary

    We give an explicit formula for local densities of integral representations of nondegenerate integral symmetric matrices of arbitrary size in the ease p not equal 2, in terms of invariants of quadratic forms, (C) 2000 Academic Press.

  • Equivalence by Iwahori subgroup and local densities on hermitian forms

    Proceedings of the Jangieon Mathematical Society(Korea)   1   51 - 73  2000.07

  • Classification of hermitian forms by the Iwahori subgroup and local densities,

    Hironaka Yumiko

    Commentarii Mathematici Universitatis Sancti Pauli   49 ( 2 ) 105 - 142  2000  [Refereed]

    CiNii

  • Local Densities of Representations of Quadratic Forms,佐藤文広氏と共著,

    第2回 整数論オータムワークショップ報告集     67 - 86  2000.01

  • Spherical functions and local densities on hermitian forms

    Y Hironaka

    JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN   51 ( 3 ) 553 - 581  1999.07  [Refereed]

     View Summary

    First we give a formula of spherical functions on certain spherical homogeneous spaces. Then, applying it, we complete the theory of the spherical functions on the space X of nondegenerate unramified hermitian forms on a p-adic number field. More precisely, we give an explicit expression for the spherical functions, prove theorems on the spherical Fourier transforms on the space of Schwartz-Bruhat functions on X, and parametrize of all spherical functions on X. Finally, as an application, we give explicit expressions of local densities of representations of hermitian forms.

  • 二次形式の局所密度の明示公式について, 佐藤文広氏と共著

    京都大学数理解析研究所講究録   1103   60 - 70  1999

  • Local densties of hermitian forms

    Contemporary Mathematics   ( 249 ) 135 - 148  1999  [Refereed]

  • Local zeta functions on hermitian forms and its application to local densities

    Y Hironaka

    JOURNAL OF NUMBER THEORY   71 ( 1 ) 40 - 64  1998.07  [Refereed]

     View Summary

    We give an explicit description of functional equations satisfied by zeta Functions on the space of unramified hermitian forms over a p-adic field. Further, as an application, we give explicit expressions of local densities of integral representations of nondegenerate unramified hermitian matrices with entries in the ring of p-adic integers. (C) 1998 Academic Press.

  • エルミート形式の球関数と局所密度

    第43回 代数学シンポジウム報告集     98 - 109  1998

  • エルミート形式に関する局所ゼータ関数と局所密度

    早稲田大学理工学総合センター研究集会報告集   IX   95 - 100  1998

  • Eisenstein series on reductive Symmetrie spaces and representations of Hecke algebras

    Yumiko Hironaka, Fumihiro Sato

    Journal fur die Reine und Angewandte Mathematik   1993 ( 445 ) 45 - 108  1993  [Refereed]

    DOI

    Scopus

    4
    Citation
    (Scopus)
  • Fourier-Eisenstein transform and Plancherel formula for rational binary quadratic forms

    Yumiko Hironaka, Fumihiro Sato

    Nagoya Mathematical Journal   128   121 - 151  1992  [Refereed]

    DOI

    Scopus

    1
    Citation
    (Scopus)
  • Spherical functions of hermitian and symmetric forms II

    Yumiko Hironaka

    Japanese Journal of Mathematics   15 ( 1 ) 15 - 51  1989  [Refereed]

    DOI

    Scopus

    7
    Citation
    (Scopus)
  • Local densities of alternating forms

    Yumiko Hironaka, Fumihiro Sato

    Journal of Number Theory   33 ( 1 ) 32 - 52  1989  [Refereed]

     View Summary

    We give an explicit formula for local densities of integral representations of non-degenerate alternating matrices with entries in the ring of p-adic integers in terms of elementary divisors. The proof of the formula is based on the local functional equation satisfied by the zeta function on the space of alternating forms and some properties of spherical functions. © 1989.

    DOI

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    10
    Citation
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  • Spherical functions of Hermitian and symmetric forms III

    Yumiko Hironaka

    Tohoku Mathematical Journal   40 ( 4 ) 651 - 671  1988  [Refereed]

    DOI

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    7
    Citation
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Research Projects

  • 球関数に基づくp進等質空間の調和解析的研究

    Project Year :

    2016.04
    -
    2020.03
     

     View Summary

    引き続き,$p$進体上の division quaternion 上のエルミート形式の空間 $X$ の研究を主に進めた.この空間に作用している群はdivision quaternion 上の一般線形群なので,これ自体の球関数は,佐武一郎の結果までさかのぼれて,最も標準的なマクドナルド多項式(むしろ Hall-Littlewood多項式と言うべき)によって記述されている.この quaternion エルミート形式の空間でも,体上のエルミート形式や対称形式などと同様に,球関数を局所密度の生成関数としてとらえられる.球関数の明示式の主要項には,エルミート形式のタイプによって,異なる形の新たなマクドナルド多項式が現れるが,正規化は共通にできる.それによる球フーリエ変換で,$X$上の急減少関数の空間は,対称ローラン多項式環の中に移される.サイズ1,2のときは全射となるが,一般には真のイデアルとなると予想される.サイズ3,4のときには,実際に球フーリエ変換像の2個からなる生成元を与えた.引き戻して,$X$上の急減少関数の空間のヘッケ環加群としての生成元が得られた.また,このようなquaternionエルミート形式の空間の研究に示唆されて, 以前扱っていた$p$進体上の分岐エルミート形式の研究を見直しが進んだ.こちらは,同じ固有関数に対応する球関数の次元が上がるので,指標を導入する必要があるが,その指標の種類によっては,quaternionエルミート形式の場合によく似た明示式が与えられ,球関数の主要部として,新たな形のマクドナルド多項式が現れる.この空間の研究は,対称形式の空間の研究に役立つと思われる.division quaternion 上の エルミート形式の空間の球関数について研究が進み,球フーリエ変換像についての計算機を用いての考察により,サイズ3,4のときの球フーリエ変換像の決定や,引き戻しての球減少関数の空間の生成系がとらえられた.この研究には,Mathematica や Grebner基底を用いる Macaulay2 などの計算ソフトが役立った.また,体上の分岐エルミート形式の研究の見直しが可能となり,この空間の球関数の表示式の考察が進んだ.division quaternion 上のエルミート形式の空間の調和解析的考察として,急減少関数の空間のヘッケ環加群としての生成元,球フーリエ変換の像,Plancherel 測度について研究する.球フーリエ変換像について,サイズ4以下では分かったが,一般サイズの場合に決定することは今後の問題である.剰余体標数が偶数(dyadic)の場合に,同じような議論は成立しない部分が多いが,どこまでできるかはやはり興味がある.$p$進体上の分岐エルミート形式の空間についても,研究を進めたい.これ以外にも類似の空間について,視野を広げて研究を進めていきたい.引き続き,立教大学の小森靖氏を連携研究者として迎えて,特にMacdonald 多項式関係の部分の研究を進める予定である.概均質ベクトル空間の理論も,等質空間上の球関数には密接につながりがある.元立教大学の佐藤文広氏に協力して早稲田大学において定期的にセミナーを開催している.ここで,概均質ベクトル空間関係の研究者との研究連絡を図っている.また,研究集会「数論女性の集まり」も引き続き世話人として関わり,自分自身の研究発表をして批評を仰ぐとともに,関連する研究者の講演や討論を踏まえて,この研究を進めていく予定である

  • Explicit formulas of $p$-adic spherical functions and their applications

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research

    Project Year :

    2012.04
    -
    2016.03
     

    Hironaka Yumiko, SATO FUMIHIRO, KOMORI YASUSHI, Rubenthaler Hubert, Boecherer Siegfried

     View Summary

    We have investigated the spaces of unitary-hermitian matrices on the basis of spherical functions as $p$-adic homogeneous spaces. We may apply a general expression formula of spherical functions which the researcher got before. The present groups have different root systems according to the parity of the size of matrices, and the Cartan decomposition of the spaces have different shapes according to the residual characteristic of the base field. We have studied at first the odd residual and even-size space, then the other cases. Finally we have a unified description for the results

  • Arithmetic study of automorphic forms of many variables by various method

    Project Year :

    2011.05
    -
    2015.03
     

     View Summary

    We obtained some fundamental results on the integral expressions and power series expressions of the A-radial parts of either Whittaker functions or spherical functions for the standard representations (i.e principal series and/or discrete series representations) of the Lie groups, GL(n,R), Sp(2,R) and SU(3,1).The formulas of Whittaker functions of non-spherical principal series put a period on the research history beginning from the studies of D. Bump and others, and we can expect various applications of this result (this is a joint works with Taku Ishii of Seikei Univ.). We obtained an explicit formulas of the matrix coefficients of the large discrete series of the Lie groups SU(2,1), SU(3,1) (joint wrok with T.Hayata, H. Koseki, and T. Miyazaki).This result gives a suggestion for study of the reproducing kernels. We push forward the investigation oh the cell-decomposition of Siegel-Gottschling fundamental domain of genus 2 (the first paper was a joint paper with T. Hayata)

  • p進球関数の明示的公式とその応用

    科学研究費助成事業(早稲田大学)  科学研究費助成事業(基盤研究(C))

    Project Year :

    2012
    -
    2015
     

     View Summary

    2013年度は主に,$p$ 進体上の不分岐ユニタリ・エルミート行列の空間$X$について研究した,但し 剰余体標数は奇数とする.逆対角線上に$1$ が $2n+1$ 個ならぶ行列を固定するユニタリ群 $G$ をとり,これを $k$ 上定義されたユニタリ群の$k$-有理点集合とみなしておく.$G$ 内のエルミート行列で,代数的閉体上で単位行列を含む軌道の有理点集合を $X$とする.$K$ を $G$ の整点からなる極大コンパクト部分群とする.$G$ が作用する空間 $X$ をヘッケ環 $H(G,K)$ の作用を通して解析したい.$X$上の球関数とは,$X$ から複素数体へのヘッケ環同時固有関数を意味し,これを具体的に求めることは基本的かつ重要な問題である.
    まず,この空間$X$ の $K$-軌道分解(カルタン分解)を与え,$X$上の典型的な球関数 $\omega(x;s)$ を定義し,その関数等式や極・零点の位置を調べた,ここで$x \in X$ で,行列のサイズ $2n+1$ に対し,$s \in \C^n$ ($n$ 次の複素変数)である.群 $G$ は$BC_n$ 型であるが,$\omega(x;s)$ の明示式は,$C_n$型のルート系に付随するマクドナルド多項式の特殊化を主要項として持つ.これを核関数に用いて $X$ 上の球フーリエ変換を定し,$X$ 上のシュバルツ関数の空間 $S(K\backslash X)$ を解析した.球フーリエ変換はヘッケ環 $H(G,K)$ の作用と可換であり,$S(K\backslash X)$ は,階数 $2^n$ の自由ヘッケ環加群であることが分かる.また,これから,$X$上のすべての球関数のなす空間が $n$次元であることが分かり,先の典型的な球関数を用いて,全体の基底を構成した.

  • Explicit formulas of $p$-adic spherical functions and their applications

    Project Year :

    2012
    -
    2015
     

     View Summary

    We have investigated the spaces of unitary-hermitian matrices on the basis of spherical functions as $p$-adic homogeneous spaces. We may apply a general expression formula of spherical functions which the researcher got before. The present groups have different root systems according to the parity of the size of matrices, and the Cartan decomposition of the spaces have different shapes according to the residual characteristic of the base field. We have studied at first the odd residual and even-size space, then the other cases. Finally we have a unified description for the results

  • Spherical functions on p-adic homogeneous spaces and those applications

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research

    Project Year :

    2008
    -
    2011
     

    HIRONAKA Yumiko, SATO Fumihiro, KAMANO Ken, OKAMOTO Akihiko

     View Summary

    We intend to investigated certain$ p$-adic homogeneous spaces together with spherical functions and have number theoretic applications. We have approached this purpose in the following way : 1) to formulate expression formulas of spherical functions which can be widely applied ; 2) to formulate explicit formulas of spherical functions for concrete examples and analyze the spaces ; 3) to obtain number theoretic interesting quantities for the spaces of symmetric forms and hermitian forms.The most important result is to complete the theory of spherical functions on the space of unramified unitary hermitian matrices

  • Analysis, geometry and arithmetic of automorphic forms of many variables and higher dimensional modular varieties

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research

    Project Year :

    2007
    -
    2010
     

    ODA Takayuki, ISHII Taku, ICHIKAWA Takashi, IBUKIYAMA Tomoyoshi, ARIYAMA Kazutoshi, KOSEKI Harutaka, SATO Fumihiro, SUGANO Takashi, TSUZUKI Masao, HAYATA Takahiro, HAMAHATA Yoshinori, HIRANO Miki, HIRONAKA Yumiko, MURASE Atsushi, WATANABE Takao

     View Summary

    About matrix coefficients of semi-simple Lie groups, we obtained a more precise result for the middle discrete series of SU(2, 2) about the asymptotic expansion. We investigated the explicit formula of the matrix coefficients of SU(3, 1)(both are joint works together with T. Hayata and H. Koseki). Utilizing the asymptotic expansion, we obtained the explicit formula of the c-functions of certain P_J-principal series representations of Sp(2, R) explicitly(joint work with M. Iida). We push forward the investigation of 0-cells of the fundamental domain of the Siegel modular group of genus 2

  • Constructive geometry of arithmetic quotients of symmetric spaces

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research

    Project Year :

    2002
    -
    2005
     

    ODA Takayuki, TSUZUKI Masao, KARIYAMA Kazutoshi, HIRONAKA Yumiko, IBUKIYAMA Tomoyshi

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    In the period of the project, we got the following results.1.Explicit formula for Whittaker functions of SL(3,R) belonging to non-spherical principal series. (Published in 2004).2.Given a pair of arithmetic quotients (X,Y), where X,Y are either type IV quotients, or quotients of the complex hyperball, and when Y is a divisor of X, we constructed the Green current of Y as an automorphic form. (Published in 2003).3.We proved the confluence from Siegel-Whittaker functions to Whittaker functions for P_J principal series representations of Sp(2,R). (In press).4.Together with Miki Hirano of Ehime University, we obtained the integral and power series expressions of the principal series Whittaker functions on GL(3,C) with minimal K-type. The project is still in progress.5.The project to get explicit formula for P_J principal series representations of Sp(3,R) : The contents are finished. We are preparing a joint paper together with Miki Hirano and Taku Ishii of Chiba Inst. of Technology.6.We maintained the monthly seminar on Automorphic Forms at Komaba

  • Harmonic analysis on weakly spherical homogeneous spaces and its application to number theory.

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research

    Project Year :

    2001
    -
    2003
     

    HIRONAKA Yumiko, SATO Fumihiro

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    We investigate Sp_2 as a spherical homogeneous Sp_2×(Sp_1)^2-space intensively. First we give explicit formulas of spherical functions explicitly as an application of the previous general formula which was given by the author. Then we determine the Hecke module structure of the space of Schwartz-Bruhat functions on Sp_2 and parametrization of all spherical functions, where the space of spherical functions attached to the same parameter has dimension 4.Extending the calculation of functional equations of spherical functions on Sp_2, we give a general method to obtain functional equations of spherical functions on certain spherical homogeneous p-adic spaces. This is based on the uniqueness of the relatively invariant distributions on a homogeneous space, and it guarantees the existance of functional equations attached to elements of Weyl group.As a joint research with F.Sato, we give an integral representation of the p-oart of the Siegel series, which is the main part of the Fourier coefficients of Siegel Eisenstein series. By using this representation, Siegel series is expresses as a finite sum of spherical functions on SO(n, n)/S(O(n)×O(n)) with respect to Siegel parabolic subgroup, which gives us a new proof of functional equations of Siegel series. The explicit calculation is now in progress

  • A study on relations between the theory of prehomogeneous vector spaces, the theory of group representations and the theory of automorphic forms

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research

    Project Year :

    2000
    -
    2003
     

    SATO Fumihiro, HIRONAKA Yumiko, YAMADA Yuji, AARAKAWA Tsuneo, IBUKIYAMA Tomoyoshi, GYOJA Akihoko

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    In this research project, we investigated zeta functions of prehomogeneous vector spaces from the view points(1)relations between functional equations and representations of general linear groups,(2)relations to automorphic L-functions,(3)generalization of the theory to non-regular prehomogeneous vector spaces.(1)We showed that the functional equations of zeta functions are closely related to intertwining operators between degenerate principal series representations of general linear groups, and, using the relation, we obtained an integral expression of Eulerian type of the gamma matrices of functional equations. This enables us to identify the variable change in functional equations as an action of an element in the Weyl group of a general linear group, and to decompose functional equations into a product of more elementary functional equations. There exists a similar results for p-adic local zeta functions. As an application of p-adic theory, we investigated the Fourier coefficients of Elsenstein series of Sp(n) and GL(n) and the theory of spherical transforms on certain spherical homogeneous spaces.(2)We identified the Koecher-Maass series of real analytic Siegel Eisenstein series with a zeta function associated with a certain prehomogeneous vector space on which the Siegel parabolic subgroup of SO(n, n) acts. It is quite probable that this result can be extended to other classical groups. A considerable progress has been made in explicit calculation of zeta functions. We obtained an explicit expression of zeta functions in terms of the Riemann zeta function and the Mellin transforms of the Cohen Eisenstein series for more than 70 percent of irreducible regular reduced prehomogeneous vector spaces.(3)For non-regular prehomogeneous vector spaces, we developed a general theory of integral representations and the functional equation of the zeta integrals, which is a formal generalization of the theory for regular prehomogeneous vector spaces. We also gave the first example of explicit functional equations for non-regular spaces

  • Theory of automorphic forms and arithmetic into functions

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research

    Project Year :

    1998
    -
    2000
     

    IBUKIYAMA Tomoyashi, ARAKAWA Tsumeo, KATSUNADA Hidenori, SATO Fumihara, ODA Takayuki, SATO Hiroshi

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    In these three years, the principal researcher of this subject organized a conference at Oberwolfach in Germany, three autumn workshops and two mini conferences in Japan, hence 6 conferences in total, and studied effectively with many foreign researchers and cooperative researchers in Japan. In particular in three autumn workshops, we took three themes, Koecher Maass series, Eisenstein series, dimension formulae of automorphic forms, as initially planned. There we studied the foundation and development of arithmetic zeta functions, and obtained satisfactory results, and besides we published three volumes of proceedings of 640 total pages which can be regarded as fundamental references of the research of this direction. More concretely, the principal researcher defined Koecher-Maass series for automorphic forms of any tube domains, proved their analytic continuation and gave functional equations. Explicit forms of Koecher-Maass series for modular forms such as Eisenstein series, or those closely related to liftings were also obtained (jointly with Katsurada) and established the meaning of Koecher-Maass series as "arithmetic" zeta functions. Also, he obtained a lifting conjecture on Siegel modular forms of half integral weights (joint with Hayashida), developed theories on modular forms of rational weights and modular varieties, and applied differential operators to the study of structures of vector valued Siegel modular forms. As you can see from the list of their papers, the other researchers of our project also actively studied on spherical functions and zeta functions of prehomogeneous vector spaces, their explicit forms and convergence, Jacobi forms, liftings, explicit density formulae, adele geometry, real analysis on automorphic forms, Siegel modular forms mod p, fundamental lemma, the Fourier expansion of modular forms on bounded symmetric domains of non-tube type and other various researches. These are all related to the development of our research project and we think our project was very successful. Now we think we are in the stage to go from fundamental research to various applications, and expecting and planning a project of next stage such as explicit theories of graded rings of automorphic forms and vertex operator algegras

  • Zeta functions on weakly spherical homogeneous spaces

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research

    Project Year :

    1997
    -
    1999
     

    SATOU Fumihiro, HIRONAKA Yumiko, ARAKAWA Tsuneo

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    1.Sato, the head investigator, studied the weakly spherical homogeneous spaces (WSHS for short) obtained from the prehomogeneous vector spaces (PV for short) (SpinィイD210ィエD2 × GLィイD23ィエD2, half-spin 【cross product】 ΛィイD21ィエD2), (SLィイD25ィエD2 × GLィイD23ィエD2, ΛィイD22ィエD2 【cross product】 ΛィイD21ィエD2) and proved the functional equations satisfied by Eisenstein series attached to the spaces, which is a joint work with T. Kimura of Tsukuba Univ. and his students. As an application we can calculate explicitly the Fourier transforms of the complex powers of relative invariants of the PV's above. This shows that the theory of WSHS is very fruitful, since the structure of these two PV's is very complicated and even the method of micro local calculus can not be applied to them. We also made several attempts to generalize the theory to WSHS of reductive groups other than the general linear group. The result is still unsatisfactory ; however several suggestive partial results have been obtained.2.Hironaka succeeded in calculating the explicit formula for spherical functions of spherical homogeneous spaces over p-adic fields in a rather general setting. Using the formula, she constructed the theory of spherical functions of the space of hermitian forms over unramified quadratic extensions of the base p-adic field and gave an explicit formula for local densities of hermitian forms. Moreover, jointly with Sato, she calculated local densities of quadratic forms over nondyadic p-adic fields explicitly in the most general setting ; the results can be extended to the space of hermitian forms over ramified quadratic extensions.3.Arakawa studied mainly Koecher-Maass zeta functions attached to Siegel modular forms and obtained some results including the following : (a) a generalization of Koecher-Maass zeta functions to Jacobi forms, (b) an explicit expression of the Koecher-Maass zeta function attached to the Nagaoka Eisenstein series of weight 1 of degree 2, which is obtained from p-adic Siegel Eisenstein series

  • Spherical functions on p-adic homogeneous spaces.

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research

    Project Year :

    1997
    -
    1999
     

    HIRONAKA Yumiko, SATO Fumihiro

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    Let G be an algebraic group defined over a p-adic field k, X a homogeneous space of G and X = X (k).Let H(G,K) be the Hecke algebra of G = G(k) with respect to a maximal compact subgroup K of G.An H(G,K)-common eigenfunction on X is called a spherical function, which is an intereted object both in Number Theory and Representation Theory.Under certain condition on orbits by a parabolic subgroup in X, we have given a method to obtain explicit formulas of spherical functions by means of functional equations and spherical functions of groups. We can apply it to the spaces of symmetric forms and hermitian forms, which are interesting objects in Number theory. In these spaces, spherical functions can be viewed as generating functions of local densities of representation, so they are closely related. It is a classical problem in Number Theory to obtain explicit formulas of local densities of representations for symmetric forms and hermitian forms.For the space X of unramified hermitian forms, we have determined an explicit formula for spherical functions by the above method and the structure of the space of Schwartz-Bruhat functions S(K\X) as Hecke module. We have also obtained two kinds of explicit formulas of local densities by using spherical functions on X.By a joint research with F. Sato, we have obtained a complete explicit formula of local densities of symmetric forms for p ≠ 2, in general sizes. The method is to classify symmetric forms by the action of Iwahori subgroup and calculate certain Gaussian sums.Extending the above method to the the case of hermitina forms, we have obtained an explicit formula of local densities of hermitian forms for p ≠ 2, in general sizes

  • Reserch on loop spaces related to mathematical physics

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research

    Project Year :

    1991
    -
    1993
     

    ASADA Akira, ABE Kojun, SAITO Shiroshi, YOKOTA Ichiro, HIRONAKA Yumiko

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    (1) Research on loop group bundlesi. Characteristic map of a loop group bundle is realized as a matrix valued function.ii. Lifting and descent of a vector bundle or a loop group bundle over M to a loop group bundle or a vector bundle over OMEGAM or MXS^1 are defined.(2) Research on characteristic classes of loop group bundles (string classes)i. Differential geometric definition of string classes is given.ii. Relations between string classes and Chern classes of a bundle and its lifting or descent are computed.iii. String classes are expressed as WZNW terms of characteristic maps.(3) By using above results and non-abelian de Rham theory, relations between Chern-Simons gauge theory and topological field theory are studied.(4) To extend above results for loop groups over exceptiohal groups, concrete realizations of exceptional groups are done.(5) To get more advanced information in this direction, Eisenstein serieses of nurmber theoretical symmetric spaces are studied.(6) Research on current group bundles , I.Noncommutative connections. The notion of noncommutative connection is introduced and several results such as reduction to U_1-bundles and noncommutative Poincare lemma, are get.(7) Research on current group bundles, II.Connections with respect to the Dirac operator. Connections with respect to the Dirac operator is defined. They give families of Dirac operator and their n-functions give bundle invariants

  • 代数多様体の研究

    科学研究費助成事業(信州大学)  科学研究費助成事業(一般研究(C))

    Project Year :

    1988
     
     
     

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    本研究は多様体、特に代数多様体を対象の中心にすえて、その構造、それに働く代数的作用、解析的作用等を数学の各分野、特に代数的、解析的、位相幾何学的分野から、総合的に究明する点にあった。より具体的には代数多様体をめぐる種々の研究、位相的不変量の計算とそのG-構造、解析的方程式の基本解の構成、スペクトル分解とその作用素で生成される半群の研究、基本解の特異領域の幾何学的構造等がそれである。これらの点に関しては、それぞれに一定の成果があげられている。高次微分の作用に関する不変環の理論、エルミートあるいは対称空間の球関数、球変換の理論、例外型リー群の対合自己同型と不変群の実現、非アーベル的ドラーム理論の構成等顕著な成果がみられる(研究発表の項を参照されたい)。また本研究を発展させる上で重要なポイントの一つとなる他大学における研究者との交流も科研費を利用して計画通り行われ、意見・情報の交換、ディスカッションを行なうことが出来た。また、本大学での、解析学、代数学、幾何学の各分野で、他大学の研究者を交えてのセミナー、シンポジウム等がもたれたことも特筆すべきことである。

  • 保型形式に関連するL-関数の研究

    科学研究費助成事業(立教大学)  科学研究費助成事業(一般研究(C))

    Project Year :

    1987
     
     
     

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    1.実2次体の量指標のL-関数のある特殊値のLambert型Dirichlet級数のある極における留数を用いた興味深いexpressionを得た. また二重ガンマ関数と上記Lambert型Dirichlet級数との間の関数等式を得た(荒川).
    2.2元2次形式のゼータ関数を核関数として球フーリエ変換の有理数体上の類似物を構成し, 有理2元2次形式の空間上のシャワルツ関数の空間のGL(2)のヘッケ環加群としての構造を決定した. H.Maassによって考えられたKoecherのゼータ函数の球関数係数への拡張を概均質ベクトル空間の枠組で一般化した(佐藤)
    3.局所体上のエルミート形式および対称形式の空間Xに球関数の類似を定義し, これらを核関数として, X上のSchwartz-Bruhat空間上の球-Fourier変換を与えた. 球関数の関数等式を研究することにより, 球Fourier変換像の情報を提供した. 特に, サイズ2の形式については, 具体的に球関数のparametrization,Fourier逆変換や, Hecke環加群としてのS-B空間の構造の決定等を行った(小林).
    4.P進体上の交代行列の空間の球函数論, 概均質ベクトル空間の理論の応用として, 交代行列の表現の局所密度の公式を得た. (佐藤, 小林)
    5.Riemann Zeta関数の零点の分布についてのいくつかの興味ある結果を導いた. 零点の虚数部分の一様分布性に関する結果を用いて, L-関数の平均値定理を得た. また, 零点の分布についてのGramの法則に関する定理を得, 予想を作った(藤井).

  • ホップ代数の構造とその周辺の研究

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    交付された補助金は、研究代表者及び研究分担者の各地で開催されるシンポジウム、研究集会、及び各地の大学の研究者との研究打合せ資料収集等に充当する予定であったが、本年度は代数学、環論、及びトポロジ-関係のシンポジウムが夏期に札幌、福島の遠隔地で開催されたため、それらに出席するための旅費に大部分が使われるという結果になってしまい,各地の大学の研究者との研究打合せのための旅費が、予定に比べて大幅に縮小し、資料収集のみを目的とした旅費に充当することが不可能になってしまった。具体的には、代数学シンポジウムが7月に北海道大学学術交流会館で、トポロジ-シンポジウムが7月に福島大学で、環論シンポジウムが8月に北海道大学百年記念会館で開催され、トポロジ-シンポジウムには、研究分担者の向井、阿部が出席し、代数学シンポジウムと環論シンポジウムには研究代表者の西川、研究分担者の二宮、岸本、小林が出席し、これら全員の旅費を賄うことが出来た。また、環論シンポジウムには二宮が「Blocks of p-solvable groups with 2 or 3 simple modules」なる題目で講演した。これらのシンポジウムに手分けして出席したことにより,研究代表者及び分担者それぞれが、Hopf対数の構造論との関連をもつ種々の分野における第一線の研究の状況、具体的には有限群のmodular表現論、環の拡大理論、射影空間のHomotopy論、可微分多様体の構造理論、Hecke代数と対称空間の表現論等の分野の研究の現状についての理解が進み、代表者、分担者それぞれの研究の準備の進展及び論文作成等の効果を得ることが出来た

  • Hecke環の表現と対称空間

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    標数Oの局所体k上のn次非退化対称行列のなす空間Xには、G=GLu(k)が自然に作用している。X上の、K=GLu(Ok)不変でコンパクトな台をもつ関数の空間をS(K\X)とすると,これには、Hecke環H(G,K)が作用している。kが奇素数pについてpーfieldである場合に、H(G,K)の表現空間としてS(K|X)の構造を調べることを以前した。今回は2ーfieldの場合を考察した。以上の対称空間の構造を調べる上でも、局所理論は是非しておくべきことである。X上の帯球関数を定義し、それを用いてS(K|X)上に球Fowier変換Fを定義し、これを通してS(K\X)のH(G,K)ー加群としての構造を調べる。size nの帯球関数は、size n未満の帯球関数と,行列の表現の局所密度で表わすことができ、帯球関数の研究と局所密度の研究は密接に結びついている。より具体的には、i)Fの単射性,ii)Fの像,iii)Fの逆変換,iv)S(K\X)上のPlruckerel測度,v)H(G,K)ー同時固有関数,の決定を問題にする。size zの時は完全に決定された。p≠2との著しい違いは、S(K\X)のH(G_1K)ー自由加群としての階数で64(p≠2のときは16),一般にも、P=2なら8^n,P≠2なら4^nと思われる。一般のsizeについても、単射性と、関数等式については決定した。以上と同様の考察を、2ーfield上のHevmete行列のなす空間についても考察し、まとめた。補助金により、他大学の研究者と交流できたのは、大変有意義であった。又、数論関係の書籍も購入できた

  • 群多元環とその根基の構造に関する研究

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    有限群の正標数の体上の群多元環の構造についての研究の観点から、その根基の巾零指数と群の構造との関連について考察することを当面の目標とした。体Kの標数をpとし、群Gのシローp部分群の位数をP^aとする。群多元環KGの根基の巾零指数をt(G)で表すことにする。Gがp可解群のとき、t(G)はp^a以下であることが知られている。さらに、p^<a-1>≦t(G)≦p^aをみたす群Gの構造は完全に決定されている。そこで、不等式p^<a-2>≦t(G)<p^<a-1>をみたす群Gを決定するべく研究を行った。Gがp群の場合、この研究を遂行するためには、指数p^<a-2>の群をすべて決定する必要があるが、これについては、Millerによる分類があり、そのような群の同型類の個数が得られている。このMillerの論法を精密化することにより、指数p^<a-2>のp群すべてについて、それらの生成元と基本関係による表示を与えることができた。この結果を用いることにより、Gがp群の場合には、不等式p^<a-2>≦t(G)<p^<a-1>をみたす群を完全に決定することができた。さらに、この問題をp可解群の場合に解決するべく研究中であるが、いくつかの例外的な群をのぞいて、ほぼ解決の見通しが立つ状況になった。さらに、この研究の過程において、永年、研究者の間で予想されていた不等式t(G)≦t(P)(Gはp可解群で、PはGのシローp部分群)に対する反例を見つけることができた。このことは、この方面の研究者の注目を集めると思われる。この反例が特異なものなのかどうかについて、さらに上記不等式t(G)≦t(P)をみたす群の特徴付けについて、研究を深める必要があると考えている

  • 対称空間上の球関数

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    対称空間のある種の作用素の同時固有関数族として得られる関数は球関数と呼ばれ、対称空間の表現論で重要な役割をになうものである。実または複素数での理論やp-進reductive群の球関数の理論はすでに研究されている。p-進体上の等質空間の球関数の理論を構成し、さらには、代数体上定義された対称空間、等質空間の調和解析の理論を構成することは、興味深い問題である。以下は標数0の非アルキメデス的局所体上定義された等質空間を考え、ヘッケ環の同時固有関数を球関数と呼ぶ。まず、球等質空間上の球関数についての表現論的な考察をし、一定の仮定の下で、球関数の公式を与えた。これを不分岐エルミート形式のなす空間について適用し、球関数の理論を構成した。具体的には、標準的な球関数をうまく定義し、その良い具体的表示式を与えること;球関数を核関数とするフーリエ変換の像、および逆変換の決定この空間上の急減少関数のなす空間のヘッケ環加群としての構造の決定すべての球関数をうまくパラメトライズすることまた、球関数は局所密度μ_p(B,A)達の母関数とみなすことができる。したがって、球関数の明示式を用いて、局所密度μ_pを引き出すことができる。実際、局所密度および原始的局所密度の、組合わせ論的量を用いた明示式を得た

  • Arithmetic of automorphic forms of many variable : reconstruction of the foundation

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    As planned in our application, at Tokyo and at Kobe we had a series of monthly seminars. Since we have already reported about them in the annual reports, here we do not write about them. We also omit the details about the workshops at RIMS, Kyoto University and the summer schools. The proceedings on these meetings are available. The head investigator Oda in a joint work with Takahiro HAYATA (Yamagata Univ.) and Harutaka KOSEKI (Mie University) obtained an explicit formula for matrix coefficients of the middle discrete series of SU(2,2). Added to this he also obtained explicit formulae of matrix coefficients for the large discrete series of SU(2,2) and Sp(2,R). Moreover in a joint paper with Masao TSUZUKI (Sophia Univ.) he constructed Green functions of modular divisors on arithmetic quotients of certain bounded symmetric domains, as automorphic forms. We also explain the outline of the fruits of the research by the investigators joined to this plan. This result has not only have applications for the dimension of spaces of automorphic forms, but also theoretically significance. Ibukiyama and Saito had remarkable results on the evaluation of the special values of the zeta functions of prehomogeneous vector spaces. This result has not only application to the explicit dimension formulae of the spaces of automorphic forms, but also theoretically very important meaning. The investigation of p-adic spherical functions by Sato and Hironaka made much progress. The new point among others is that they can handle also the case of "ramified" spherical functions. The joint work of Murase and Sugano also advanced. Their work of the primitive theta function of SU(2,1) is one interesting result. But also the is a progress in the theory of automorphic L-functions on orthogonal groups A joint work with Shin-ichi Kato (Kyoto Univ.) on p-adic spherical functions is one of fruits. Watanabe together with Masanori Morishita (Kanazawa Univ.) pushed forward the investigation of Hermite constants for algebraic groups, Which is so to speak a non-abelian version of "Geometry of Numbers". The result of Katsurada is also interesting by regarding it as an investigation of ramified p-adic spherical functions

  • Explicit study on automorphic structure in algebra and zeta functions.

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    In these four years, the principal researcher of this subject organized a conference at Oberwol-fach, three Spring Confecences, two Autumn Workshops (one of which is Japan-Germany Seminar supported by JSPS), Mini-workshop on Number Theory, and Meeting on Modular Forms and Zeta Functions, hence 8 conferences in total, and studied effectively with many foreign researchers and cooperative researchers in Japan, In particular, in Spring Conference, we focused to communicate with wide areas arround modular forms, and took Graded rings of modular forms, Vertex operator algebras in first and second one, and more mixed areas in the third one. We published five Proceedings of 970 total pages. More concretely, the principal researcher proposed a conjecture on Shimura type correspondence between Siegel modular forms of integral and half integral weight, studied structures of scalar valued and vector valued Siegel modular forms and differential operators, Borcherds product expression, holonomic system coming from differential operators on modular forms, positivity of eta products, Koecher-Maass series of forms obtained by lifting, modular forms of rational weight. As it can be seen from the list of their papers, the other researchers of our project also actively studied on conformal field theory, spherical and Whittaker functions, prehomogenous vector spaces, modular forms on unitary groups, elliptic root systems, p-adic modular forms, L-packet, adele geometry, converse theorem, vertex operator algebras, combinatorial designs, weakly spherical spaces, inverse Galois problem and CAP forms

  • Relation between automorphic forms and zeta functions associated with prehomogeneous vector spaces

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    The main problems we investigated in this research project are(1) To identify the zeta functions associated with prehomogeneous vector spaces with some kind of zeta functions attached to automorphic forms(2) To construct a theory of(local) functional equations which is not covered by the theory of prenomogeneous vector spaces. The results we obtained are as follows :(1) According to the classification theory due to Sato and Kimura, irreducible regular prehomogeneous vector spaces are classified into 5 series of classical type and 24 spaces of sporadic type. We identified the zeta functions associated with 4 series of prehomogeneous vector spaces of classical type with the standard L-functions or Koecher-Maass zeta functions of certain real analytic Eisenstein series. One of the results which are necessary for the proof of these results is a new integral representation of the Siegel series (= p-part of the Fourier coefficients of Eisenstein series). As another application of the new integral representation, we proved a formula which connects the Siegel series to spherical functions on a p-adic semisimple symmetric space of the orthogonal groups.(2) We proved that, given a pair of homogeneous polynomials on Ra satisfying a local functional equation and a pair of nondegenerate dual quadratic mappings of R^m to R^n, then, the pull backs of the polynomials by the quadratic mappings also satisfy a local functional equation. This generalizes a result due to Faraut-Koranyi-Clerc and we can construct many examples of functional equations which are not covered by the theory of prehomogeneous vector spaces. We also classified nondegenerate dual quadratic mappings over quadratic spaces and proved that such quadratic mappings are in one to one correspondence to representations of a tensor product of 2 Clifford algebras

  • Researches on zeta functions of prehomogeneous vector spaces

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    Theory of prehomogeneous vector spaces gives a systematic method of constructing zeta functions from polynomial invariants of prehomogeneous group actions. We investigated zeta functions of prehomogeneous vector spaces from the following 3 view points : (1) relations to Eisenstein-periods, (2) relations to the Koecher-Maass zeta functions of automorphic forms, (3) zeta functions of invariants of non-prehomogeneous group-actions. The most important result is the construction of zeta functions of non-degenerate quadratic mappings, which include zeta functions of certain non-prehomogeneous polynomials of degree 4

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Overseas Activities

  • p進弱球等質空間の球関数の研究

    2005.03
    -
    2006.03

    ドイツ   マンハイム大学数学研究所

    フランス   ストラスブール大学高等数学研究所

Internal Special Research Projects

  • 弱球等質空間上のp進球関数とその数論的応用

    2007  

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    極小放物型部分群が開軌道をもつような$p$進体上定義されている等質空間を考え,この上の球関数を研究する.それらは,典型例としては,この放物型部分群に関する相対不変式を極大コンパクト開部分群について平均した関数として得られる.独立な相対不変式の個数が放物型部分群の階数より小さくとも,もとの群からの情報と球関数の関数等式を組み合わせて,球関数を定式化することができる.さらに,かなり多くの例を含むような類の等質空間について,その関数等式が,特殊な形の小さな次元の概均質ベクトル空間の局所ゼータ関数の関数等式に帰着することが分かる.これらの結果の一部を,$p$進群上の球関数ワークショップ(玉原国際セミナーハウス, 2007.7.29 -- 8.3),Modulformen(ドイツ Oberwolfach高等数学研究所,2007.10.28 -- 11.3), ゼータ関数と$L$関数(日仏冬の学校,三浦海岸,2008.1.8 --11) で講演した.出張旅費をこの研究経費でまかなえたことにより,研究発表ができ,また,研究集会に出席して議論することにより新たな知見を得ることもできて感謝している.関連する書籍の購入もできて,研究推進に役立った.

  • p進等質空間の球関数とその応用

    2006  

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    Mannheim大学の Siegfried Boecherer 氏と立教大学の佐藤文広氏とは,引き続き密接な連絡をとりつつ共同研究を行い,以下のような結果をまとめることができた.対称形式の局所密度は,その生成関数が典型的な球関数であるが,これを表現される対称形式についての関数とみて,一次独立になるような,元になる対称形式のよい系列を与えた.対称形式からJacobi形式を構成することにより,大域的なSiegel保型形式の理論に応用される.Siegel保型形式の空間の中で,Jacobi形式によって張られる部分空間を考える.それは,考えている空間のlevelが平方因子をもたなければ,全空間に一致し,平方因子をもつときは,別の言葉で特徴付けされる部分空間に含まれることが分かる.これらの結果の一部は,京都大学数理解析研究所での保型形式研究集会(2006.1.15 -- 19)や,浜松で開かれた保型形式周辺分野スプリングコンフェランス(2006.2.5 -- 9)で紹介された.出張旅費をこの研究経費でまかなえたことにより,研究連絡や発表ができ,また,研究集会に出席し,討論することにより新たな知見を得ることもできて感謝している.関連する書籍の購入もできて,研究推進に役立った.

  • p進等質空間の球関数と局所密度

    2001  

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    以前,指標和(ガウス和)を用いて合同部分群の作用に関する対称形式の代表系に関する和として局所密度を表示する方法を立教大学の佐藤文広氏との共同研究で開発している.対称形式の場合に適用し,合同部分群として,岩堀部分群をとることにより,局所密度の明示式を剰余体標数が奇数の場合に完全に求めた.今回は,この方法をエルミート形式にも拡張して,やはり,剰余体標数が奇数の場合に,一般に求めた.分岐拡大から得られるエルミート形式に関しては,初めての明示式である.不分岐拡大の場合には,3通りの明示式が得られたことになる.二次形式の局所密度に付随する Kitaoka 級数(これは有理関数となる)の分母について,以前の結果を見直し改良した.表現する行列のサイズが偶数のときは,unimodular行列のときと同様に,以前に比べてほぼ半分の分母となることが分かった.二次形式の居所密度に関しては,以前、立教大学の佐藤文広氏との共同研究において,ある明示式を与えているが,それから局所密度相互の関係を読み取ることは困難である.この Kitaoka 級数に関する結果は,局所密度の明示式が改良されるべきものであることも示唆している.また,対称空間ではない球等質空間として,Sp_2 x (Sp_1)^2 が作用する空間 Sp_2 を取り上げて研究した.この空間上の球関数論を構成する端緒として,この空間のカルタン分解を考察し,球関数の明示式を与えた.

  • p進等質空間の球関数

    1998  

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    p-進体k上定義された代数群Gの等質空間Xのk-有理点の全体をXとし、Gのk-有理点全体Gとその極大コンパクト部分群Kに関するヘッケ環H(G,K)を考える。ヘッケ環に関する同時固有関数となるX上の関数は、X上の球関数と呼ばれ、整数論的にも表現論的にも興味深い研究対象である。 Gの放物部分群に関する相対不変式となるX上の正則関数からX上の球関数の典型例が構成できる。さらに、一定の仮定の下に、Xの球関数の明示式を、その関数等式と群上の球関数の明示式の双方を組み合わせる事で与える事ができている。それは、たとえば整数論的に興味深い対象である対称形式やエルミート形式の空間に適用できる。これらの空間では、球関数と、表現の局所密度とは密接な関係を持ち、後者を求めることは整数論の古典的な問題である。 不分岐エルミート行列の空間では、球関数の明示式を与え、K-不変でコンパクトな台のX上の関数のなす空間S(K\X)のヘッケ環加群としての構造や、すべての球関数のパラメトライズもなされた。また、局所密度についても明示式が得られた。 対称形式の場合も次数が少ない場合には球関数の明示式を得ることができるが、一般には困難である。分岐エルミート形式の場合も球関数、局所密度とも残された部分は多い。 一方、局所密度については、指標和(ガウス和)を用いた別のアプローチも可能である。佐藤文広氏との共同研究により、こちらの方法で対称形式の局所密度の明示式を得ることができた。これを球関数の理論に逆に応用できないかどうかは今後の課題である。