Spherical functions on p-adic homogeneous spaces and those applications

• Project/Area Number: 20540029
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Waseda University
• Principal Investigator: HIRONAKA Yumiko 早稲田大学, 教育・総合科学学術院, 教授 (10153652)
• Co-Investigator(Renkei-kenkyūsha): SATO Fumihiro 立教大学, 理学部, 教授 (20120884) ; KAMANO Ken 大阪工業大学, 工学部, 特任講師 (50409611) ; OKAMOTO Akihiko 早稲田大学, 教育総合科学学術院, 助手 (50608825)
• Project Period (FY): 2008 – 2011
• Project Status: Completed (Fiscal Year 2011)
• Budget Amount: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000); Fiscal Year 2011: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2010: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2009: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2008: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
• Keywords: 数論 / $p$進球関数 / 球等質空間 / 概均質ベクトル空間 / 対称空間 / ヘッケ環 / カルタン分解 / Macdonald対称多項式 / Schwartz空間 / 局所密度 / Siegel特異級数 / Eisenstein級数

Research Abstract

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.

Research Products

• [Journal Article] Spherical functions on $ U(2n)/U(n)\times U(n)$ and hermitian Siegel series (2011)
  • Author(s): Y. Hironaka
  • Journal Title: Series on Number Theory and Its Applications Volume: vol.7 Pages: 120-159
  • Peer Reviewed
• [Journal Article] Construction fo non-prehomogeneous polynomials with local functional equations from representations of Clifford algebras (2011)
  • Author(s): F Sato and T. Kogiso
  • Journal Title: Tambov University Report Volume: vol.16 Pages: 1744-1750
• [Journal Article] On a characterization of a q-analogue of the gamma function (2011)
  • Author(s): K. Kamano
  • Journal Title: International Journal of Applied Mathematics & Statistics Volume: vol.21 Pages: 110-117
  • Peer Reviewed
• [Journal Article] On expressions of theta series by eta-products (2011)
  • Author(s): A. Okamoto
  • Journal Title: Tokyo Journal of Mathematics Volume: vol.34 Pages: 319-326
  • Peer Reviewed
• [Journal Article] Spherical functions on $U(2n)/U(n) ￥times U(n)$ and hermitian Sisgel series (2011)
  • Author(s): Yumiko Hironaka
  • Journal Title: "Geometry and Analysis of Automorphic Forms of Several Variables", Series on Number Theory and Its Applications Volume: 7 Pages: 120-159
  • Peer Reviewed
• [Journal Article] $p$進ユニタリ・エルミート行列の空間のカルタン分解 (2011)
  • Author(s): 広中由美子
  • Journal Title: 「数論女性の集まり」報告集 Volume: 4 Pages: 39-47
• [Journal Article] Spherical functions on $U(n)/U(n) ￥times U(n)$ and hermitian Siegel series (2011)
  • Author(s): Yumiko Hironaka
  • Journal Title: Series on Number Theory and Its Applications (World Scientific) Volume: (to appear)
  • Peer Reviewed
• [Journal Article] Spherical functions on $ p$-adic homogeneous spaces (2010)
  • Author(s): Y. Hironaka
  • Journal Title: MSJ memoirs Volume: vol.21 Pages: 50-72
  • Peer Reviewed
• [Journal Article] Multiple $ p$-adic lo g f-gamma functions and their characterization theorem (2010)
  • Author(s): K. Kamano
  • Journal Title: Acta Arithmetica Volume: vol.145 Pages: 109-122
  • Peer Reviewed
• [Journal Article] Sums of products of hypergeometric Bernoulli numbers (2010)
  • Author(s): K. Kamano
  • Journal Title: Jounal of Number Theory Volume: vol.130 Pages: 2259-2271
  • Peer Reviewed
• [Journal Article] Spherical functions on $p$-adic homogeneous spaces (2010)
  • Author(s): Y.Hironaka
  • Journal Title: MSJ memoirs vol.21 Pages: 50-72
• [Journal Article] Multiple $p$-adic logf-gamma functions and their characterization theorem (2010)
  • Author(s): K.Kamano
  • Journal Title: Acta Arithmetica vol.145 Pages: 109-122
• [Journal Article] Sums of products of hypergeometric Bernoulli numbers (2010)
  • Author(s): K.Kamano
  • Journal Title: Jounal of Number Theory vol.130 Pages: 2259-2271
• [Journal Article] Spherical functions on $p$-adic homogeneous spaces (2010)
  • Author(s): Yumiko Hironaka
  • Journal Title: MSJ Memoirs Volume: 21 Pages: 50-72
  • Peer Reviewed
• [Journal Article] Spherical functions on $U(2n)/(U(n) ￥times U(n))$ and hermitian Siegel series (2010)
  • Author(s): Yumiko Hironaka
  • Journal Title: 京都大学数理解析研究所講究録 Volume: 1722 Pages: 126-145
• [Journal Article] $U(n,n)/U(n) ￥times U(n)$の球関数,その明示式や球フーリエ変換など (2010)
  • Author(s): 広中由美子
  • Journal Title: 第3回「数論女性の集まり」報告集 Volume: 3 Pages: 65-75
• [Journal Article] Linear independence of local densities of quadratic forms and its application to the theory of Siegel modular forms (2009)
  • Author(s): S. Boecherer, Y. Hironaka and F. Sato
  • Journal Title: Contemporary Math. AMS Volume: vol.493 Pages: 51-82
  • Peer Reviewed
• [Journal Article] Linear independence of local densities of quadratic forms and its application to the theory of Siegel modular forms (2009)
  • Author(s): S.Boecherer, Y.Hironaka, F.Sato
  • Journal Title: Contemporary Math.AMS vol.493 Pages: 51-82
• [Journal Article] Linear independence of local densities of quadratic forms and its application to the theory of Siegel modular forms (2009)
  • Author(s): S.Boecherer, Y.Hironaka, F.Sato
  • Journal Title: Contemporary Mathematics 493 Pages: 51-82
  • Peer Reviewed
• [Journal Article] Spherical functions on $p$-adic homogeneous spaces (2009)
  • Author(s): Yumiko Hironaka
  • Journal Title: MSJ Memoirs 50 Pages: 50-72
  • Peer Reviewed
• [Journal Article] Spherical functions on $U(n, n)\big{/}(U(n)\times U(n))$ and hermitian Siegel series (2009)
  • Author(s): Yumiko Hironaka
  • Journal Title: 京都大学数理解析研究所講究録 1659 Pages: 141-156
• [Journal Article] U(2n)/(U(n)xU(n))上の球関数とエルミートSiegel級数について (2009)
  • Author(s): Yumiko Hironaka
  • Journal Title: 「第2回 数諭女性の集まり」報告集 2 Pages: 50-57
• [Journal Article] Linear independence of local densities of quadratic forms and its application to the theory of Siegel modular forms (2009)
  • Author(s): S. Boecherer, Y. Hironaka, F. Sato
  • Journal Title: Contemporary Mathematics, American Mathematical Society 掲載確定(査読済み)
  • Peer Reviewed
• [Journal Article] Spherical functions on $ U(2n)\big{/}(U(n)\times U(n))$ and hermitian Siegel series
  • Author(s): Y. Hironaka
  • Journal Title: 京都大学数理解析研究所講究録 Volume: 1722巻 Pages: 126-145
• [Journal Article] Spherical functions on $U(2n)/U(n)\times U(n)$ and hermitian Siegel series
  • Author(s): Y.Hironaka
  • Journal Title: to appear in Geometry and Analysis of Automorphic Forms of Several Variables, World Scientific 40
• [Presentation] Spherical functions on the space of $ p$-adic unitary hermitian forms (2012)
  • Author(s): Y. Hironaka
  • Organizer: RIMS研究集会「保型形式と保型的$ L$函数の研究」
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2012-01-18
• [Presentation] ある$C_n$型$p$進等質空間上の球関数 (2011)
  • Author(s): 広中由美子
  • Organizer: 第4回「数論女性の集まり」
  • Place of Presentation: 早稲田大学(東京都)(招待講演)
  • Year and Date: 2011-05-28
• [Presentation] Spherical functions and Siegel singular series for$ p$-adic case,"Explicit theory of automorphic forms, applications and computations" (2011)
  • Author(s): Y. Hironaka
  • Organizer: Conference at Centre International de Rencontres Math\'{e} matiques
  • Place of Presentation: Luminy, France
  • Year and Date: 2011-05-13
• [Presentation] Spherical functions and Siegal singular series for $p$-adic case (2011)
  • Author(s): Yumiko Hironaka
  • Organizer: Explicit theory of automorphic forms, applications and computations
  • Place of Presentation: Luminy (France)(招待講演)
  • Year and Date: 2011-05-13
• [Presentation] Spherical functions on certain $ p$-adic homogeneous spaces (2011)
  • Author(s): Y. Hironaka
  • Organizer: Ueda Memorial Conference on Automorphic Forms
  • Place of Presentation: 奈良女子大学
  • Year and Date: 2011-01-26
• [Presentation] Spherical functions on certain $p$-adic homogeneous spaces, (2011)
  • Author(s): Y.Hironaka
  • Organizer: Ueda Memorial Conference on Automorphic Forms
  • Place of Presentation: 奈良女子大学
  • Year and Date: 2011-01-26
• [Presentation] Spherical functions on certain $p$-adic homogeneous spaces (2011)
  • Author(s): Yumiko Hironaka
  • Organizer: Ueda Memorial Conference on Automorphic Forms
  • Place of Presentation: 奈良女子大学(招待講演)
  • Year and Date: 2011-01-26
• [Presentation] Spherical functions on $U(2n)/U(n) ￥times U(n)$ (2010)
  • Author(s): Yumiko Hironaka
  • Organizer: Sendai Number Theory Conference
  • Place of Presentation: 東北大学数理科学記念館(招待講演)
  • Year and Date: 2010-10-09
• [Presentation] $U(H_n)/U(T) x U(T)$上の球関数について (2010)
  • Author(s): Y. Hironaka
  • Organizer: RIMS研究集会「等質空間と非可換調和解析」
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2010-06-16
• [Presentation] $U(H_n)/U(T) ￥times U(T)$上の球関数について (2010)
  • Author(s): Yumiko Hironaka
  • Organizer: RIMS研究集会「等質空間と非可換調和解析」
  • Place of Presentation: 京都大学数理科学研究所(招待講演)
  • Year and Date: 2010-06-16
• [Presentation] $U(H n)/U(T) ￥times U(T)$上の球関数,その明示式や球フーリエ変換など (2010)
  • Author(s): 広中由美子
  • Organizer: 第三回数論女性の集まり
  • Place of Presentation: 早稲田大学教育学部
  • Year and Date: 2010-05-22
• [Presentation] $p$進多重ログガンマ関数とその一意性定理 (2010)
  • Author(s): 鎌野健
  • Organizer: 第3回ゼータ値.ゼータ関数見にセミナー
  • Place of Presentation: 九州大学
  • Year and Date: 2010-03-05
• [Presentation] $p$進多重ログガンマ関数とその-意性定理 (2010)
  • Author(s): 鎌野健
  • Organizer: 第3回ゼータ値.ゼータ関数見にセミナー
  • Place of Presentation: 九州大学
  • Year and Date: 2010-03-05
• [Presentation] 概均質ベクトル空間の保型超関数とゼータ関数 (2010)
  • Author(s): 佐藤文広
  • Organizer: RIMS研究集会「保型形式・保型表現およびそれに伴うL関数と周期の研究」
  • Place of Presentation: 東京大学
  • Year and Date: 2010-01-19
• [Presentation] 概均質ベクトル空間の保型超関数とゼータ関数 (2010)
  • Author(s): Y.Hironaka
  • Organizer: RIMS研究集会「保型形式・保型表現およびそれに伴うL関数と周期の研究
  • Place of Presentation: 東京大学
  • Year and Date: 2010-01-19
• [Presentation] Spherical functions and(hermitian) Siegel series (2009)
  • Author(s): Y. Hironaka
  • Organizer: Arithmetic of Quadratic Forms of the KMS-AMS Joint Meeting
  • Place of Presentation: Ewha Womans University, Seoul, Korea.
  • Year and Date: 2009-12-16
• [Presentation] Spherical functions and (hermitian) Siegel series (2009)
  • Author(s): Y.Hironaka
  • Organizer: Arithmetic of Quadratic Forms of the KMS-AMS Joint Meeting
  • Place of Presentation: Ewha Womans University, Seoul, Korea.
  • Year and Date: 2009-12-16
• [Presentation] Spherical functions and(hermitian)Siegel series (2009)
  • Author(s): Yumiko Hironaka
  • Organizer: Joint meeting of KMS and AMS
  • Place of Presentation: Seoul, Korea
  • Year and Date: 2009-12-16
• [Presentation] $p$-adic Spherical functions and singular series (2009)
  • Author(s): Y. Hironaka
  • Organizer: Geometry and Analysis of Automorphic Forms of Several Variables
  • Place of Presentation: 東京大学
  • Year and Date: 2009-09-14
• [Presentation] $p$-adic spherical functions and singular series (2009)
  • Author(s): Yumiko Hironaka
  • Organizer: 織田孝幸教授還暦記念国際研究集会 "Geometry and Analysis of Automorphic Forms of Several Variables"
  • Place of Presentation: 東京大学数理科学研究科
  • Year and Date: 2009-09-14
• [Presentation] $p$進球関数の関数等式と表示式--$C_n$型の例-- (2009)
  • Author(s): Yumiko Hironaka
  • Organizer: 「第2回 数論女性の集まり」研究集会
  • Place of Presentation: 早稲田大学
  • Year and Date: 2009-05-09
• [Presentation] Spherical functions on $U(n, n)/\left(U(n)\times U(n)\right) and hermitian Siegel series (2009)
  • Author(s): Yumiko Hironaka
  • Organizer: Automorphic representations, automorphio $L$-functions and arithmetic
  • Place of Presentation: RIMS
  • Year and Date: 2009-01-21
• [Presentation] Spherical functions and hermitian Siegel serieson certain homogeneous spaces of U(n, n) (2008)
  • Author(s): Yumiko Hironaka
  • Organizer: The 11th Autumn Workshop on Number Theory
  • Place of Presentation: Hakuba
  • Year and Date: 2008-09-11
• [Presentation] Spherical functions and hermitian Siegel series on certain homogeneous spaces of $U(n, n)$ (2008)
  • Author(s): Yumiko Hironaka
  • Organizer: The 5th Japan-China Seminar on Number Theory
  • Place of Presentation: Kinki Univ.
  • Year and Date: 2008-08-28