1999 年度実績報告書

多変数保型形式の数論:基礎の再構築

研究課題

研究課題/領域番号	09440007
研究機関	東京大学
研究代表者	織田孝幸東京大学, 大学院・数理科学研究科, 教授 (10109415)
研究分担者	齋藤裕京都大学, 大学院・人間環境学研究科, 教授 (20025464) 伊吹山知義大阪大学, 大学院・理学研究科, 教授 (60011722) 荒川恒男立教大学, 理学部, 教授 (60097219) 広中由美子早稲田大学, 教育学部, 教授 (10153652) 高瀬幸一宮城教育大学, 教育学部, 助教授 (60197093)
キーワード	保型形式 / L関数 / 球関数 / モジュラー形式 / 特殊関数
研究概要	本計画の最後の年度である.本年度実施分を中心に述べる.但し「研究発表」の項は3年を通してみる. <集会等について> 99年7月に長野県において、第7回整数論サマースクールを分担者の渡部隆夫が「代数群」を主題として行った。この集会に関わる旅費のそうとうの分と、記録の印刷代を本計画より支出した. 分担者の伊吹山が代表となる関連する研究計画の中で、同年9月に長野県において第二回保型形式のオータムスクールを行った.これの旅費の一部を本計画より支出した. 分担者の佐藤文広が東京の日仏会館で同年10月に組織した概均質ベクトル空間のフランス側参加者の一人の旅費等を本計画より支出した. <研究成果等> 代表者の織田は、早田孝博(山形大学工学部)・古関春隆(三重大学教育学部)との共同研究であるSU(2,2)のmiddle discrete seriesの行列係数の明示公式の研究を完成した. また都築正男(上智大学)と共同研究である種の有界対称領域の算術商のモジュラー埋入によって得られる代数的な因子でlogarithmicに振舞う保型的なグリーン関数の構成に成功した.これはモジュラー多様体で算術的交点理論(Arakelov,Gillet-Soule'の意味で)を展開するための第一歩と考えられる.この二つの研究はいずれもプレプリントとしてまとめ関係者に送付した段階である. 分担者の成果も著しい.主題別に述べよう. ヤコビ形式と半整数重さのジーゲル保型形式:高瀬はこれのアデール群上の一般理論を展開し、荒川はこれらの保型形式のKoecher-Maass級数の解析的な性質を詳しく調べた. Koecher-Maass級数: 荒川の上記の研究の他に、伊吹山はJordan代数の言葉を用いて、これを一般のtube領域上の保型形式に一般化した. 概均質ベクトル空間のゼータ関数:齋藤による特殊値の明示的な計算の研究が著しい.これはまたSelberg跡公式のunipotent項とも関連して、伊吹山・高瀬も研究している. 局所密度公式関係:桂田・伊吹山のEisenstein級数のフーリェ展開係数に関連する研究と、佐藤・広中の「分岐する」球関数に向かう二つの互いに関連する方向で成果があがっている. 村瀬・菅野の原始テータ級数の研究は最初の論文が印刷中で、加藤信一(京都大学)とのp-進球関数の明示公式の研究も直交群のときはプレプリントの形でまとめられた.橋本のmodular O-curveの研究、渡部の研究については最終報告に譲る.

研究成果

(47件)

すべてその他

すべて文献書誌 (47件)

[文献書誌] Takahiro HAYATA: "Matrix coefficients of the principal P_j - series and the middle deiscrete series of SU(2.2)"Advanced Studies in pure mathematics. 26. 49-75 (1999)
[文献書誌] Toshiyuki KOBAYASHI: "A vanishing theorem for modular symbols on locally symmetric spaces"Commentarii Math Helvetia. 73. 45-70 (1998)
[文献書誌] Takahiro HAYATA: "An explicit integral representation of whittaker funchons for the representations of the discrete series -the case of SU(2.2)"J. of Math., Kyoto University. 37. 519-530 (1997)
[文献書誌] Takuya MIYAZAKI: "Principal series whittaker functions on Sp(2;IR),II"Tohoku Math. J.. 50. 243-260 (1997)
[文献書誌] 織田孝幸: "保型形式の数論のための実解析"数学(日本数学会、岩波書店). 50-4. 350-357 (1998)
[文献書誌] Tsuneo ARAKAWA: "A note on the restriction map for Jacobi forms"Abh. Math. Sem. Univ. Hamburg. 69. 309-317 (1999)
[文献書誌] Tsuneo ARAKAWA: "On Poly-Bernoulli numbers"Commentarii Math.. Univ. St. Pauli. 48. 159-167 (1999)
[文献書誌] Tsuneo ARAKAWA: "Multiple zeta valumes Poty-Bernoulii numbers, and related zeta functions"Nagoya Math. J. 153. 189-209 (1999)
[文献書誌] Tsuneo ARAKAWA: "Kocher-MaaB Dirichlet series corresponding to Jacobi forms and cohen Eisenstein series"Commentarii Math. Univ. St. Pauli. 47. 93-122 (1998)
[文献書誌] Tomoyoshi IBUKIYAMA: "On squared Mobius function for half integral matrices and its application"J. of Number Theory. (印刷中).
[文献書誌] Tomoyoshi IBUKIYAMA: "On differential operators on automorphic forms and invariant pluriharmonic polynomials"Commertarii Math. Univ. St. Pauli. 48. 103-118 (1999)
[文献書誌] Tomoyoshi IBUKIYAMA: "On some elementaly character sums"Commentarii Math. Univ. St. Pauli. 47. 7-13 (1998)
[文献書誌] W. Eholzer: "Rankin-Cohen type differential operators for Siegel modular forms"International J. Math.. 9-4. 443-463 (1998)
[文献書誌] Tomoyoshi IBUKIYAMA: "On the graded ring of modular forms of the Siegel paramodular gronp of level 2"Abh. Math. Sem. Univ. Hamburg. 67. 297-305 (1997)
[文献書誌] Tomoyoshi IBUKIYAMA: "On zeta functions associated to symmetric matrices III. An explicit form of L-functions."Nagoya Math. J. 146. 149-183 (1997)
[文献書誌] 伊吹山知義: "「やさしい」ゼータ関数について"数学(日本数学会、岩波書店). 50-1. 1-11 (1998)
[文献書誌] Masao KOIKE: "Hypergeometric polynomials over finite fields"Tohoku Math. J.. 51. 75-79 (1999)
[文献書誌] Masao KOIKE: "A note on hypergeometric polymomials over finite field"Advanced studies in contemporary Math. (Algebraic Number Theory) in Memoirs of the Janjun Math. Soc.. 1. 35-43 (1999)
[文献書誌] Masao KOIKE: "Special Functions over Finite Fields"上智大学数学講究録講究録No.41 Threr Lectures in Algebra. 1-24 (1999)
[文献書誌] Hidenori KATSURADA: "Square Mobius function for half-integral matrices"Contemporary Math.. 249. 235-248 (1999)
[文献書誌] Hidenori KATSURADA: "A short proof of Kitaoka-Feit induction lemma for Siegel series"Abh. Math. Sem. Hamburg. 68. 297-304 (1998)
[文献書誌] Hidenori KATSURADA: "A power series attached to local densities of forms of higher degree"Hong Kong Math. J.. 2. 297-304 (1999)
[文献書誌] Hidenori KATSURADA: "An explicit formula for Siegel series"Amer. J. Math.. 121. 415-452 (1999)
[文献書誌] Hidenori KATSURADA: "A recursion formula for local densities with congruence conditions"Mem. Muroran Inrt. Tech.. 47. 97-102 (1997)
[文献書誌] Hidenori KATSURADA: "A recursion formula for local densities of quachatic forms"J. Number Theory. 64. 183-210 (1997)
[文献書誌] Hidenori KATSURADA: "A remark on Kitaoka formal power series attached to local density"Manuscripta Math.. 92. 477-486 (1997)
[文献書誌] Hidenori KATSURADA: "An explicit formula for the Fourier coefficients of Siegel-Eisenctein series of degree 3"Nagoya Math.. J.. 146. 199-223 (1997)
[文献書誌] Hiroshi SAITO: "Explicit form of the zeta functions of prehogeneous vector spaces"Math. Annal.. (印刷中).
[文献書誌] Hiroshi SAITO: "Explicit formula of orbital p-adic zeta functions associated to symmetric and hermitian matrices"Commentarii Math. Univ. Santi Pauli. 46. 175-216 (1997)
[文献書誌] Fumihiro SATO: "Local densities of representations of quadratic foums over p-adic integers : the non-dyadic case"J. of Number Theory. (印刷中).
[文献書誌] Fumihiro SATO: "b-Functions of prehomogeneous vector spaces attached to flag manifoldo of the general linear gronp"Comentarii Math. Univ. St. Pauli. 48. 129-136 (1999)
[文献書誌] Fumihiro SATO: "Eisenstein series on weakly spherical homogeneous spaces of GL(n)"Tohoku Math. J.. 50. 23-69 (1998)
[文献書誌] Atushi MURASE: "Local theory of primitive theta functions"Compositio Math.. (印刷中).
[文献書誌] Atushi MURASE: "On standard L-functions attached to automorphic forms on definite orthogoral groups"Nagoya Math. J.. 152. 57-96 (1998)
[文献書誌] Ki-ichiro HASHIMOTO: "Q-curves of degree 5 and jacobian surfaces of GL_2-type"Manuscripta Math.. 98. 165-182 (1999)
[文献書誌] Ki-ichiro HASHIMOTO: "On the Sato-Tate conjecture for QM-curves of genus two"Mathematics of Computation. 68-228. 1649-1662 (1999)
[文献書誌] Yumiko HIRONAKA: "Local densities of hermition forms"Contemporary Math. (Integral Quadratic Forms and Lattices). 249. 135-148 (1999)
[文献書誌] Yumiko HIRONAKA: "Spherical function and local densities on hermition forms"Journal of Math. Soc. Japan. 51. 553-581 (1999)
[文献書誌] Yumiko HIRONAKA: "Local zeta functions on hermition forms and its application to local density"Journal of Number Theory. 71. 40-64 (1998)
[文献書誌] Masanori MORISHITA: "Adele geometry of numbers"Advanced Studies in pure math.. (印刷中).
[文献書誌] Takao WATANABE: "A comparison of automorphic L-functions in a theta series lifting of unitary groups"Israel J. of Math.. (印刷中).
[文献書誌] Takao WATANABE: "Upper bounds of Hermite constants for orthogonal groups"Commentarii Math. Univ. Sanoti Pauli. 48. 25-33 (1999)
[文献書誌] 菅野孝史(編集者): "数理解析研究所講究録1052"保型形式と整数論""京都大学数理解析研究所. 230 (1998)
[文献書誌] 都築正男(編集者): "数理解析研究所講究録1094"Sp(2;1R)とSU(2,2)上の保型形式II""京都大学数理解析研究所. 117 (1999)
[文献書誌] 水本信一郎(編): "数理解析研究所講究録1103"保型形式とL関数の研究""京都大学数理解析研究所. 199 (1999)
[文献書誌] 村瀬篤・広中由美子(編): "整数論サマースクール報告集(第5回)"Siegel保型形式入門""218 (1997)
[文献書誌] 橋本喜一郎・百瀬文之(編): "第6回整数論サマースクール報告集「楕円曲線とそのArithmetic moduli」"205 (1998)

1999 年度 実績報告書

多変数保型形式の数論:基礎の再構築

研究代表者

織田 孝幸 東京大学, 大学院・数理科学研究科, 教授 (10109415)

研究成果

[文献書誌] Takahiro HAYATA: "Matrix coefficients of the principal P_j - series and the middle deiscrete series of SU(2.2)"Advanced Studies in pure mathematics. 26. 49-75 (1999)

[文献書誌] Toshiyuki KOBAYASHI: "A vanishing theorem for modular symbols on locally symmetric spaces"Commentarii Math Helvetia. 73. 45-70 (1998)

[文献書誌] Takahiro HAYATA: "An explicit integral representation of whittaker funchons for the representations of the discrete series -the case of SU(2.2)"J. of Math., Kyoto University. 37. 519-530 (1997)

[文献書誌] Takuya MIYAZAKI: "Principal series whittaker functions on Sp(2;IR),II"Tohoku Math. J.. 50. 243-260 (1997)

[文献書誌] 織田 孝幸: "保型形式の数論のための実解析"数学(日本数学会、岩波書店). 50-4. 350-357 (1998)

[文献書誌] Tsuneo ARAKAWA: "A note on the restriction map for Jacobi forms"Abh. Math. Sem. Univ. Hamburg. 69. 309-317 (1999)

[文献書誌] Tsuneo ARAKAWA: "On Poly-Bernoulli numbers"Commentarii Math.. Univ. St. Pauli. 48. 159-167 (1999)

[文献書誌] Tsuneo ARAKAWA: "Multiple zeta valumes Poty-Bernoulii numbers, and related zeta functions"Nagoya Math. J. 153. 189-209 (1999)

[文献書誌] Tsuneo ARAKAWA: "Kocher-MaaB Dirichlet series corresponding to Jacobi forms and cohen Eisenstein series"Commentarii Math. Univ. St. Pauli. 47. 93-122 (1998)

[文献書誌] Tomoyoshi IBUKIYAMA: "On squared Mobius function for half integral matrices and its application"J. of Number Theory. (印刷中).

[文献書誌] Tomoyoshi IBUKIYAMA: "On differential operators on automorphic forms and invariant pluriharmonic polynomials"Commertarii Math. Univ. St. Pauli. 48. 103-118 (1999)

[文献書誌] Tomoyoshi IBUKIYAMA: "On some elementaly character sums"Commentarii Math. Univ. St. Pauli. 47. 7-13 (1998)

[文献書誌] W. Eholzer: "Rankin-Cohen type differential operators for Siegel modular forms"International J. Math.. 9-4. 443-463 (1998)

[文献書誌] Tomoyoshi IBUKIYAMA: "On the graded ring of modular forms of the Siegel paramodular gronp of level 2"Abh. Math. Sem. Univ. Hamburg. 67. 297-305 (1997)

[文献書誌] Tomoyoshi IBUKIYAMA: "On zeta functions associated to symmetric matrices III. An explicit form of L-functions."Nagoya Math. J. 146. 149-183 (1997)

[文献書誌] 伊吹山 知義: "「やさしい」ゼータ関数について"数学(日本数学会、岩波書店). 50-1. 1-11 (1998)

[文献書誌] Masao KOIKE: "Hypergeometric polynomials over finite fields"Tohoku Math. J.. 51. 75-79 (1999)

[文献書誌] Masao KOIKE: "A note on hypergeometric polymomials over finite field"Advanced studies in contemporary Math. (Algebraic Number Theory) in Memoirs of the Janjun Math. Soc.. 1. 35-43 (1999)

[文献書誌] Masao KOIKE: "Special Functions over Finite Fields"上智大学数学講究録講究録No.41 Threr Lectures in Algebra. 1-24 (1999)

[文献書誌] Hidenori KATSURADA: "Square Mobius function for half-integral matrices"Contemporary Math.. 249. 235-248 (1999)

[文献書誌] Hidenori KATSURADA: "A short proof of Kitaoka-Feit induction lemma for Siegel series"Abh. Math. Sem. Hamburg. 68. 297-304 (1998)

[文献書誌] Hidenori KATSURADA: "A power series attached to local densities of forms of higher degree"Hong Kong Math. J.. 2. 297-304 (1999)

[文献書誌] Hidenori KATSURADA: "An explicit formula for Siegel series"Amer. J. Math.. 121. 415-452 (1999)

[文献書誌] Hidenori KATSURADA: "A recursion formula for local densities with congruence conditions"Mem. Muroran Inrt. Tech.. 47. 97-102 (1997)

[文献書誌] Hidenori KATSURADA: "A recursion formula for local densities of quachatic forms"J. Number Theory. 64. 183-210 (1997)

[文献書誌] Hidenori KATSURADA: "A remark on Kitaoka formal power series attached to local density"Manuscripta Math.. 92. 477-486 (1997)

[文献書誌] Hidenori KATSURADA: "An explicit formula for the Fourier coefficients of Siegel-Eisenctein series of degree 3"Nagoya Math.. J.. 146. 199-223 (1997)

[文献書誌] Hiroshi SAITO: "Explicit form of the zeta functions of prehogeneous vector spaces"Math. Annal.. (印刷中).

[文献書誌] Hiroshi SAITO: "Explicit formula of orbital p-adic zeta functions associated to symmetric and hermitian matrices"Commentarii Math. Univ. Santi Pauli. 46. 175-216 (1997)

[文献書誌] Fumihiro SATO: "Local densities of representations of quadratic foums over p-adic integers : the non-dyadic case"J. of Number Theory. (印刷中).

[文献書誌] Fumihiro SATO: "b-Functions of prehomogeneous vector spaces attached to flag manifoldo of the general linear gronp"Comentarii Math. Univ. St. Pauli. 48. 129-136 (1999)

[文献書誌] Fumihiro SATO: "Eisenstein series on weakly spherical homogeneous spaces of GL(n)"Tohoku Math. J.. 50. 23-69 (1998)

[文献書誌] Atushi MURASE: "Local theory of primitive theta functions"Compositio Math.. (印刷中).

[文献書誌] Atushi MURASE: "On standard L-functions attached to automorphic forms on definite orthogoral groups"Nagoya Math. J.. 152. 57-96 (1998)

[文献書誌] Ki-ichiro HASHIMOTO: "Q-curves of degree 5 and jacobian surfaces of GL_2-type"Manuscripta Math.. 98. 165-182 (1999)

[文献書誌] Ki-ichiro HASHIMOTO: "On the Sato-Tate conjecture for QM-curves of genus two"Mathematics of Computation. 68-228. 1649-1662 (1999)

[文献書誌] Yumiko HIRONAKA: "Local densities of hermition forms"Contemporary Math. (Integral Quadratic Forms and Lattices). 249. 135-148 (1999)

[文献書誌] Yumiko HIRONAKA: "Spherical function and local densities on hermition forms"Journal of Math. Soc. Japan. 51. 553-581 (1999)

[文献書誌] Yumiko HIRONAKA: "Local zeta functions on hermition forms and its application to local density"Journal of Number Theory. 71. 40-64 (1998)

[文献書誌] Masanori MORISHITA: "Adele geometry of numbers"Advanced Studies in pure math.. (印刷中).

[文献書誌] Takao WATANABE: "A comparison of automorphic L-functions in a theta series lifting of unitary groups"Israel J. of Math.. (印刷中).

[文献書誌] Takao WATANABE: "Upper bounds of Hermite constants for orthogonal groups"Commentarii Math. Univ. Sanoti Pauli. 48. 25-33 (1999)

[文献書誌] 菅野孝史(編集者): "数理解析研究所講究録1052"保型形式と整数論""京都大学数理解析研究所. 230 (1998)

[文献書誌] 都築 正男(編集者): "数理解析研究所講究録1094"Sp(2;1R)とSU(2,2)上の保型形式II""京都大学数理解析研究所. 117 (1999)

[文献書誌] 水本 信一郎(編): "数理解析研究所講究録1103"保型形式とL関数の研究""京都大学数理解析研究所. 199 (1999)

[文献書誌] 村瀬篤・広中由美子(編): "整数論サマースクール報告集(第5回)"Siegel保型形式入門""218 (1997)

[文献書誌] 橋本喜一郎・百瀬文之(編): "第6回整数論サマースクール報告集「楕円曲線とそのArithmetic moduli」"205 (1998)

1999 年度実績報告書

織田孝幸東京大学, 大学院・数理科学研究科, 教授 (10109415)

[文献書誌] 織田孝幸: "保型形式の数論のための実解析"数学(日本数学会、岩波書店). 50-4. 350-357 (1998)

[文献書誌] 伊吹山知義: "「やさしい」ゼータ関数について"数学(日本数学会、岩波書店). 50-1. 1-11 (1998)

[文献書誌] 都築正男(編集者): "数理解析研究所講究録1094"Sp(2;1R)とSU(2,2)上の保型形式II""京都大学数理解析研究所. 117 (1999)

[文献書誌] 水本信一郎(編): "数理解析研究所講究録1103"保型形式とL関数の研究""京都大学数理解析研究所. 199 (1999)