Study on automorphic L-functions and related spherical functions

• Project/Area Number: 18740016
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Algebra
• Research Institution: Seikei University (2008); Chiba Institute of Technology (2006-2007)
• Principal Investigator: ISHII Taku Seikei University, 理工学部, 准教授 (60406650)
• Project Period (FY): 2006 – 2008
• Project Status: Completed (Fiscal Year 2008)
• Budget Amount: ¥3,400,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥300,000); Fiscal Year 2008: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2007: ¥1,000,000 (Direct Cost: ¥1,000,000); Fiscal Year 2006: ¥1,100,000 (Direct Cost: ¥1,100,000)
• Keywords: 保型形式 / 保型L関数 / Whittaker関数 / Rankin-Selberg法 / Whittaker模型 / 主系列表現

Research Abstract

多変数非正則保型形式のフーリエ展開を記述するために、実半単純リー群上の球関数の研究を様々な場合に行った。これら球関数は偏微分方程式系によって特徴付けられる。表現論的手法によりその方程式系を導出することから始め、保型形式への応用に耐えうる形でその解の積分表示を求めた。さらに、保型L関数への応用として、これら球関数の積分変換である、ゼータ積分の無限素点における計算を実行し、L関数の関数等式、正則性という大域的な結果を得た。

Research Products

• [Journal Article] Spinor L-functions for generic cusp forms on GSp(2) belonging to principal series representations (2008)
  • Author(s): Taku Ishii and Tomonori Moriyama
  • Journal Title: Transaction of the American Mathematical Society 360 Pages: 5683-5709
  • Peer Reviewed
• [Journal Article] Lie群上の球関数と保型L関数 (2008)
  • Author(s): Taku Ishii
  • Journal Title: 第53回代数学シンポジウム報告集 Pages: 190-201
• [Journal Article] 一変数保型形式に付随する保型L関数 (2008)
  • Author(s): Taku Ishii
  • Journal Title: 第16回整数論サマースクール報告集 Pages: 3-36
• [Journal Article] Spinor L-functions for generic cusp forms on GSp(2)belonging to princi pal series representations (2008)
  • Author(s): T. Ishii and T. Moriyama
  • Journal Title: Transaction of the American Mathematical Society 360
  • Peer Reviewed
• [Journal Article] New formulas for Whittaker functions on GL(n,R) (2007)
  • Author(s): Taku Ishii and Eric Stade
  • Journal Title: Journal of Functional Analysis 244 Pages: 289-314
  • Peer Reviewed
• [Journal Article] Whittaker functions for P_J principal series representations of Sp(3,R) (2007)
  • Author(s): Miki Hirano, Taku Ishii and Takayuki Oda
  • Journal Title: Advances in Mathematics 215 Pages: 734-765
  • Peer Reviewed
• [Journal Article] Whittaker functions for P_J-principal series representations of Sp(3,R) (2007)
  • Author(s): M. Hirano, T. Ishii, T. Oda
  • Journal Title: Advances in Mathematics 215 Pages: 734-765
  • Peer Reviewed
• [Journal Article] New formulas for Whittaker functions on GL (n. IR) (2007)
  • Author(s): Taku Ishii, Eric Stade
  • Journal Title: Journal of Functional Analysis 244 Pages: 289-314
• [Journal Article] Confluence from Siegel-Whittaker functions to Whittaker functions on Sp(2,R) (2006)
  • Author(s): Miki Hirano, Taku Ishii and Takayuki Oda
  • Journal Title: Mathematical Proceedings of the Cambridge Philosophical Society 141 Pages: 15-31
  • Peer Reviewed
• [Journal Article] Class one Whittaker functions on real semisimple Lie groups (2006)
  • Author(s): Taku Ishii
  • Journal Title: 京都大学数理解析研究所講究録, 保型表現, L函数, 周期の研究 1523 Pages: 70-78
• [Journal Article] Archimedean zeta integrals on GSp(4), Automorphic Forms on GSp(4) (2006)
  • Author(s): Taku Ishii
  • Journal Title: Proceedings of the 9th Autumn Workshop on Number Theory Pages: 65-74
• [Journal Article] Confluence from Siegel-Whittaker functions to Whittaker functions on sp(2, IR) (2006)
  • Author(s): Miki Hirano, Taku Ishii, Takayuki Oda
  • Journal Title: Matrematical Proceedings of the Cambridge Philosophical Society 141 Pages: 15-31
• [Journal Article] Calculus of the principal series Whittaker functions on SL(4,R)
  • Author(s): Tatsuo Hina, Taku Ishii and Takayuki Oda
  • Journal Title: 京都大学数理解析研究所講究録別冊 (to appear)
  • Peer Reviewed
• [Journal Article] Whittaker functions on real semisimple Lie groups of rank two
  • Author(s): Taku Ishii
  • Journal Title: Canadian Journal of Mathematics (to appear)
  • Peer Reviewed
• [Presentation] Archimedean Whittaker functions and archimedean zeta integrals (2009)
  • Author(s): Taku Ishii
  • Organizer: 愛媛整数論集会
  • Place of Presentation: 愛媛大学
  • Year and Date: 2009-02-20
• [Presentation] 一変数保型形式に付随する保型L関数 (2008)
  • Author(s): Taku Ishii
  • Organizer: 第16回整数論サマースクール
  • Place of Presentation: 幕張メッセ国際会議場
  • Year and Date: 2008-08-18
• [Presentation] Lie群上の球関数と保型L関数 (2008)
  • Author(s): Taku Ishii
  • Organizer: 第53回代数学シンポジウム
  • Place of Presentation: いわて県民情報交流センター
  • Year and Date: 2008-08-07
• [Presentation] Explicit formulas of Whittaker functions on real semisimple Lie groups (2007)
  • Author(s): Taku Ishii
  • Organizer: Representation Theory, Systems of Differential Equations and their Related Topics
  • Place of Presentation: 北海道大学
  • Year and Date: 2007-07-02
• [Presentation] Calculus of the principal series Whittaker functions on SL(4,R) (2007)
  • Author(s): Taku Ishii
  • Organizer: 日本数学会年会
  • Place of Presentation: 埼玉大学
  • Year and Date: 2007-03-30
• [Presentation] Principal series Whittaker functions on SL(4,R) (2007)
  • Author(s): Taku Ishii
  • Organizer: 保型形式およびその周期の構成と応用
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2007-01-15
• [Presentation] Archimedean zeta integrals on GSp(4) (2006)
  • Author(s): Taku Ishii
  • Organizer: 第9回整数論オータムワークショップ
  • Place of Presentation: 白馬ハイマウントホテル
  • Year and Date: 2006-11-09
• [Presentation] Class one Whittaker functions on real semisimple Lie groups (2006)
  • Author(s): Taku Ishii
  • Organizer: Kempner Colloquium
  • Place of Presentation: University of Colorado at Boulder
  • Year and Date: 2006-09-11