Explicit study on spherical functions on reductive groups and archimedean zeta integrals

• Project/Area Number: 15K04796
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Seikei University
• Principal Investigator: Ishii Taku 成蹊大学, 理工学部, 教授 (60406650)
• Project Period (FY): 2015-04-01 – 2019-03-31
• Project Status: Completed (Fiscal Year 2018)
• Budget Amount: ¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000); Fiscal Year 2018: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: 保型形式 / 保型L関数 / アルキメデスゼータ積分 / Whittaker関数 / Shalika関数 / 主系列表現 / 一般化Whittaker関数 / 標準Ｌ関数 / スピノールＬ関数

Outline of Final Research Achievements

Among various integral representations for automorphic L-functions, we compute the following arhimedean zeta integrals: (a)Godement-Jacquet integrals for standard L-functions on GL(n) over the complex local field, (b)Pollach-Shah integrals for product of three L-functions on GL(4) and product of two L-functions on GSp(2)×GL(2), (c)Jacquet-Shalika integrals for exterior square L-functions on GL(4), (d) Novodvorsky integrals for spinor L-functions on GSp(2),（e）Bump-Friedberg-Ginzburg integrals for product of spinor and standard L-functions on GSp(2).

Academic Significance and Societal Importance of the Research Achievements

保型形式という高い対称性を兼ね備えた関数に対するゼータ関数（保型L関数）は、様々な整数論的なゼータ関数と結びつくと考えられている重要な研究対象である。保型L関数を積分表示式によって研究する上でネックとなるのが「悪い素点」における解析であり、そのうち無限素点における研究をゼータ積分を直接計算することで実行した。

Research Products

• [Journal Article] Whittaker functions for generalized principal series representations on GSp(2,R) (2019)
  • Author(s): Taku Ishii
  • Journal Title: Journal of Functional Analysis Volume: 276 Issue: 1 Pages: 284-327
  • DOI: 10.1016/j.jfa.2018.09.011
  • Peer Reviewed
• [Journal Article] The computation of multivariate archimedean zeta integrals on GL_2×GSp_4 and GL_4 (2019)
  • Author(s): Taku Ishii
  • Journal Title: Proceedings of the American Mathematical Society Volume: 147 Issue: 1 Pages: 103-114
  • DOI: 10.1090/proc/14211
  • Peer Reviewed
• [Journal Article] Godement-Jacquet integrals on GL(n,C) (2018)
  • Author(s): Taku Ishii
  • Journal Title: Ramanujan Journal Volume: 印刷中 Issue: 1 Pages: 129-139
  • DOI: 10.1007/s11139-018-0025-8
  • Peer Reviewed
• [Journal Article] Archimedean zeta integrals for the exterior square L-functions on GL_n (2018)
  • Author(s): Taku Ishii
  • Journal Title: Journal of Number Theory Volume: 186 Pages: 304-345
  • DOI: 10.1016/j.jnt.2017.10.007
  • Peer Reviewed
• [Journal Article] The computation of multivariate archimedean zeta integrals on GL_2 ×GSp_4 and GL_4 (2018)
  • Author(s): Taku Ishii
  • Journal Title: Proceedings of the American Mathematical Society Volume: 印刷中
  • Peer Reviewed
• [Journal Article] The archimedean zeta integrals for GL(3)×GL(2) (2016)
  • Author(s): Miki Hirano, Taku Ishii and Tadashi Miyazaki
  • Journal Title: Proc. Japan Acad. Ser. A Math. Sci. Volume: 92 Issue: 2 Pages: 27-32
  • DOI: 10.3792/pjaa.92.27
  • NAID: 40020756994
  • Peer Reviewed / Open Access
• [Presentation] Whittaker functions on GSp(2,R) and archimedean zeta integrals (2016)
  • Author(s): Taku Ishii
  • Organizer: 19th Autumn Workshop on Number Theory ``Archimedean local theory of automorphic forms"
  • Place of Presentation: 白馬ハイマウントホテル
  • Year and Date: 2016-11-03
  • Int'l Joint Research
• [Presentation] Archimedean zeta integrals on GSp（2） (2016)
  • Author(s): Taku Ishii
  • Organizer: Modular forms and period integrals
  • Place of Presentation: 東京大学数理科学研究科
  • Year and Date: 2016-09-14
• [Presentation] Whittaker functions on Sp(2,R) and archimedean zeta integrals (2016)
  • Author(s): Taku Ishii
  • Organizer: 保型形式、保型的Ｌ関数とその周辺
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2016-02-01
• [Presentation] Whittaker functions on Sp(2,R) and archimedean zeta integrals (2015)
  • Author(s): Taku Ishii
  • Organizer: Moduli spaces of abelian varieties and curves, and related analysis
  • Place of Presentation: 東京大学数理科学研究科
  • Year and Date: 2015-12-17
• [Presentation] Archimdean zeta integrals for generic cusp forms on GSp(2) (2015)
  • Author(s): Taku Ishii
  • Organizer: Kyoto conference on automorphic forms
  • Place of Presentation: 京都大学
  • Year and Date: 2015-06-28
• [Funded Workshop] 19th Autumn Workshop on Number Theory ``Archimedean local theory of automorphic forms" (2016)
  • Place of Presentation: 白馬ハイマウントホテル
  • Year and Date: 2016-11-02