Theory of automorphic forms and quadratic forms

• Project/Area Number: 17H02834
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Kyoto University
• Principal Investigator: Ikeda Tamotsu 京都大学, 理学研究科, 教授 (20211716)
• Project Period (FY): 2017-04-01 – 2022-03-31
• Project Status: Completed (Fiscal Year 2022)
• Budget Amount: ¥17,420,000 (Direct Cost: ¥13,400,000、Indirect Cost: ¥4,020,000); Fiscal Year 2021: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000); Fiscal Year 2020: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000); Fiscal Year 2019: ¥3,770,000 (Direct Cost: ¥2,900,000、Indirect Cost: ¥870,000); Fiscal Year 2018: ¥2,990,000 (Direct Cost: ¥2,300,000、Indirect Cost: ¥690,000); Fiscal Year 2017: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000)
• Keywords: 保型形式 / 二次形式 / エルミート形式 / 保形形式 / 保型表現 / 保型的L関数 / 退化Whittaker関数 / Gross-Keating不変量 / 保型的Ｌ関数 / 退化Whittaka関数 / ジーゲル級数

Outline of Final Research Achievements

The Siegel series is an important invariant appearing in the Fourier coefficients of the Siegel-Eisenstein series. In this research, we study the Gross-Keating invariants of quadratic forms and their refinement, namely the extended Gross-Keating data, and give an explicit formula for the Siegel series. As applications, we also gave lifting of Hilbert-Siegel modular forms and evaluation of the Fourier coefficients of Eisenstein series. We also studied the Gross-Keating invariants for Hermite forms.

Academic Significance and Societal Importance of the Research Achievements

整数論において，保型形式，とくにヘッケ作用素の同時固有形式の不変量を調べることは重要な課題である．筆者の過去の研究では一変数のヘッケ同時固有形式から高次のジーゲル保型形式へのリフティングが存在することを示したが，このリフティングはまたヘッケ同時固有形式となる．
本研究ではリフティングの研究で重要な役割を果たしたジーゲル級数を詳細に研究した．ジーゲル級数が二次形式のグロス・キーティング不変量とその精密化である拡大グロス・キーティングデータを用いて表すことができることを示し，その明示的公式を与えた．また，応用としてヒルベルトジーゲル級数のリフティングを与え，そのフーリエ級数の評価なども与えた．

Research Products

• [Journal Article] Estimates for the Fourier coefficients of the Duke-Imamoglu-Ikeda lift (2023)
  • Author(s): Tamotsu Ikeda and Hidenori Katsurada
  • Journal Title: Forum Math. Volume: - Issue: 4 Pages: 975-990
  • DOI: 10.1515/forum-2022-0197
  • Peer Reviewed
• [Journal Article] An inductive formula of the Gross-Keating invariant of a quadratic form (2023)
  • Author(s): Sungmun Cho, Tamotsu Ikeda, Hidenori Katsurada, Chul-hee Lee and Takuya Yamauchi
  • Journal Title: Tohoku Mathematical Journal Volume: -
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] On the lifting of Hilbert cusp forms to Hilbert-Siegel cusp forms (2020)
  • Author(s): Ikeda, Tamotsu; Yamana, Shunsuke
  • Journal Title: Annales Scientifiques de l'Ecole Normale Superieure Volume: 53 Issue: 5 Pages: 1121-1181
  • DOI: 10.24033/asens.2442
  • Peer Reviewed
• [Journal Article] On the Gross-Keating invariant of a quadratic form over a non-archimedean local field (2018)
  • Author(s): T. Ikeda and H. Katsurada
  • Journal Title: Amer. J. Math. Volume: 140 Issue: 6 Pages: 1521-1565
  • DOI: 10.1353/ajm.2018.0046
  • Peer Reviewed
• [Presentation] On the theory of the liftings (2023)
  • Author(s): Tamotsu Ikeda
  • Organizer: 保型形式と数論
  • Int'l Joint Research / Invited
• [Presentation] An explicit formula for the Siegel series (2018)
  • Author(s): T. Ikeda
  • Organizer: Representation theory of reductive Lie groups and algebras
  • Int'l Joint Research / Invited
• [Presentation] Hilbert-Siegel modular forms on adele groups (2018)
  • Author(s): T. Ikeda
  • Organizer: 21st autumn workshop on number theory
• [Presentation] Algebraic automorphic forms and Hilbert-Siegel modular forms (2018)
  • Author(s): T. Ikeda
  • Organizer: 21st autumn workshop on number theory
• [Presentation] Gross-Keating invariants of Hermitian forms (2018)
  • Author(s): T. Ikeda
  • Organizer: Pan Asia Number Theory Conference 2018
• [Presentation] On the Gross-Keating invariant for hermitian forms (2018)
  • Author(s): 池田保・桂田英典
  • Organizer: 保型形式の解析的・数論的研究
  • Int'l Joint Research
• [Presentation] On the Gross-Keating invariant of a quadratic form and its application to Siegel series (2017)
  • Author(s): Tamotsu Ikeda
  • Organizer: Automorphic forms and related topics
  • Int'l Joint Research
• [Presentation] On the Gross-Keating invariant of a quadratic form and its applicationto Siegel series (2017)
  • Author(s): Tamotsu Ikeda
  • Organizer: Special values of automorphic L-functions, periods of automorphic forms and related topics
  • Int'l Joint Research
• [Funded Workshop] アジア地域における数論研究（Pan Asian Number Thoery Conference） (2021)
• [Funded Workshop] 21st Autumn Workshop on Number Theory (2018)