Explicit and constructive research of automorphic forms of several variables

• Project/Area Number: 16K05076
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Tokyo University of Science
• Principal Investigator: AOKI Hiroki 東京理科大学, 理工学部数学科, 准教授 (10333189)
• Research Collaborator: HAYASHIDA Shuichi
• Project Period (FY): 2016-04-01 – 2019-03-31
• Project Status: Completed (Fiscal Year 2018)
• Budget Amount: ¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000); Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: 保型形式 / ヤコビ形式 / ヒルベルト保型形式 / 数論

Outline of Final Research Achievements

Although automorphic forms of several variables are important research subjects, their research is generally more difficult than in the case of one variable. This research work was conducted with the purpose of obtaining more knowledge about automorphic forms of several variables by the researcher's own research method using Jacobi forms. The results obtained during this research period are as follows:
`Struture theorem of Jacobi forms of real weights and indces' `Determination of the structures of mixed weight Hilbert modular forms' `A construction of Maass lift of weak Jacobi forms'.

Academic Significance and Societal Importance of the Research Achievements

多変数の保型形式は、整数論のみならず幾何学においても重要な研究対象であり、さらに近年では素粒子論においても使われはじめている。本研究では、いくつかの多変数の保型形式について、それはどのくらいあるのか、また具体的にどのような形をしているのかという問いに回答を与えた。これは保型形式そのものに関する疑問を解決したにとどまらず、上述の諸分野の研究に進展をもたらすものである。

Research Products

• [Int'l Joint Research] ハイデルベルク大学/マンハイム大学/ジーゲン大学(ドイツ)
• [Int'l Joint Research] Max-Planck-Institut fuer Mathematik(ドイツ)
• [Int'l Joint Research] University of Nottingham(英国)
• [Int'l Joint Research] ハイデルベルク大学/マンハイム大学(ドイツ)
• [Journal Article] The structure of mixed weight Hilbert modular forms (2019)
  • Author(s): Aoki Hiroki、Takemori Sho
  • Journal Title: International Journal of Mathematics Volume: 30 Issue: 02 Pages: 1950008-1950008
  • DOI: 10.1142/s0129167x19500083
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] On mixed weight Hilbert modular forms and their structure theorem (2019)
  • Author(s): Aoki Hiroki
  • Journal Title: 数理解析研究所講究録 Volume: 2100 Pages: 65-79
  • Open Access
• [Journal Article] On Jacobi forms of real weights and indices (2017)
  • Author(s): Hiroki Aoki
  • Journal Title: Osaka Journal of Mathematics Volume: 3 Pages: 569-585
  • Peer Reviewed
• [Presentation] On Maass lifts of weak Jacobi forms (2019)
  • Author(s): Aoki Hiroki
  • Organizer: Seminar on automorphic forms in Mannheim University
  • Int'l Joint Research
• [Presentation] A short survey on elliptic modular forms (2019)
  • Author(s): Aoki Hiroki
  • Organizer: IPMU Special Seminar `Elliptic Artin groups I, II'
  • Int'l Joint Research
• [Presentation] 弱ヤコビ形式のマースリフトについて (2018)
  • Author(s): 青木宏樹
  • Organizer: 京都大学数論合同セミナー
• [Presentation] ワイエルシュトラスのペー関数の一般化と弱ヤコビ形式のマースリフトについて (2018)
  • Author(s): 青木宏樹
  • Organizer: 2018大分鹿児島整数論研究集会
• [Presentation] On mixed weight Hilbert modular forms and their structure theorem (2018)
  • Author(s): Hiroki Aoki
  • Organizer: 保型形式の解析的・数論的研究
  • Int'l Joint Research
• [Presentation] On the structures of mixed weight Hilbert modular forms on real quadratic fields (2018)
  • Author(s): Hiroki Aoki
  • Organizer: Workshop On Jacobi Forms and Applications
  • Int'l Joint Research
• [Presentation] 混合重みヒルベルト保型形式の構成と全体構造の決定について (2018)
  • Author(s): Hiroki Aoki
  • Organizer: Automorphic forms/representaions on covering groups
• [Presentation] 簡単な代数学の知識だけで進むヒルベルト保型形式の話 (2017)
  • Author(s): Hiroki Aoki
  • Organizer: 九大代数学セミナー
• [Presentation] ヤコビ形式の簡単な紹介 (2017)
  • Author(s): Hiroki Aoki
  • Organizer: 神楽坂代数セミナー
• [Presentation] On the structure of mixed weight Hilbert modular forms (2017)
  • Author(s): Hiroki Aoki
  • Organizer: Seminar Aachen-Bonn-Koeln-Lille-Siegen on Automorphic Forms
• [Presentation] On the structure theorem for modular forms ... Igusa’s result and beyond (2017)
  • Author(s): Hiroki Aoki
  • Organizer: Local zeta functions and the arithmetic of moduli spaces: A conference in memory of Jun-Ichi Igusa
  • Place of Presentation: ボルチモア（アメリカ）
  • Invited
• [Presentation] On the structure of Hilbert modular forms with discriminant 5 (2016)
  • Author(s): Hiroki Aoki
  • Organizer: Oberseminar Modulfunktionen (in Heidelberg University)
  • Place of Presentation: ハイデルベルク（ドイツ）
  • Year and Date: 2016-11-20
  • Invited
• [Presentation] 実数重み・実数指数のヤコビ形式について (2016)
  • Author(s): 青木宏樹
  • Organizer: 大阪大学整数論保型形式セミナー
  • Place of Presentation: 大阪大学（大阪府・豊中市）
  • Invited
• [Presentation] ヤコビ形式の無限積表示について (2016)
  • Author(s): 青木宏樹
  • Organizer: 愛知数論セミナー
  • Place of Presentation: 名古屋工業大学（愛知県・名古屋市）
  • Invited
• [Presentation] ヤコビの三重積を保型形式の立場から見てみる (2016)
  • Author(s): 青木宏樹
  • Organizer: 新潟代数セミナー
  • Place of Presentation: 新潟大学（新潟県・新潟市）
• [Presentation] A remark on the structure of mixed weight Hilbert modular forms (2016)
  • Author(s): Hiroki Aoki
  • Organizer: Seminar on automorphic forms in Mannheim University
  • Place of Presentation: マンハイム（ドイツ）
• [Remarks] 研究代表者の研究紹介ページ
  • URL: https://www.ma.noda.tus.ac.jp/u/ha/index-j.html
• [Remarks] 東京理科大学 研究者情報データベース RIDAI
  • URL: https://www.tus.ac.jp/ridai/doc/ji/RIJIA01.php
• [Remarks] RIDAI（東京理科大学研究者情報データベース）
  • URL: https://www.tus.ac.jp/ridai/doc/ji/RIJIA01.php
• [Remarks] 研究代表者のwebページ（研究成果の紹介あり）
  • URL: http://www.ma.noda.tus.ac.jp/u/ha/index.html
• [Funded Workshop] 第１９回整数論オータムワークショップ 「Archimedean local theory of automorphic forms」 (2016)
  • Place of Presentation: 白馬ハイマウントホテル（長野県・白馬村）
  • Year and Date: 2016-11-02