Lifting for Siegel modular forms of half-integral weight

• Project/Area Number: 23740018
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: Joetsu University of Education (2012-2013); Osaka University (2011)
• Principal Investigator: HAYASHIDA SHUICHI 上越教育大学, 学校教育研究科(研究院), 准教授 (80597766)
• Project Period (FY): 2011 – 2013
• Project Status: Completed (Fiscal Year 2013)
• Budget Amount: ¥3,250,000 (Direct Cost: ¥2,500,000、Indirect Cost: ¥750,000); Fiscal Year 2013: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2012: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2011: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
• Keywords: ジーゲル・アイゼンシュタイン級数 / マース関係式 / 重さ半整数の保型形式 / リフティング / Zharkovskaya's theorem / 保型形式 / Fourier-Jacobi 展開 / Maass 関係式 / Ikeda lift / Ikeda-Miyawaki lift / Zharkovskaya の定理 / 国際研究者交流 / ヤコビ形式 / ジーゲル保型形式 / ヒルベルト保型形式 / ドイツ / アメリカ

Research Abstract

Modular forms are functions of complex variables which have nice transformation formula. So-called "liftings" are certain maps between the space of modular forms of several variables and "liftings" are useful to investigate the structure of the space of modular forms and the properties of the L-functions.
In this study I obtained lifting maps from two elliptic modular forms to Siegel modular forms of half-integral weight of even degrees under the assumption that the constructed form does not vanish. To show the lifting I obtained a certain generalization of the Maass relation for Siegel modular forms of half-integral weight of even degrees.

Research Products

• [Journal Article] On the spinor L-funcion of Miyawaki-Ikeda lifts (2014)
  • Author(s): Shuichi Hayashida
  • Journal Title: International Journal of Number Theory Volume: 10 Issue: 02 Pages: 297-307
  • DOI: 10.1142/s1793042113500930
  • Peer Reviewed
• [Journal Article] On generalized Maass relations and their application to Miyawaki-Ikeda lifts (2013)
  • Author(s): S. Hayashida
  • Journal Title: Comment. Math. Univ. St. Pauli. Volume: Vol 62, No 1 Pages: 59-90
  • URL: https://rikkyo.repo.nii.ac.jp/?action=pages_view_main&active_action=repository_view_main_item_detail&item_id=7398&item_no=1&page_id=13&block_id=49
  • Peer Reviewed
• [Journal Article] On generalized Maass relations and their application to Miyawaki-Ikeda lifts (2013)
  • Author(s): Shuichi Hayashida
  • Journal Title: Commentarii Mathematici Universitatis Sancti Pauli Volume: 62 Pages: 59-90
  • Peer Reviewed
• [Journal Article] 二つの楕円保型形式から重さ半整数Siegel保型形式へのリフト (2011)
  • Author(s): 林田秀一
  • Journal Title: 第19回整数論サマースクール報告集 Pages: 302-321
  • URL: http://www.sci.kumamoto-u.ac.jp/~narita/ss2011_proceedings.htm
• [Journal Article] 次数2の重さ半整数ジーゲル保型形式へのリフティング (2011)
  • Author(s): 林田秀一
  • Journal Title: 第56回代数学シンポジウム報告集 Pages: 14-14
  • URL: http://server03.mathsoc.jp/section/algebra/algsymp_past/algsymp11.html
• [Presentation] Lifts from two elliptic modular forms to Siegel modular forms of half-integral weight of even degrees (2014)
  • Author(s): S. Hayashida
  • Organizer: Explicit Theory of Automorphic Forms
  • Place of Presentation: 同済大学(中国・上海市)
  • Year and Date: 2014-03-28
• [Presentation] Relations among Fourier coefficients of holomorphic Siegel-Eisenstein series (2013)
  • Author(s): S. Hayashida
  • Organizer: Workshop on modular forms and Jacobi forms
  • Place of Presentation: 上越教育大学
  • Year and Date: 2013-06-11
• [Presentation] Generalized Maass relations and Miyawaki-Ikeda lifts (2013)
  • Author(s): S. Hayashida
  • Organizer: 27th Automorphic Forms Workshop
  • Place of Presentation: University of College Dublin (アイルランド・ダブリン市)
  • Year and Date: 2013-03-13
• [Presentation] Generalized Maass relations and Miyawaki-Ikeda lifts. (2013)
  • Author(s): 林田秀一
  • Organizer: 27th Automorphic Forms Workshop
  • Place of Presentation: University of College Dublin, Ireland
• [Presentation] Numerical examples of Euler factors of Siegel modular forms of half-integral weight of degree three (2012)
  • Author(s): S. Hayashida
  • Organizer: 26th Automorphic Forms Workshop
  • Place of Presentation: University of British Columbia (カナダ・バンクーバー市)
  • Year and Date: 2012-04-28
• [Presentation] Miyawaki-Ikedaリフトのスピノル L関数について (2012)
  • Author(s): 林田秀一
  • Organizer: 第5回ゼータ若手研究集会
  • Place of Presentation: 名古屋大学多元数理
  • Year and Date: 2012-02-11
• [Presentation] Numerical examples of Euler factors of Siegel modular forms of half-integral weight of degree three. (2012)
  • Author(s): 林田秀一
  • Organizer: 26th Automorphic Forms Workshop
  • Place of Presentation: University of British Columbia, Canada
• [Presentation] 二つの楕円保型形式からの重さ半整数Siegel保型形式へのリフト (2011)
  • Author(s): 林田秀一
  • Organizer: 第19回整数論サマースクール
  • Place of Presentation: 富士箱根ランド・スコーレプラザホテル
  • Year and Date: 2011-09-09
• [Presentation] Cohen type Eisenstein series of degree two and a relation among Fourier coefficients (2011)
  • Author(s): S. Hayashida
  • Organizer: 2011 International Conference on Number Theory
  • Place of Presentation: 上海交通大学(中国・上海市)
  • Year and Date: 2011-08-17
• [Presentation] 次数2の重さ半整数ジーゲル保型形式へのリフティング (2011)
  • Author(s): 林田秀一
  • Organizer: 第56回代数学シンポジウム
  • Place of Presentation: 岡山大学
  • Year and Date: 2011-08-11
• [Presentation] 次数2の重さ半整数ジーゲル保型形式へのリフティング (2011)
  • Author(s): 林田秀一
  • Organizer: 第56回代数学シンポジウム(招待講演)
  • Place of Presentation: 岡山大学
• [Presentation] Cohen type Eisenstein series of degree two and a relation among Fourier coefficients. (2011)
  • Author(s): 林田秀一
  • Organizer: 2011 International Conference on Number Theory
  • Place of Presentation: 上海交通大学
• [Presentation] 二つの楕円保型形式からの重さ半整数Siegel保型形式へのリフト (2011)
  • Author(s): 林田秀一
  • Organizer: 第19回整数論サマースクール
  • Place of Presentation: 富士箱根ランド
• [Presentation] Lifts from two elliptic modular forms to Siegel modular forms of half-integral weight of even degrees.
  • Author(s): Shuichi Hayashida
  • Organizer: Explicit Theory of Automorphic Forms
  • Place of Presentation: 同済大学（中華人民共和国・上海市）
  • Invited
• [Remarks]
  • URL: http://www.juen.ac.jp/lab/~hayasida/index_jp.html