Automorphic L-functions by using the generalized Maass relations

• Project/Area Number: 26400007
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Joetsu University of Education
• Principal Investigator: Hayashida Shuichi 上越教育大学, 大学院学校教育研究科, 准教授 (80597766)
• Co-Investigator(Renkei-kenkyūsha): AOKI Hiroki 東京理科大学, 理工学部, 准教授 (10333189)
• Project Period (FY): 2014-04-01 – 2018-03-31
• Project Status: Completed (Fiscal Year 2017)
• Budget Amount: ¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000); Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2016: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2015: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2014: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
• Keywords: ランキン・セルバーグ型ディリクレ級数 / ヤコビ形式のランキン・セルバーグ型ディリクレ級数 / 重さ半整数のコーネン・スコルッパ型ディリクレ級数 / コーネン・スコルッパ・山崎型ディリクレ級数 / 保型形式 / エル関数 / ジーゲル保型形式 / ヤコビ形式 / マース空間 / プラス空間 / マース関係式 / ディリクレ級数 / ジーゲルアイゼンシュタイン級数 / 保型L関数 / Ikeda lift / Ikeda-Miyawaki lift / ヤコビアイゼンシュタイン級数 / 重さ半整数の保型形式 / 保型形式の持ち上げ

Outline of Final Research Achievements

Siegel modular forms and Jacobi forms are modular forms of several variables. In this research some properties of Siegel modular forms and Jacobi forms were investigated. More precisely, the following results are obtained. The isomorphism between the space of Jacobi forms of matrix index of integral weight and of half-integral weight. The meromorphic continuation and functional equation for certain Rankin-Selberg type Dirichlet series (Kohnen-Skoruppa-Yamazaki type Dirichlet series). An explicit formula for a Kohnen-Skourppa-Yamazaki type Dirichlet series which is constructed from half-integral weight version of the Ikeda lift.

Research Products

• [Int'l Joint Research] Siegen大学(ドイツ)
• [Int'l Joint Research] GUtech大学(オマーン)
• [Journal Article] On Kohnen plus-space of Jacobi forms of half integral weight of matrix index (2018)
  • Author(s): Shuichi Hayashida
  • Journal Title: Osaka Journal of Mathematics Volume: 55
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Lifting from two elliptic modular forms to Siegel modular forms of half-integral weight of even degree (2016)
  • Author(s): Shuichi Hayashida
  • Journal Title: Documenta Mathematica Volume: 21 Pages: 125-196
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Generalized Maass relations and lifts (2015)
  • Author(s): Shuichi Hayashida
  • Journal Title: 数理解析研究所考究録 Volume: 1973 Pages: 1-9
• [Presentation] Kohnen-Skoruppa-Yamazaki type Dirichlet series with respect to certain Siegel cusp forms of half-integral weight (2018)
  • Author(s): Shuichi Hayashida
  • Organizer: Workshop on Jacobi forms and applications
  • Int'l Joint Research / Invited
• [Presentation] Generalized Maass relations and lifts. (2015)
  • Author(s): 林田秀一
  • Organizer: RIMS 研究集会「モジュラー形式と保型表現」
  • Place of Presentation: 数理解析研究所
  • Year and Date: 2015-02-02
• [Presentation] Relations among Fourier-Jacobi coefficients of Siegel-Eisenstein series and their applications. (2014)
  • Author(s): 林田秀一
  • Organizer: The 17th Hakuba Autumn Workshop on Number Theory
  • Place of Presentation: 白馬ハイマウントホテル
  • Year and Date: 2014-11-08
• [Presentation] ジーゲル保型形式の一般化マース関係式について. (2014)
  • Author(s): 林田秀一
  • Organizer: 日本数学会2014年度秋季総合分科会
  • Place of Presentation: 広島大学
  • Year and Date: 2014-09-25
• [Funded Workshop] Joetsu Number Theory Seminar (2017)
• [Funded Workshop] RIMS 研究集会「保型形式・保型的L関数とその周辺」 (2016)
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2016-02-01
• [Funded Workshop] 第18回白馬整数論オータム ワークショップ「Geometrical Applications of Modular Forms of Several Variables」 (2015)
  • Place of Presentation: 長野県北安曇郡白馬村白馬ハイマウントホテル
  • Year and Date: 2015-11-06