Study of automorphic forms and zeta functions by using generalized or refined resolvent type trace formulas

• Project/Area Number: 23540020
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Kyushu University
• Principal Investigator: GON Yasuro 九州大学, 数理(科)学研究科(研究院), 准教授 (30302508)
• Co-Investigator(Renkei-kenkyūsha): TSUZUKI Masao 上智大学, 理工学部, 准教授 (80296946)
• Project Period (FY): 2011 – 2013
• Project Status: Completed (Fiscal Year 2013)
• Budget Amount: ¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000); Fiscal Year 2013: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2012: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2011: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000)
• Keywords: 数論 / 保型形式 / ヒルベルトモジュラー曲面 / セルバーグ型ゼータ関数 / スペクトルゼータ関数 / 正規化行列式 / セルバーグ型ディリクレ級数 / セルバーグ跡公式 / ヒルベルトモジュラー群 / ルエル型ゼータ関数 / 素測地線定理 / ワイルの法則

Research Abstract

We investigated a generalization or refinement of resolvent type trace formulas, which is useful for studying automororphic forms and zeta functions. Based on our results, we proved analytic properties of zeta functions of Ruelle and Selberg type in one or two variables for Hilbert modular surfaces. Besides, we also proved a prime geodesic type theorem and a regularized determinant formula for restricted Laplacians on Hilbert-Maass forms.

Research Products

• [Journal Article] Selberg type zeta function for the Hilbert modular group of a real quadratic field (2012)
  • Author(s): Y. Gon
  • Journal Title: Proc. Japan Acad Volume: 88A Issue: 9 Pages: 145-148
  • DOI: 10.3792/pjaa.88.145
  • NAID: 40019504453
  • Peer Reviewed
• [Journal Article] Spectral average of central values of automorphic L-functions for holomorphic cusp forms on SO_0(m,2), I (2012)
  • Author(s): MasaoTsuzuki
  • Journal Title: Journal of Number Theory Volume: 132 Issue: 11 Pages: 2407-2454
  • DOI: 10.1016/j.jnt.2012.04.020
  • Peer Reviewed
• [Journal Article] セルバーグ跡公式, セルバーグゼータ関数 (2011)
  • Author(s): 権寧魯
  • Journal Title: 第18回整数論サマースクール報告集「アーサー・セルバーグ跡公式入門」 Pages: 21-50
• [Presentation] Zeta functions of Ruelle and Selberg for Hilbert modular surfaces (2013)
  • Author(s): Y. Gon
  • Organizer: 研究集会" Geometric zeta functions and related topics"
  • Place of Presentation: 佐賀大学
  • Year and Date: 2013-10-30
• [Presentation] The square root of Selberg type zeta functions and determinants of restricted Laplacians (2013)
  • Author(s): Y. Gon
  • Organizer: 研究集会"Zeta functions in OKINAWA 2013"
  • Place of Presentation: 沖縄コンベンションセンター
  • Year and Date: 2013-10-20
• [Presentation] Selberg type Dirichlet series in two variables (2012)
  • Author(s): Y. Gon
  • Organizer: 金沢数論ミニ集会
  • Place of Presentation: 金沢大学
  • Year and Date: 2012-11-30
• [Presentation] Selberg type Dirichlet series in two variables (2012)
  • Author(s): Y. Gon
  • Organizer: 研究集会"Zetas and Limit Laws 2012"
  • Place of Presentation: フェストーネ, 沖縄県宜野湾市
  • Year and Date: 2012-11-19
• [Presentation] Class numbers of binary quadratic forms and Ruelle type zeta functions for Hilbert modular varieties (2012)
  • Author(s): Y. Gon
  • Organizer: Kyoto conference on automorphic forms
  • Place of Presentation: 京都大学
  • Year and Date: 2012-10-06
• [Presentation] Zeta functions of Ruelle and Selberg types for Hilbert modular groups (2012)
  • Author(s): Y. Gon
  • Organizer: 研究集会"Zeta Function 2012"
  • Place of Presentation: 東京工業大学
  • Year and Date: 2012-09-27
• [Presentation] Zeta functions defined by class numbers of binary quadratic forms over rings of algebraic integers (2012)
  • Author(s): Y. Gon
  • Organizer: 研究集会"Zeta Function 2012"
  • Year and Date: 2012-09-27
• [Presentation] 2元2次形式の類数とヒルベルトモジュラー群に対するルエル型ゼータ関数について (2012)
  • Author(s): 権 寧魯
  • Organizer: 大阪大学整数論・保型形式セミナー
  • Place of Presentation: 大阪大学
  • Year and Date: 2012-07-06
• [Presentation] 2元2次形式の類数とヒルベルトモジュラー群に対するルエル型ゼータ関数 (2012)
  • Author(s): 権寧魯
  • Organizer: 大岡山談話会
  • Place of Presentation: 東京工業大学
  • Year and Date: 2012-01-25
• [Presentation] ２元２次形式の類数とヒルベルトモジュラー群に対するルエル型ゼータ関数 (2012)
  • Author(s): 権 寧魯
  • Organizer: 大岡山談話会(招待講演)
  • Place of Presentation: 東京工業大学（東京都）
• [Presentation] Class numbers of binary quadratic forms and Ruelle type zeta functions for Hilbert modular groups (2011)
  • Author(s): Y. Gon
  • Organizer: 研究集会"Zetas and Limit Laws 2011"
  • Place of Presentation: フェストーネ, 沖縄県宜野湾市
  • Year and Date: 2011-11-28
• [Presentation] Mahler measures and their generalization (2011)
  • Author(s): Y. Gon
  • Organizer: 香川セミナー
  • Place of Presentation: 香川大学
  • Year and Date: 2011-10-15
• [Presentation] Selberg type zeta functions for the Hilbert modular group of a real quadratic field (2011)
  • Author(s): Y. Gon
  • Organizer: Workshop on L-functions
  • Place of Presentation: 九州大学
  • Year and Date: 2011-04-23
• [Presentation] Zeta functions of Ruelle and Selberg for Hilbert modular surfaces
  • Author(s): Yasuro Gon
  • Organizer: Geometric zeta functions and related topics
  • Place of Presentation: 佐賀大学
  • Invited
• [Presentation] Zeta functions of Ruelle and Selberg types for Hilbert modular groups
  • Author(s): Yasuro Gon
  • Organizer: Zeta Function 2012
  • Place of Presentation: 東京工業大学
  • Invited