Potential role of the elliptic modular j-function in the "Kronecker's dream of youth" for real quadratic fields

• Project/Area Number: 23654013
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: Kyushu University
• Principal Investigator: KANEKO Masanobu 九州大学, 数理(科)学研究科(研究院), 教授 (70202017)
• 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: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2012: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2011: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: 楕円モジュラー関数 / 実二次体 / 類体 / マルコフ数 / 虚数乗法 / caliber / マルコフ二次無理数 / モジュラー形式

Research Abstract

Although we have not yet succeeded in obtaining any arithmetic quantities from the "values" of the elliptic modular j-function, we have obtained on the way some arithmetic results concerning Markoff numbers. We have formulated a kind of Diophantine continuity of the values of j-function at Markoff irrationalities in terms of Farey parametrization of Markoff numbers.

Research Products

• [Journal Article] Double zeta values, double Eisenstein series, and modular forms of leve 12 (2013)
  • Author(s): M. Kaneko, K. Tasaka
  • Journal Title: Math. Ann Volume: 357(3) Issue: 3 Pages: 1091-1118
  • DOI: 10.1007/s00208-013-0930-5
  • URL: http://catalog.lib.kyushu-u.ac.jp/recordID/27531
  • Peer Reviewed
• [Journal Article] The Ramanujan-Serre differential operators and certain elliptic curves (2013)
  • Author(s): Masanobu Kaneko, Yuichi Sakai
  • Journal Title: Proceedings of the American Mathematical Society Volume: 141 (10) Issue: 10 Pages: 3421-3429
  • DOI: 10.1090/s0002-9939-2013-11917-9
  • NAID: 120005323553
  • URL: http://catalog.lib.kyushu-u.ac.jp/recordID/27241
  • Peer Reviewed
• [Journal Article] Modular Forms and Second Order Ordinary Differential Equations : Applications to Vertex Operator Algebras (2013)
  • Author(s): Masanobu Kaneko, Kiyokazu Nagatomo, Yuichi Sakai
  • Journal Title: Letters in Mathematical Physics Volume: 103 (4) Issue: 4 Pages: 439-453
  • DOI: 10.1007/s11005-012-0602-5
  • NAID: 120005227214
  • URL: http://catalog.lib.kyushu-u.ac.jp/recordID/26122
  • Peer Reviewed
• [Journal Article] On Fourier coefficients of some meromorphic modular forms (2012)
  • Author(s): Y. Honda, M. Kaneko
  • Journal Title: Bulletin of the Korean Mathematical Society Volume: 49(6)(DOI:0.4134/BKMS.2012.49.6.1349) Pages: 1349-1357
  • NAID: 120004951632
  • URL: http://catalog.lib.kyushu-u.ac.jp/recordID/25521
  • Peer Reviewed
• [Journal Article] On Fourier coefficients of some meromorphic modular forms (2012)
  • Author(s): Yutaro Honda, Masanobu Kaneko
  • Journal Title: Bulletin of the Korean Mathematical Society Volume: 49(6) Issue: 6 Pages: 1349-1357
  • DOI: 10.4134/bkms.2012.49.6.1349
  • NAID: 120004951632
  • Peer Reviewed
• [Presentation] On the elliptic modular function j, Modular functions and Quadratic forms -- Number theoretic delights (2013)
  • Author(s): M. Kaneko
  • Place of Presentation: 大阪大学中之島センター
  • Year and Date: 2013-12-21
• [Presentation] On the elliptic modular function j (2013)
  • Author(s): Masanobu Kaneko
  • Organizer: Modular functions and Quadratic forms -- Number theoretic delights
  • Place of Presentation: 大阪大学中之島センター
  • Invited
• [Presentation] mod p 多重ゼータ値について (2012)
  • Author(s): M. Kaneko
  • Organizer: 関西多重ゼータ研究会
  • Place of Presentation: 大阪工業大学
  • Year and Date: 2012-11-17
• [Presentation] Double zeta values and modular forms (2012)
  • Author(s): M. Kaneko
  • Organizer: Modular form seminar
  • Place of Presentation: National Center for Theoretical Sciences, Taiwan
  • Year and Date: 2012-11-07
• [Presentation] The Ramanujan-Serre differential operators and certain elliptic curves (2012)
  • Author(s): M. Kaneko
  • Organizer: Number theory seminar
  • Place of Presentation: National Center for Theoretical Sciences, Taiwan
  • Year and Date: 2012-11-07
• [Presentation] Markoff 数について (2012)
  • Author(s): M. Kaneko
  • Organizer: 整数論・力学系セミナー
  • Place of Presentation: 弘前大学理工学部
  • Year and Date: 2012-04-27
• [Presentation] The Ramanujan-Serre differential operators and certain elliptic curves (2012)
  • Author(s): M. Kaneko
  • Organizer: Hawaii Conference in Algebraic Number Theory, Arithmetic Geometry and Modular Forms
  • Place of Presentation: University of Hawaii at Manoa, USA
  • Year and Date: 2012-03-06
• [Presentation] Congruences of Markoff numbers via Farey parametrization (2012)
  • Author(s): M. Kaneko
  • Organizer: Special Session of the AMS 2012 Spring Western Section Meeting
  • Place of Presentation: University of Hawaii at Manoa, USA
  • Year and Date: 2012-03-04
• [Presentation] Markoff数について (2012)
  • Author(s): 金子昌信
  • Organizer: 整数論・力学系セミナー
  • Place of Presentation: 弘前大学理工学部
  • Invited
• [Presentation] Congruences of Markoff numbers via Farey parametrization (2012)
  • Author(s): Masanobu Kaneko
  • Organizer: Special Session of the AMS 2012 Spring Western Section Meeting(招待講演)
  • Place of Presentation: University of Hawaii at Manoa, USA
• [Presentation] Markoff numbers and Farey sequences (2011)
  • Author(s): M. Kaneko
  • Organizer: The 1st International Congress on Natural Sciences (ICNS2011)
  • Place of Presentation: Pukyong National University, Korea
  • Year and Date: 2011-08-26
• [Remarks]
  • URL: http://www2.math.kyushu-u.ac.jp/~mkaneko/paper.html
• [Remarks]
  • URL: http://www2.math.kyushu-u.ac.jp/~mkaneko/lecture.html
• [Remarks]
  • URL: http://www2.math.kyushu-u.ac.jp/~mkaneko/english/list.html
• [Remarks] 金子昌信 論文・著書等
  • URL: http://www2.math.kyushu-u.ac.jp/~mkaneko/paper.html
• [Remarks] 金子昌信 講演等
  • URL: http://www2.math.kyushu-u.ac.jp/~mkaneko/lecture.html
• [Remarks] 金子昌信 論文・著書等（英語版）
  • URL: http://www2.math.kyushu-u.ac.jp/~mkaneko/english/list.html
• [Remarks] 金子昌信 論文・著者等
  • URL: http://www2.math.kyushu-u.ac.jp/~mkaneko/paper.html
• [Remarks] 金子昌信 講演等
  • URL: http://www2.math.kyushu-u.ac.jp/~mkaneko/lecture.html
• [Remarks]
  • URL: http://www2.math.kyushu-u.ac.jp/~mkaneko/thesis.html
• [Remarks]
  • URL: http://www2.math.kyushu-u.ac.jp/~mkaneko/lecture.html