Research on the theory of modular forms and modular functions using Mathematica

• Project/Area Number: 19540040
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Kyushu University
• Principal Investigator: KOIKE Masao Kyushu University, 大学院・数理学研究院, 教授 (20022733)
• Co-Investigator(Kenkyū-buntansha): BANNAI Eiichi 九州大学, 大学院・数理学研究院, 学術研究者 (10011652)
• Project Period (FY): 2007 – 2010
• Project Status: Completed (Fiscal Year 2010)
• Budget Amount: ¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000); Fiscal Year 2010: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2009: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2008: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2007: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
• Keywords: 保型形式 / 保型関数 / p進整数 / p進的保型形式 / マッカイ・トンプソン級数 / extremalな保型形式 / 2進的保型形式 / アトキンの方法 / 3進的保型形式 / トンプソン級数 / フリッケ群 / 超特異多項式 / 2進的な保型形式 / 級数展開 / フーリエ係数 / テータ級数 / 合同式

Research Abstract

We find the congruences between extremal modular forms and theta series of special types modulo powers of 2 and 3.This assertion enables us to prove that 2n-th root of the extremal modular forms of weight n/2 have at least one non integer coefficient.
Let f(z) be a Hauptmodul for level 2 congruence subgroup that has zero at the cusp infinity. We construct the 2-adic modular form F from f(z) by using Atkin's method. We expand F as a formal power series in f(z). We can compute explicitly the 2-adic ordinal of these coefficients of the f-expansion of F.

Research Products

• [Journal Article] On 2-adic modular forms (2010)
  • Author(s): M.Koike
  • Journal Title: Kyushu J.Math. 64 Pages: 199-214
  • NAID: 130000441856
  • Peer Reviewed
• [Journal Article] Toy models for D.H.Lehmer's conjecture (2010)
  • Author(s): E.Bannai, T.Miezaki
  • Journal Title: J.Math.Soc.Japan 62 Pages: 687-705
  • Peer Reviewed
• [Journal Article] On 2-adic modular forms (2010)
  • Author(s): M.Koike
  • Journal Title: Kyushu Mathematical Journal Volume: 64 Pages: 199-214
  • NAID: 130000441856
  • Peer Reviewed
• [Journal Article] Toy models for D.H.Lehmer's conjecture (2010)
  • Author(s): E.Bannai, T.Miezaki
  • Journal Title: Journal of Mathematical Society of Japan Volume: 62 Pages: 687-705
  • Peer Reviewed
• [Journal Article] Congruences between extremal modular forms and theta series of special type modulo powers of 2 and 3 (2009)
  • Author(s): M.Koike
  • Journal Title: Kyushu J.Math. 63 Pages: 123-132
  • NAID: 130000135274
  • Peer Reviewed
• [Journal Article] Congruences between extremal modular forms and theta series of special types modulo powers of 2 and 3 (2009)
  • Author(s): M. Koike
  • Journal Title: Kyushu J.Math. 63 Pages: 123-132
  • NAID: 130000135274
  • Peer Reviewed
• [Journal Article] A survey on spherical designs and algebraic combinatorics on spheres (2009)
  • Author(s): E.Bannai, E.Bannai
  • Journal Title: European J.Combinatorics 30 Pages: 1392-1425
  • Peer Reviewed
• [Journal Article] Congruences between extremal modular forms and theta series of special types modulo powers of 2 and 3 (2009)
  • Author(s): M. Koike
  • Journal Title: Kyushu J. Math. 63 Pages: 123-132
  • NAID: 130000135274
  • Peer Reviewed
• [Journal Article] On 2-adic modular forms
  • Author(s): M. Koike
  • Journal Title: Kyushu J.Math. (掲載決定)
  • NAID: 130000441856
  • Peer Reviewed
• [Journal Article] On 2-adic modular forms
  • Author(s): M.Koike
  • Journal Title: Kyushu Journal of Mathematics (掲載決定)
  • NAID: 130000441856
  • Peer Reviewed
• [Presentation] Toy models of D.H.Lehmer's conjecture (2009)
  • Author(s): E.Bannai
  • Organizer: Discrete Geometry and Statistics of Configurations, Steklov Institute of Math.
  • Place of Presentation: Moscow Russia
  • Year and Date: 2009-06-01
• [Presentation] 2進的な保型形式について (2009)
  • Author(s): 小池正夫
  • Organizer: ミニ研究集会「代数的組合せ論」
  • Place of Presentation: 九州大学
  • Year and Date: 2009-03-18
• [Presentation] On the late work of D.G. Higman on combinatorics and groups, Finite groups (2009)
  • Author(s): 坂内英一
  • Organizer: vertex operator algebra and combinatorics
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2009-01-09
• [Presentation] On the late work of D. G. Higman on combinatorics and groups (2009)
  • Author(s): 坂内 英一
  • Organizer: Finite groups, vertex operator algebras, and combinatorics
  • Place of Presentation: 数理研、京都大学
  • Year and Date: 2009-01-09
• [Presentation] Congruences between extremal modular forms and theta series of special type modulo powers of 2 and 3 (2008)
  • Author(s): 小池正夫
  • Organizer: East Asia Number Theory Conference
  • Place of Presentation: KAIST, Daejeon Korea
  • Year and Date: 2008-01-22
• [Presentation] Congruences between extremal modular forms and theta series of special types modulo powers of 2 and 3 (2008)
  • Author(s): M. koike
  • Organizer: East Asia Number Theory Conference
  • Place of Presentation: Kaist, Daejeon, Korea
  • Year and Date: 2008-01-22
• [Presentation] Congruences between extremal modular forms and theta series of special types modulo powers of 2 and 3 (2008)
  • Author(s): 小池 正夫
  • Organizer: East Asia NumLer Theory Conference
  • Place of Presentation: KAIST, Daejeon, Korea
  • Year and Date: 2008-01-22