Differential equations satisfied by modular forms

• Project/Area Number: 17K14213
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Basic analysis
• Research Institution: Niihama National College of Technology
• Principal Investigator: Matsuda Kazuhide 新居浜工業高等専門学校, 数理科, 准教授 (20550106)
• Project Period (FY): 2017-04-01 – 2021-03-31
• Project Status: Completed (Fiscal Year 2020)
• Budget Amount: ¥2,210,000 (Direct Cost: ¥1,700,000、Indirect Cost: ¥510,000); Fiscal Year 2020: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2019: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2018: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2017: ¥260,000 (Direct Cost: ¥200,000、Indirect Cost: ¥60,000)
• Keywords: モジュラー形式 / テータ関数 / 有理指標 / Ramanujan の微分方程式 / cubic theta 関数 / Ramanujan の方程式 / 微分体 / Jacobi の微分公式

Outline of Final Research Achievements

We concretely construct examples of differential equations satisfied by modular forms. In particular, we treat modular forms of levels 3-6. We obtain applications to quadratic forms. Moreover, we discover high level versions of Jacobi's derivative formula and Jacobi's quartic identity.

Academic Significance and Societal Importance of the Research Achievements

モジュラー形式が満たす微分方程式は、これまで可積分系および整数論の両分野から研究されてきた。本研究により新たな例が得られたことにより、両分野の新たな発展が期待できる。

Research Products

• [Journal Article] Differential equations involving cubic theta functions and Eisenstein series. (2020)
  • Author(s): Matsuda, Kazuhide
  • Journal Title: Osaka J. Math. Volume: 57 Pages: 521-542
  • NAID: 120006871530
• [Journal Article] Note on a theorem of Farkas and Kra (2020)
  • Author(s): Matsuda Kazuhide
  • Journal Title: The Ramanujan Journal Volume: 53 Issue: 2 Pages: 319-356
  • DOI: 10.1007/s11139-020-00301-x
• [Journal Article] On certain quaternary quadratic forms. (2018)
  • Author(s): K. Matsuda
  • Journal Title: Integers Volume: 18
  • Peer Reviewed / Open Access
• [Journal Article] Analogues of Jacobi's derivative formula II. (2017)
  • Author(s): Kazuhide Matsuda
  • Journal Title: Ramanujan J. Volume: 44 Issue: 1 Pages: 37-62
  • DOI: 10.1007/s11139-016-9803-3
  • Peer Reviewed