Study on Hecke operators for cusp forms and topological invariants

• Project/Area Number: 19540101
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Geometry
• Research Institution: Tsuda College
• Principal Investigator: FUKUHARA Shinji Tsuda College, 学芸学部, 教授 (20011687)
• Co-Investigator(Kenkyū-buntansha): MIYAZAWA Haruko 津田塾大学, 計数研, 研究員 (40266276)
• Project Period (FY): 2007 – 2009
• Project Status: Completed (Fiscal Year 2009)
• Budget Amount: ¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000); Fiscal Year 2009: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2008: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2007: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: 位相幾何 / 保型形式 / 位相不変量 / 結び目 / デデキント和 / 尖点形式 / ヘッケ作用素 / 周期多項式 / 周期 / ヘッケ作用

Research Abstract

Cusp forms on the complex upper half plane have been studied for the connection with number theory. The feature of our research is focusing on natural correspondences between cusp forms, periods and Dedekind symbols. We have introduced the notion of elliptic Apostol-Dedekind sums and showed these sums generate Dedekind symbols with polynomial reciprocity laws. We also studied how we can apply these sums to knot theory.

Research Products

• [Journal Article] The elliptic Apostol-Dedekind sums generate odd Dedekind symbols with Laurent polynomial reciprocity laws (2010)
  • Author(s): Shinji Fukuhara
  • Journal Title: Math. Ann 346 Pages: 769-794
  • Peer Reviewed
• [Journal Article] Twisted Hecke L-value and period polynomials (2010)
  • Author(s): Shinji Fukuhara
  • Journal Title: J.Number Theory 130 Pages: 976-999
  • Peer Reviewed
• [Journal Article] Period polynomials and explicit formulas for Hecke operators on \Gamma_0(2) (2009)
  • Author(s): Shinji Fukuhara, Yifan Yang
  • Journal Title: Proc. Cambridge Philos. Soc. 146 Pages: 321-350
  • Peer Reviewed
• [Journal Article] C_n-moves and V_n-equivalence for links (2009)
  • Author(s): Haruko Aida Miyazawa
  • Journal Title: Tokyo J. Math 32 Pages: 381-393
  • Peer Reviewed
• [Journal Article] SC_n-moves and the (n+1)-st coefficients of the Conway polynomials of links (2009)
  • Author(s): Haruko Aida Miyazawa
  • Journal Title: Tokyo J. Math 32 Pages: 395-408
  • Peer Reviewed
• [Journal Article] C_n-moves and V_n-equivalence for links (2009)
  • Author(s): Haruko Aida Miyazawa
  • Journal Title: Tokyo J.Math. 32 Pages: 381-393
  • Peer Reviewed
• [Journal Article] SC_n-moves and the (n+1)-st coefficients of the Conway polynomials of links (2009)
  • Author(s): Haruko Aida Miyazawa
  • Journal Title: Tokyo J.Math. 32 Pages: 395-408
  • Peer Reviewed
• [Journal Article] Period polynomials and explicit formulas for Hecke operators on \Garama_0(2) (2009)
  • Author(s): Shinji Fukuhara Yifan Yang
  • Journal Title: Math. Proc. Cambridge Philos. Soc. 146 Pages: 321350-321350
  • Peer Reviewed
• [Journal Article] Dedekind symbols with plus reciprocity laws (2008)
  • Author(s): Shinji Fukuhara
  • Journal Title: J. Number Theory 128 Pages: 781-795
  • Peer Reviewed
• [Journal Article] Explicit formulas for Hecke operators on cusp forms, Dedekind symbols and period polynomials (2007)
  • Author(s): Shinji Fukuhara
  • Journal Title: J. Reine Angew. Math 607 Pages: 163-216
  • Peer Reviewed
• [Journal Article] Explicit formulas for Hecke operators on cusp forms, Dedekind symbols and period polynomials (2007)
  • Author(s): Shinji Fukuhara
  • Journal Title: J. refine angew. Math. 607 Pages: 163-216
  • Peer Reviewed
• [Journal Article] C_n-moves and V_n-equivalence for links
  • Author(s): Haruko Aida Miyazawa
  • Journal Title: Tokyo J. Math, (to appear)
  • Peer Reviewed
• [Journal Article] SC_n-moves and the (n+1) -st coefficients of the Conway polynomials of links
  • Author(s): Haruko Aida Miyazawa
  • Journal Title: Tokyo J. Math, (to appear)
  • Peer Reviewed