Knot invariants, modular forms and elliptic Dedekind sums

• Project/Area Number: 22540096
• 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 津田塾大学, 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): 2010-04-01 – 2014-03-31
• Project Status: Completed (Fiscal Year 2013)
• Budget Amount: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000); Fiscal Year 2013: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2012: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2011: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2010: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: 位相幾何学 / 保型形式 / デデキント和 / 周期 / 不変量 / 結び目 / 周期多項式 / 局所変形 / 楕円デデキント和 / 結び目不変量 / L-関数

Research Abstract

The feature of our work is that we study Dedekind symbols in the relation with modular forms. Thus it is a significant step that we have constructed nice bases for the vector spaces of modular forms. We also obtained formulas to express powers of the theta function in terms of Eisenstein series. These formulas give us formulas for the numbers of representations of integers as sums of squares. These seem interesting from number theoretical and geometrical view points.

Research Products

• [Journal Article] Bases for S_k(¥Gamma_1(4)) and formulas for even powers of the Jacobi theta function (2013)
  • Author(s): Shinji Fukuhara, Yifan Yang
  • Journal Title: Int. J. Number Thoery Volume: 9 Pages: 1973-1993
  • Peer Reviewed
• [Journal Article] Bases for S_k(\Gamma_1(4)) and formulas for even powers of the Jacobi theta function (2013)
  • Author(s): Shinji Fukuhara, Yifan Yang
  • Journal Title: Int. J. Number Theory Volume: 9 Issue: 08 Pages: 1973-1993
  • DOI: 10.1142/s1793042113500693
  • Peer Reviewed
• [Journal Article] A basis for S_k(¥Gamma_0(4)) and representations of integers as sums of squares (2012)
  • Author(s): Shinji Fukuhara, Yifan Yang
  • Journal Title: Ramanujan J. Volume: 28 Pages: 25-43
  • Peer Reviewed
• [Journal Article] A basis for the space of modular forms (2012)
  • Author(s): Shinji Fukuhara
  • Journal Title: Acta Arith. Volume: 151 Pages: 421-427
  • Peer Reviewed
• [Journal Article] A basis for S_k(Gamma_0(4)) and representations of integers as sums of squares (2012)
  • Author(s): Shinji Fukuhara
  • Journal Title: Ramanujan J. Volume: 28 Issue: 1 Pages: 25-43
  • DOI: 10.1007/s11139-011-9341-y
  • Peer Reviewed
• [Journal Article] A basis for the space of modular forms (2012)
  • Author(s): Shinji Fukuhara
  • Journal Title: Acta Arithmetica Volume: 151 Pages: 421-427
  • Peer Reviewed
• [Journal Article] A basis for S_k(Gamma_0(4)) and representations of integers as sums of squares (2012)
  • Author(s): Shinji Fukuhara, Yifan Yang
  • Journal Title: Ramanujan J Volume: 28 Pages: 25-43
  • Peer Reviewed
• [Journal Article] Twisted Hecke L-values and period polynomials (2010)
  • Author(s): Shinji Fukuhara, Yifan Yang
  • Journal Title: J. Number Theory Volume: 130 Pages: 976-999
  • Peer Reviewed
• [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. Volume: 346 Pages: 769-794
  • Peer Reviewed
• [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. Volume: 346 Pages: 769-794
  • Peer Reviewed
• [Journal Article] Twisted Hecke L-values and period polynomials (2010)
  • Author(s): Shinji Fukuhara
  • Journal Title: J.Number Theory Volume: 130 Pages: 976-999
  • Peer Reviewed