The non-abelian topological torsion and the Iwasawa polynomial

• Project/Area Number: 22540068
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Geometry
• Research Institution: Chiba University
• Principal Investigator: SUGIYAMA Ken-ichi 千葉大学, 理学(系)研究科(研究院), 教授 (90206441)
• Project Period (FY): 2010-04-01 – 2015-03-31
• Project Status: Completed (Fiscal Year 2014)
• Budget Amount: ¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000); Fiscal Year 2014: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); 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: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
• Keywords: 結び目群 / 双曲結び目 / ３次元双曲多様体 / A多項式 / 結び目 / 不変量 / Weil予想 / 虚数乗法 / 基本群 / 超幾何方程式 / 特性曲線 / Jones汎関数 / 楕円曲線 / L関数 / 特殊値 / 岩澤理論 / Birch & Swinnerton-Dyer予想 / Tate予想 / 測度 / オイラー系 / 岩澤多項式 / 非可換類体論 / 写像類群 / 多重対数関数 / Magnus展開

Outline of Final Research Achievements

The geometric structure of the complement of a knot in the three dimensional sphere is determined by its fundamental group. The group is called the knot group. If the complement admits a complete hyperbolic structure of finite volume the knot group is nothing but the Kleinian group which is a discrete subgroup of the 2×2 special linear group. It is an important object both in geometry and in number theory. Our research is to investigate how the knot group changes if one alters a crossing of a knot. If moreover the complement admits a complete hyperbolic metric of finite volume we have also studied the change of the hyperbolic structure. We also study a similarity between the Alexander polynomial and the Hasse-Weil congruent zeta function.

Research Products

• [Journal Article] On a generalization of Deuring's results (2014)
  • Author(s): Sugiyama, K
  • Journal Title: Finite Fields and Their Applications Volume: 26C Pages: 69-85
  • DOI: 10.1016/j.ffa.2013.11.004
  • Peer Reviewed
• [Presentation] RamanujanグラフとHasse-Weil合同ゼータ関数 (2015)
  • Author(s): 杉山健一
  • Organizer: 数理学談話会
  • Place of Presentation: 金沢大学理工研究領域数物科学系
  • Year and Date: 2015-02-04
• [Presentation] On a generalization of Deuring's results (2014)
  • Author(s): Ken-ichi Sugiyama
  • Organizer: Low dimensional topology and number theory VI
  • Place of Presentation: Soft Research Park Center, Fukuoka
• [Presentation] 高次元線型符号について (2012)
  • Author(s): 杉山健一
  • Organizer: 金沢大学数学教室談話会
  • Place of Presentation: 金沢大学理学部
  • Year and Date: 2012-02-03