Hyperbolic structures on manifolds and their deformations

本文主要研究了具有非空奇异集的三维双曲锥流形M的变形问题。首席研究员Fujii和研究员Ochiai构建了一个算法，用于求解描述M.该算法发表在Interdisciplinary Information Sciences 9（2003）期刊上。Fujii发现了双曲结上双曲结构的退化与椭圆曲线上某些有理点之间的关系。这一结果发表在京都大学数学杂志45（2005）上.藤井成功地在椭圆曲线的模曲线上画出了这样一个有理点。这一结果在2004年12月东京工业大学的“Riemann Surfaces and Discontinuous Groups”会议上发表。此外，Fujii发现了这些有理点位于构成模曲线某些基本区域边界的某个圆上的可能性。这一结果在2005年11月大坂产业大学的“Topology and Computer 2005”会议上发表。Ue证明了Seifert三维流形的Fukumoto-Furuta不变量与Neumann-Siebenmann不变量一致，并证明了它的自旋有理同调配边不变量。UE获得的条件塞弗特3-流形获得Dehn手术结在3球。调查齐藤成功地给一个具体的描述可接受的陈述非阿基米德当地领域的交织运营商。

The main purpose of this research was to study deformations of a 3-dimensional hyperbolic cone manifold M with non-empty singular set. The head investigator Fujii and the investigator Ochiai constructed an algorithm for solving ordinary differential equations of Fuchsian type which describe deformations of M. This algorithm was reported in the journal, Interdisciplinary Information Sciences 9(2003). Fujii found some relation between the degeneration of hyperbolic structures on a hyperbolic knot and some rational points of an elliptic curve. This result was reported in the journal, J.Math.Kyoto Univ.45(2005). Fujii succeeded in drawing such a rational point on the modular curve of the elliptic curve. This result was reported in the conference, "Riemann Surfaces and Discontinuous Groups" at Tokyo Institute of Technology in December 2004. Furthermore, Fujii found the possibility that these rational points are located on some circle which constitute the boundary of some fundamental region of the modular curve. This result was reported at the conference, "Topology and Computer 2005", at Osaka Sangyo University in November 2005.The investigator Ue proved that the Fukumoto-Furuta invariant for Seifert 3-manifolds coincides with the Neumann-Siebenmann invariant, and also proved its spin rational homology cobordism invariance. Ue obtained the conditions for Seifert 3-manifolds to be obtained by Dehn surgery on knots in the 3-sphere. The investigator Saito succeeded to give a concrete description of admissible representations on non-Archimedes local fields in terms of intertwining operators.