Hyperbolic structures on manifolds and their deformations

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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.