Degeneration of hyperbolic structures on the figure-eight knot complement and points of finite order on an elliptic curve

Abstract

In this paper, we consider a hyperbolic structure on a manifold to be a Riemannian metric of constant sectional curvature −1 which is not necessarily complete. A hyperbolic 3-manifold is an orientable 3-manifold with a hyperbolic structure. It is well known that the complement of the figure-eight knot K in the 3-dimensional sphere S admits a complete, finite volume hyperbolic structure σ∞. By Mostow-Prasad rigidity, such a hyperbolic structure on S − K is unique. Incomplete hyperbolic structures are also of interest. Indeed, Thurston [8] analyzed the flexibility of hyperbolic structures on S − K by allowing incomplete hyperbolic structures. In fact, he showed that there are deformations of σ∞ on S3−K that are not complete hyperbolic structures. Such deformations are holomorphically parametrized by points in an open subset U of a complex affine plane curve C. The curve C and this subset U are given in (1) and (2) below. When the parameter of deformation becomes close to the boundary ∂U or enters C−U , degeneration of hyperbolic structures on S3−K occurs. There are cases in which such degeneration results in closed 3-manifolds obtained by Dehn fillings of S −K with other types of geometric structure. In fact, twenty such cases have been identified. In each case, this resultant closed 3-manifold is one of the following types: a Sol-manifold, a PSL2(R)-manifold, a Haken manifold which is decomposed into a PSL2(R)-manifold and a Euclidean manifold along an embedded torus, or a Euclidean orbifold. The genus of the curve C is one, because C is of degree four and has two ordinary double points. In fact, we give an explicit form of a birational map from C to a non-singular plane cubic curve in Weierstrass form E, which is given in (4) below. The curve E is an example of what is called an ‘elliptic curve defined over Q’. It is well known that any elliptic curve is an abelian group under an addition law. In this paper, we see that there is a concrete