Linking numbers and modular forms for the triangle groups

Abstract: The coset space SL(2,Z)\SL(2,R) is diffeomorphic to the complement of the trefoil knot in the 3-sphere. For each hyperbolic matrix in SL(2,Z) or real quadratic irrationality, we can naturally construct a simple closed orbit in this space, which is called a modular knot. At ICM 2006, Ghys showed a beautiful relation that the linking number of the modular knot and the missing trefoil is equal to the Rademacher invariant. This invariant classically appears in the transformation law of the Dedekind eta function, and has the expression as a geodesic cycle integral of the Eisenstein series of weight 2. In this talk, we generalize Ghys’ result to the knot complement of the torus knots. To get a similar relation between linking numbers and cycle integrals, modular forms for triangle groups have crucial roles. This is joint work with Jun Ueki (Tokyo Denki University).

number theory

Audience: researchers in the topic

Japan Europe Number Theory Exchange Seminar

Series comments: The purpose of the Japan Europe Number Theory Exchange Seminar is to give a opportunity for researchers in Japan and Europe to exchange their research projects by giving short talks (30 min). The target audience are researchers of any level in the area of number theory.

Start for Fall 2021: 26th October