Linking Numbers and Modular Cocycles

It is known that the 3-manifold SL(2,Z)\ SL(2,R) is diffeomorhic to the complement of the trefoil knot in S3. As is shown by E. Ghys the linking number of the trefoil with a modular knot associated to a hyperbolic conjugacy is related to the classical Dedekind symbol. These symbols arose historically in the transformation property of the logarithm of Dedekind’s eta function. In this paper we study the linking numbers between modular knots associated to two hyperbolic conjugacy classes. To this end we give a generalization of the Dedekind symbol. These new symbols appear in the transformation property of analogs of Dedekind’s eta function. They are also related to the special values of a certain Dirichlet series associated to weight 2 modular integrals.