Abstract

In 1939 Rademacher derived a conditionally convergent series expression for the elliptic modular invariant. This motivated investigations by various authors in to the problem of constructing modular functions, and even modular forms of negative weight, for discrete groups of isometries of the hyperbolic plane.

We will describe a generalization of Rademacher’s construction that furnishes spanning sets for a certain subspace of the space of meromorphic modular forms of even integral weight, for any group commensurable with the modular group. In the course of this we are led to an analytic continuation of the elliptic modular invariant, and an association of Dirichlet series to groups commensurable with the modular group.

The expressions we obtain behave well under the actions of Hecke operators and have simple branching rules. These facts lead to applications in monstrous moonshine and three dimensional quantum gravity.