Rademacher sums, moonshine and gravity

摘要

In 1939 Rademacher derived a conditionally convergent series expression for the elliptic modular invariant, and used this expres-sion - the first Rademacher sum-to verify its modular invari-ance. By generalizing Rademacher's approach we construct bases for the spaces of automorphic integrals of arbitrary even integer weight, for groups commensurable with the modular group. Our methods provide explicit expressions for the Fourier expansions of the Rademacher sums we construct at arbitrary cusps, and illu-minate various aspects of the structure of the spaces of automor-phic integrals, including the actions of Hecke operators. We give a moduli interpretation for a class of groups commensurable with the modular group which includes all those that are associated to the Monster via monstrous moonshine.