  The construction of $SL(2,Z)$ invariant amplitudes that generalizethe
Virasoro amplitude is investigated in detail. We describe a number of
mathematical properties that characterize the simplest example, and present
pieces of evidence that it represents the tree-level four-graviton scattering
amplitude in membrane theory on ${\bf R}^9\times T^2$ in the limit that the
torus area goes to zero. In particular, we show that the poles of the $S$-dual
amplitude are in precise correspondence with the states of membrane theory that
survive in the type IIB limit. These are shown to be the states that span the
Cartan subspaces of area preserving diffeomorphisms of the 2-torus; all other
states become infinitely massive, and membrane world-volume theory acquires the
structure of a free theory.
