S and U - duality Constraints on IIB S - matrices

S and U-duality dictate that graviton scattering amplitudes in IIB superstring theory be automorphic functions on the appropriate fundamental domain which describe the inequiv-alent vacua of (compactified) theories. A constrained functional form of graviton scattering is proposed using Eisenstein series and their generalizations compatible with: a) two-loop supergravity, b) genus one supersting theory, c) the perturbative coupling dependence of the superstring, and d) with the unitarity structure of the massless modes. The form has a perturbative truncation in the genus expansion at a given order in the derivative expansion. Comparisons between graviton scattering S-matrices and effective actions for the first quan-tized superstring are made at the quantum level. Possible extended finiteness properties of maximally extended quantum supergravity theories in different dimensions is implied by the perturbative truncation of the functional form of graviton scattering in IIB superstring theory.