  S and U-duality dictate that graviton scattering amplitudes in IIB
superstring theory be automorphic functions on the appropriate fundamental
domain which describe the inequivalent 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 superstring 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 quantized 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.
