  The low-momentum expansion of the two-loop four-graviton scattering amplitude
in eleven-dimensional supergravity compactified on a circle and a two-torus is
considered up to terms of order S^6R^4 (where S is a Mandelstam invariant and R
is the linearized Weyl curvature). In the case of the toroidal compactification
the coefficient of each term in the low energy expansion is generically a sum
of a number of SL(2,Z)-invariant functions of the complex structure of the
torus. Each such function satisfies a separate Poisson equation on moduli space
with particular source terms that are bilinear in coefficients of lower order
terms, consistent with qualitative arguments based on supersymmetry. Comparison
is made with the low-energy expansion of type II string theories in ten and
nine dimensions. Although the detailed behaviour of the string amplitude is not
generally expected to be reproduced by supergravity perturbation theory to all
orders, for the terms considered here we find agreement with direct results
from string perturbation theory. These results point to a fascinating pattern
of interrelated Poisson equations for the IIB coefficients at higher orders in
the momentum expansion which may have a significance beyond the particular
methods by which they were motivated.
