  This letter is an attempt to carry out a first-principle computation in
M-theory using the point of view that the eleven-dimensional membrane gives the
fundamental degrees of freedom of M-theory. Our aim is to derive the exact BPS
$R^4$ couplings in M-theory compactified on a torus $T^{d+1}$ from the toroidal
BPS membrane, by pursuing the analogy with the one-loop string theory
computation. We exhibit an $Sl(3,\Zint)$ modular invariance hidden in the
light-cone gauge (but obvious in the Polyakov approach), and recover the
correct classical spectrum and membrane instantons; the summation measure
however is incorrect. It is argued that the correct membrane amplitude should
be given by an exceptional theta correspondence lifting $Sl(3,\Zint)$ modular
forms to $\exc(\Zint)$ automorphic forms, generalizing the usual theta lift
between $Sl(2,\Zint)$ and $SO(d,d,\Zint)$ in string theory. The exceptional
correspondence $Sl(3)\times E_{6(6)}\subset E_{8(8)}$ offers the interesting
prospect of solving the membrane small volume divergence and unifying membranes
with five-branes.
