  We present a 1-loop toroidal membrane winding sum reproducing the conjectured
$M$-theory, four-graviton, eight derivative, $R^4$ amplitude. The $U$-duality
and toroidal membrane world-volume modular groups appear as a Howe dual pair in
a larger, exceptional, group. A detailed analysis is carried out for $M$-theory
compactified on a 3-torus, where the target-space $Sl(3,\Zint)\times
Sl(2,\Zint)$ $U$-duality and $Sl(3,\Zint)$ world-volume modular groups are
embedded in $E_{6(6)}(\Zint)$. Unlike previous semi-classical expansions,
$U$-duality is built in manifestly and realized at the quantum level thanks to
Fourier invariance of cubic characters. In addition to winding modes, a pair of
new discrete, flux-like, quantum numbers are necessary to ensure invariance
under the larger group. The action for these modes is of Born-Infeld type,
interpolating between standard Polyakov and Nambu-Goto membrane actions. After
integration over the membrane moduli, we recover the known $R^4$ amplitude,
including membrane instantons. Divergences are disposed of by trading the
non-compact volume integration for a compact integral over the two variables
conjugate to the fluxes -- a constant term computation in mathematical
parlance. As byproducts, we suggest that, in line with membrane/fivebrane
duality, the $E_6$ theta series also describes five-branes wrapped on $T^6$ in
a manifestly U-duality invariant way. In addition we uncover a new action of
$E_6$ on ten dimensional pure spinors, which may have implications for ten
dimensional super Yang--Mills theory. An extensive review of $Sl(3)$
automorphic forms is included in an Appendix.
