  A theoretical foundation for a generalization of the elliptic difference
Painlev\'e equation to higher dimensions is provided in the framework of
birational Weyl group action on the space of point configurations in general
position in a projective space. By introducing an elliptic parametrization of
point configurations, a realization of the Weyl group is proposed as a group of
Cremona transformations containing elliptic functions in the coefficients. For
this elliptic Cremona system, a theory of $\tau$-functions is developed to
translate it into a system of bilinear equations of Hirota-Miwa type for the
$\tau$-functions on the lattice.
