  We study a decoupling limit of M-theory where the three-form gauge potential
becomes critical. This limit leads to nonrelativistic M-theory coupled to a
non-Lorentzian spacetime geometry. Nonrelativistic M-theory is U-dual to
M-theory in the discrete light cone quantization, a non-perturbative approach
related to the Matrix theory description of M-theory. We focus on the
compactification of nonrelativistic M-theory over a two-torus that exhibits
anisotropic behaviors due to the foliation structure of the spacetime geometry.
We develop a frame covariant formalism of the toroidal geometry, which provides
a geometrical interpretation of the recently discovered polynomial realization
of SL(2,Z) duality in nonrelativistic type IIB superstring theory. We will show
that the nonrelativistic IIB string background fields transform as polynomials
of an effective Galilean "boost velocity" on the two-torus. As an application,
we construct an action principle describing a single M5-brane in
nonrelativistic M-theory and study its compactification over the anisotropic
two-torus. This procedure leads to a D3-brane action in nonrelativistic IIB
string theory that makes the SL(2,Z) invariance manifest in the polynomial
realization.
