  We construct a rational $T^2$-equivariant elliptic cohomology theory for the
2-torus $T^2$, starting from an elliptic curve C over the complex numbers and a
coordinate data around the identity. The theory is defined by constructing an
object $EC_{T^2}$ in the algebraic model category $dA(T^2)$, which by Greenlees
and Shipley is Quillen-equivalent to rational $T^2$-spectra. This result is a
generalization to the 2-torus of the construction [Gre05] for the circle. The
object $EC_{T^2}$ is directly built using geometric inputs coming from the
Cousin complex of the structure sheaf of the surface CxC.
