  We calculate the elliptic genus of two dimensional abelian gauged linear
sigma models with (2,2) supersymmetry using supersymmetric localization. The
matter sector contains charged chiral multiplets as well as Stueckelberg fields
coupled to the vector multiplets. These models include theories that flow in
the infrared to non-linear sigma models with target spaces that are non-compact
Kahler manifolds with U(N) isometry and with an asymptotically linear dilaton
direction. The elliptic genera are the modular completions of mock Jacobi forms
that have been proposed recently using complementary arguments. We also compute
the elliptic genera of models that contain multiple Stueckelberg fields from
first principles.
