  We consider the Chern-Simons-Schr\"odinger model in 1+2 dimensions, and prove
scattering for small solutions of the Cauchy problem in the Coulomb gauge. This
model is a gauge covariant Schr\"odinger equation, with a potential decaying
like $r^{-1}$ at infinity. To overcome the difficulties due to this long range
decay we start by performing $L^2$-based estimates covariantly. This gives
favorable commutation identities so that only curvature terms, which decay
faster than $r^{-1}$, appear in our weighted energy estimates. We then select
the Coulomb gauge to reveal a genuinely cubic null structure, which allows us
to show sharp decay by Fourier methods.
