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The optimal $L^4$-Strichartz estimate for the Schr{\"o}dinger equation on thetwo-dimensional rational torus $\mathbb{T}^2$ is proved, which improves anestimate of Bourgain. A new method based on incidence geometry is used. Theapproach yields a stronger $L^4$ bound on a logarithmic time scale, whichimplies global existence of solutions to the cubic (mass-critical) nonlinearSchr\"odinger equation in $H^s(\mathbb{T}^2)$ for any $s>0$ and data which issmall in the critical norm.

Strichartz estimates and global well-posedness of the cubic NLS on $\mathbb{T}^{2}$