Periodic Cubic Hyperbolic Schrodinger equation on $T^2$

We consider the cubic Hyperbolic Schrodinger equation eqref{eq:nls} on torus $T^2$. We prove that sharp $L^4$ Strichartz estimate, which implies that eqref{eq:nls} is analytic locally well-posed in in $H^s(T^2)$ with $s>1/2$, meanwhile, the ill-posedness in $H^s(T^2)$ for $s<1/2$ is also obtained. The main difficulty comes from estimating the number of representations of an integer as a difference of squares.