Ampleness of Automorphic Line Bundles on $U(2)$ Shimura Varieties

Let $F$ be a totally real field in which $p$ is unramfied and let $S$ denote the integral model of the Hilbert modular variety with good reduction at $p$. Consider the usual automorphic line bundle $\mathcal{L}$ over $S$. On the generic fiber, it is well known that $\mathcal{L}$ is ample if and only if all the coefficients are positive. On the special fiber, it is conjectured in \citep{Tian-Xiao} that $\mathcal{L}$ is ample if and only if the coefficients satisfy certain inequalities. We prove this conjecture for $U(2)$ Shimura varieties in this paper and deduce a similar statement for Hilbert modular varieties from this.