On rational singularities and counting points of schemes over finite rings

We study the connection between the singularities of a finite type $\mathbb{Z}$-scheme X and the asymptotic point count of X over various finite rings. In particular, if the generic fiber $X_{\mathbb{Q}}=X{\times }_{\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}\mathbb{Z}}\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}\mathbb{Q}$ is a local complete intersection, we show that the boundedness of $\frac{\mid X(\mathbb{Z}\mathrm{/}{p}^n\mathbb{Z})\mid }{p^{n\mathrm{d}\mathrm{i}\mathrm{m}{X}_{\mathbb{Q}}}}$ in p and n is in fact equivalent to the condition that $X_{\mathbb{Q}}$ is reduced and has rational singularities. This paper completes a result of Aizenbud and Avni.