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 $X_{ \mathbb{ Q}}=X\times_{ \mathrm{ Spec}\mathbb{ Z}}\mathrm{ Spec}\mathbb{ Q}$ is a local complete intersection, we show that the boundedness of $\frac{ \left|X(\mathbb{ Z}/p^{ n}\mathbb{ Z})\right|}{ p^{ n\mathrm{ dim}X_{ \mathbb{ Q}}}}$ 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.
Categories: math.AG math.NT