On rational singularities and counting points of schemes over finite rings

Abstract

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 $X_{\mathbb{Q}}=X{\times }_{Spec\mathbb{Z}}Spec\mathbb{Q}$ is a local complete intersection, we show that the boundedness of $\left|X\left(\mathbb{Z}∕p^n\mathbb{Z}\right)\right|∕p^{ndim{X}_{\mathbb{Q}}}$ 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 recent result of Aizenbud and Avni.