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 $X_{\mathbb{Q}}=X{\times }_{\mathrm{Spec}\mathbb{Z}}\mathrm{Spec}\mathbb{Q}$ is a local complete intersection, we show that the boundedness of $\frac{|X(\mathbb{Z}/p^n\mathbb{Z})|}{p^{n\dim X_{\mathbb{Q}}}}$ 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.

17 pages

References in corpus (2)

• Counting points of schemes over finite rings and counting representations of arithmetic lattices
• On singularity properties of convolutions of algebraic morphisms

Cited by in corpus (3)

• On singularity properties of convolutions of algebraic morphisms -- the general case (with an appendix joint with Gady Kozma)
• Rational singularities for moment maps of totally negative quivers
• A number theoretic characterization of $E$-smooth and (FRS) morphisms: estimates on the number of $\mathbb{Z}/p^k\mathbb{Z}$-points