\'Etale cohomology, Lefschetz Theorems and Number of Points of Singular Varieties over Finite Fields

TL;DR

This paper extends estimates for counting points on complete intersections over finite fields, generalizing classical inequalities and proving new bounds for singular and affine varieties using étale cohomology and Lefschetz theorems.

Contribution

It introduces a general inequality for point counts on singular complete intersections and proves the Lang-Weil inequality for affine and projective varieties with explicit bounds.

Findings

Extended Deligne's inequality to singular complete intersections

Proved Lang-Weil inequality for affine and projective varieties with explicit bounds

Confirmed a conjecture relating Picard varieties and étale cohomology

Abstract

We prove a general inequality for estimating the number of points of arbitrary complete intersections over a finite field. This extends a result of Deligne for nonsingular complete intersections. For normal complete intersections, this inequality generalizes also the classical Lang-Weil inequality. Moreover, we prove the Lang-Weil inequality for affine as well as projective varieties with an explicit description and a bound for the constant appearing therein. We also prove a conjecture of Lang and Weil concerning the Picard varieties and \'etale cohomology spaces of projective varieties. The general inequality for complete intersections may be viewed as a more precise version of the estimates given by Hooley and Katz. The proof is primarily based on a suitable generalization of the Weak Lefschetz Theorem to singular varieties together with some Bertini-type arguments and the Grothendieck-Lefschetz Trace Formula. We also describe some auxiliary results concerning the \'etale cohomology spaces and Betti numbers of projective varieties over finite fields and a conjecture along with some partial results concerning the number of points of projective algebraic sets over finite fields.