Congruence for rational points over finite fields and coniveau over local fields

TL;DR

This paper explores the relationship between the cohomological support conditions of varieties over local fields and the existence of rational points over finite fields, establishing congruences and conditions for their existence.

Contribution

It extends previous results by linking cohomological support in codimension to rational point existence and congruences, emphasizing the role of regularity in these properties.

Findings

Varieties with cohomology supported in codimension ≥ 1 have rational points over finite fields.

Regular models satisfy a congruence for the number of rational points modulo the size of the residue field.

Dropping regularity can violate the established congruence.

Abstract

If the $\ell$-adic cohomology of a projective smooth variety, defined over a local field $K$ with finite residue field $k$, is supported in codimension $\ge 1$, then every model over the ring of integers of $K$ has a $k$-rational point. For $K$ a $p$-adic field, this is math/0405318, Theorem 1.1. If the model $\sX$ is regular, one has a congruence $|\sX(k)|\equiv 1 $ modulo $|k|$ for the number of $k$-rational points 0704.1273, Theorem 1.1. The congruence is violated if one drops the regularity assumption.