# Coniveau over $p$-adic fields and points over finite fields

## TL;DR

The paper proves that certain geometric conditions on varieties over p-adic fields guarantee the existence of rational points over finite residue fields, refining previous results by weakening regularity assumptions.

## Contribution

It improves earlier results by showing that support in codimension ≥ 1 suffices for the existence of rational points, reducing regularity requirements.

## Key findings

- Supports in codimension ≥ 1 imply existence of rational points over finite fields.
- Refines previous results by weakening regularity assumptions.
- Provides a more general criterion for rational points over finite fields.

## Abstract

If the $\ell$-adic cohomology of a projective smooth variety, defined over a $\frak{p}$-adic field $K$ with finite residue field $k$, is supported in codimension $\ge 1$, then any model over the ring of integers of $K$ has a $k$-rational point. This slightly improves our earlier result math/0405318: we needed there the model to be regular (but then our result was more general: we obtained a congruence for the number of points, and $K$ could be local of characteristic $p>0$).