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

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 proved in (Esnault, 2007, Theorem 1.1). If the model $\mathcal {X}$ is regular, one has a congruence $|\mathcal {X}(k)|\equiv 1$ modulo $|k|$ for the number of $k$-rational points (Esnault, 2006, Theorem 1.1). The congruence is violated if one drops the regularity assumption.