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

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