Deligne's integrality theorem in unequal characteristic and rational points over finite fields; and Appendix (w/Pierre Deligne)

If $V$ is a smooth projective variety defined over a local field $K$ with finite residue field, so that its 'etale cohomology over the algebraic closure $\bar{K}$ is supported in codimension 1, then the mod $p$ reduction of a projective regular model carries a rational point. As a consequence, if the Chow group of 0-cycles of $V$ over a large algebraically closed field is trivial, then the mod $p$ reduction of a projective regular model carries a rational point.