  Over a perfect field $k$ of characteristic $p > 0$, we construct a ``Witt
vector cohomology with compact supports'' for separated $k$-schemes of finite
type, extending (after tensorisation with $\mathbb{Q}$) the classical theory
for proper $k$-schemes. We define a canonical morphism from rigid cohomology
with compact supports to Witt vector cohomology with compact supports, and we
prove that it provides an identification between the latter and the slope $< 1$
part of the former. Over a finite field, this allows one to compute congruences
for the number of rational points in special examples. In particular, the
congruence modulo the cardinality of the finite field of the number of rational
points of a theta divisor on an abelian variety does not depend on the choice
of the theta divisor. This answers positively a question by J.-P. Serre.
