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Let $R$ be a discrete valuation ring of mixed characteristics $(0,p)$, withfinite residue field $k$ and fraction field $K$, let $k'$ be a finite extensionof $k$, and let $X$ be a regular, proper and flat $R$-scheme, with genericfibre $X_K$ and special fibre $X_k$. Assume that $X_K$ is geometricallyconnected and of Hodge type $\geq 1$ in positive degrees. Then we show that thenumber of $k'$-rational points of $X$ satisfies the congruence $|X(k')| \equiv1$ mod $|k'|$. Thanks to \cite{BBE07}, we deduce such congruences from avanishing theorem for the Witt cohomology groups $H^q(X_k, W\sO_{X_k,\Q})$, for$q > 0$. In our proof of this last result, a key step is the construction of atrace morphism between the Witt cohomologies of the special fibres of two flatregular $R$-schemes $X$ and $Y$ of the same dimension, defined by a surjectiveprojective morphism $f : Y \to X$.

Rational points over finite fields for regular models of algebraic varieties of Hodge type $\geq 1$