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

Abstract

Let $R$ be a discrete valuation ring of mixed characteristics $(0,p)$, with finite residue field $k$ and fraction field $K$, let $k’$ be a finite extension of $k$, and let $X$ be a regular, proper and flat $R$-scheme, with generic fibre $X_K$ and special fibre $X_k$. Assume that $X_K$ is geometrically connected and of Hodge type $\geq 1$ in positive degrees. Then we show that the number of $k’$-rational points of $X$ satisfies the congruence $|X(k’)| \equiv 1$ mod $|k’|$. We deduce such congruences from a vanishing theorem for the Witt cohomology groups $H^q(X_k, W\mathcal{O}_{X_k,\mathbb{Q}})$ for $q > 0$. In our proof of this last result, a key step is the construction of a trace morphism between the Witt cohomologies of the special fibres of two flat regular $R$-schemes $X$ and $Y$ of the same dimension, defined by a surjective projective morphism $f : Y \to X$.