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

TL;DR

This paper proves a congruence relation for the number of rational points over finite fields on regular models of algebraic varieties of Hodge type at least 1, using Witt cohomology and trace morphisms.

Contribution

It establishes a new congruence for rational points on certain algebraic varieties over finite fields, based on vanishing theorems for Witt cohomology and trace morphism constructions.

Findings

Number of rational points satisfies |X(k')| ≡ 1 mod |k'|.

Vanishing theorem for Witt cohomology groups H^q(X_k, W O_{X_k,Q}) for q > 0.

Construction of trace morphisms between Witt cohomologies of special fibres.

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 $|k'|$. Thanks to \cite{BBE07}, we deduce such congruences from a vanishing 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 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 $f : Y \to X$.