Rational Singularities and Rational Points

Abstract

If $X$ is a projective, geometrically irreducible variety defined over a finite field $\F_q$, such that it is smooth and its Chow group of 0-cycles fulfills base change, i.e. $CH_0(X\times_{\F_q}\bar{\F_q(X)})=\Q$, then the second author's theorem asserts that its number of rational points satisfies $|X(\F_q)| \equiv 1$ modulo $q$. If $X$ is not smooth, this is no longer true. Indeed J. Koll\'ar constructed an example of a rationally connected surface over $\F_q$ without any rational points. Based on the work by Berthelot-Bloch and the second author computing the slope $<1$ piece of rigid cohomology, we define a notion of Witt-rational singularities in characteristic $p>0$. The theorem is then that if $X/\F_q$ is a projective, geometrically irreducible variety, such that it has Witt-rational singularities and its Chow group of 0-cycles fulfills base change, then $|X(\F_q)| \equiv 1$ modulo $q$.