Rationally connected varieties over finite fields

Abstract

Let $X$ be a geometrically rational (or, more generally, separably rationally connected) variety over a finite field $K$. We prove that if $K$ is large enough, then $X$ contains many rational curves defined over $K$. As a consequence we prove that $R$-equivalence is trivial on $X$ if $K$ is large enough. These results imply the following conjecture of J.-L. Colliot-Thélène: Let $Y$ be a rationally connected variety over a number field $F$. For a prime $P$, let $Y_P$ denote the corresponding variety over the local field $F_P$. Then, for almost all primes $P$, the Chow group of 0-cycles on $Y_P$ is trivial and $R$-equivalence is trivial on $Y_P$.