  Let $f: X \to Y$ be a dominant morphism of smooth, proper and geometrically
integral varieties over a number field $k$, with geometrically integral generic
fibre. We give a necessary and sufficient geometric criterion for the induced
map $X(k_v) \to Y(k_v)$ to be surjective for almost all places $v$ of $k$. This
generalizes a result of Denef which had previously been conjectured by
Colliot-Th\'el\`ene, and can be seen as an optimal geometric version of the
celebrated Ax-Kochen theorem.
