Let f : X → 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 ) → Y ( k v ) to be surjective for almost all places v of k. This generalises 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.