Pseudo-split fibres and arithmetic surjectivity

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.