Pseudo-split fibres and arithmetic surjectivity

TL;DR

This paper provides a precise geometric criterion for the surjectivity of local points maps induced by dominant morphisms between certain algebraic varieties over number fields, generalizing previous results and connecting to the Ax-Kochen theorem.

Contribution

It establishes a necessary and sufficient geometric condition for surjectivity of local points maps, extending Denef's result and confirming a conjecture by Colliot-Thélène.

Findings

Provides a geometric criterion for surjectivity at almost all places

Generalizes Denef's earlier result on local points

Connects to the Ax-Kochen theorem as an optimal geometric version

Abstract

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.