| Résume | We establish certain connections between perfectoid geometry and model theory of henselian fields. On one hand, we prove an Ax-Kochen/Ershov principle for perfectoid fields (and generalizations thereof). As an application, we show that the perfect hull of Fp(t)^h is an elementary substructure of the perfect hull of Fp((t)). On the other hand, we prove a model theoretic generalization of the Fontaine-Wintenberger theorem. This reveals that the relation between a perfectoid field and its tilt is analogous to that of well-understood valued fields (viz., henselian defectless with divisible value group) and their residue fields. As a new arithmetic application, we provide some of the first few examples in mixed characteristic verifying the Lang-Manin conjecture on the existence of rational points on nearly rational varieties over C1 fields. Joint work with Franziska Jahnke. |
