Abstract: We study first-order properties in henselian valued fields building on classical work by Ax-Kochen and Ershov. The guiding principle is that the first-order theories of (sufficiently) well-behaved henselian valued fields are governed by those of the residue field and of the value group. We show Ax-Kochen/Ershov type theorems for unramified henselian valued fields with imperfect residue field and for perfectoid fields. As a consequence, we obtain a generalization of the Theorem of Fontaine-Wintenberger, allowing the transfer a multitude of elementary properties between a perfectoid field of characteristic 0 and its tilt (and vice versa).
The talk contains results obtained in joint work with Sylvy Anscombe and Konstantinos Kartas.
univienna.zoom.us/j/63166383248