Sylvy Anscombe — The model theory of perfectoid fields, after Jahnke and Kartas

samedi 31 janv. 2026, 10:00 → 11:00 Europe/Paris

Amphithéâtre Charles Hermite (IHP - Bâtiment Borel)

The ``AKE principles'' of Ax, Kochen, and Ershov,
have at their heart an axiomatization of the first-order theories of henselian valued fields of residue characteristic zero.
This work yields the famous asymptotic transfer principle for first-order sentences between the local fields of mixed and positive characteristics. Fontaine's tilt of a ring $R$ is the inverse limit $R^{\flat}$ of copies of the ring $R/p$ with transition maps given by Frobenius $x\mapsto x^{p}$, and in celebrated work, Scholze prove that tilting is then an equivalence
between the category of perfectoid rings over $K$ and the category of perfectoid rings over $K^{\flat}$.

The aim of this talk is to explain a new contribution of 
Jahnke and Kartas, who have introduced an elementary class $\mathcal{C}$ of valued fields, containing the perfectoid fields, and proved a range of AKE priciples in this context, including for the first time valued fields admitting nontrivial defect extensions. From their work one obtains a non-standard version of the Almost Purity theorem of Scholze, and the Fontaine--Wintenberger theorem. This should be seen in the context of Kuhlmann's tame valued fields are another setting in which there is an Ax--Kochen/Ershov-like theory, as well as important earlier work of Kuhlmann and Rzepka on the valuation theory of deeply ramified fields, and of Kartas which showed the transfer of decidability problems via tilting. More recently,  Rideau-Kikuchi, Scanlon, and Simon have formalized the tilt as a bi-interpretation in continuous logic.