Algèbre et géométrie

Hensel minimality is a new axiomatic framework for doing tame geometry in non-Archimedean fields, aimed to mimic o-minimality. It is designed to be broadly applicable while having strong consequences. We will give a general overview of the theory of Hensel minimality. Afterwards, we discuss arithmetic applications to counting rational points on definable sets in valued fields. This is partially joint work with R. Cluckers, I. Halupczok and S. Rideau-Kikuchi, and partially with V. Cantoral-Farfan and K. Huu Nguyen.

We discuss new decidability and undecidability results for mixed characteristic henselian fields, whose proof goes via reduction to positive characteristic. The reduction uses extensively the theory of perfectoid fields and also the earlier Krasner-Kazhdan-Deligne principle. Our main results will be: (1) A relative decidability theorem for perfectoid fields. Using this, we obtain decidability of certain tame fields of mixed characteristic. (2) An undecidability result for the asymptotic theory of all finite extensions of ℚ_p (fixed p) with cross-section. We will also discuss a tentative step towards understanding the underlying model […]