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The model theory of perfectoid fields [after Jahnke and Kartas]

Sylvy Anscombe

Abstract

This text was written to support a Bourbaki seminar given in January 2026 on the subject of the model theory of perfectoid fields, especially on the work of Jahnke and Kartas in their paper "Beyond the Fontaine-Wintenberger theorem", J. Amer. Math. Soc. 38 (4), pp. 997-1047, 2025.

The model theory of perfectoid fields [after Jahnke and Kartas]

Abstract

This text was written to support a Bourbaki seminar given in January 2026 on the subject of the model theory of perfectoid fields, especially on the work of Jahnke and Kartas in their paper "Beyond the Fontaine-Wintenberger theorem", J. Amer. Math. Soc. 38 (4), pp. 997-1047, 2025.
Paper Structure (14 sections, 34 theorems, 47 equations, 3 figures)

This paper contains 14 sections, 34 theorems, 47 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Some model theory of fields
  3. Valued fields
  4. First-order languages of valued fields
  5. Residue characteristic zero and limits as $p\rightarrow\infty$
  6. Two other well-understood settings: finitely ramified and tame valued fields
  7. Finitely ramified henselian valued fields
  8. Tame valued fields
  9. Singletons, pseudo-uniformizers, and the standard decomposition
  10. Taming perfectoid fields: the class of "almost tame" valued fields
  11. Ax--Kochen/Ershov principles for almost tame valued fields
  12. The Main Theorem of Jahnke and Kartas
  13. The role of deeply ramified fields
  14. Building on these foundations

Key Result

Theorem 1.2

There is a computableA function $f:\mathbb{N}\rightarrow\mathbb{N}$ is computable if there is a Turing machine that halts with output $f(n)$ on input $n$, for each $n\in\mathbb{N}$, with appropriate coding of natural numbers. function $d\mapsto N_{d}$, such that for any $p>N_{d}$ and any form $F\in

Figures (3)

  • Figure 1: Underlying embedding lemma
  • Figure 2: Standard decomposition at $\pi$
  • Figure 3: Illustration of the proof of Theorem \ref{['thm:first']}

Theorems & Definitions (62)

  • Conjecture 1.1: Artin's Conjecture on forms over $\mathbb{Q}_{p}$
  • Theorem 1.2: AxKochen-IErshov65
  • Theorem 1.3: FontaineWintenberger
  • Remark 1.4
  • Example 1.6
  • Definition 1.7
  • Lemma 1.8: Hensel's Lemma
  • Example 1.10
  • Lemma 1.11: Fundamental equality
  • Lemma 1.12: Abhyankhar's inequality, Abhyankar
  • ...and 52 more