Decidability via the tilting correspondence

Kartas, Konstantinos

doi:10.2140/ant.2024.18.209

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Decidability via the tilting correspondence

Konstantinos Kartas

Vol. 18 (2024), No. 2, 209–248

DOI: 10.2140/ant.2024.18.209

Abstract

We prove a relative decidability result for perfectoid fields. This applies to show that the fields $ℚ_{p} (p^{1 ∕ p^{\infty}})$ and $ℚ_{p} (ζ_{p^{\infty}})$ are (existentially) decidable relative to the perfect hull of $𝔽_{p} ((t))$ and $ℚ_{p}^{a b}$ is (existentially) decidable relative to the perfect hull of ${\bar{𝔽}}_{p} ((t))$ . We also prove some unconditional decidability results in mixed characteristic via reduction to characteristic $p$ .

Keywords

perfectoid fields, model theory, valued fields, decidability, tilting, tame fields

Mathematical Subject Classification 2010

Primary: 11U05

Secondary: 11U09

Milestones

Received: 14 February 2020

Revised: 16 December 2022

Accepted: 13 April 2023

Published: 6 February 2024

Authors

Konstantinos Kartas
Institut Mathématique de Jussieu CNRS/IMJ-PRG Paris France

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Vol. 18, No. 2, 2024