Decidability via the tilting correspondence

TL;DR

This paper establishes relative decidability results for certain perfectoid fields using the tilting correspondence, and also provides unconditional decidability results in mixed characteristic by reduction to characteristic p.

Contribution

It introduces a novel approach to decidability in perfectoid fields through tilting correspondence, extending known results to new classes of fields.

Findings

Fields $ ext{Q}_p(p^{1/p^{inite}})$ and $ ext{Q}_p( ext{zeta}_{p^{inite}})$ are decidable relative to their perfect hulls.

The field $ ext{Q}_p^{ab}$ is decidable relative to the perfect hull of algebraic closure of $ ext{F}_p( ext{(t)})$.

Unconditional decidability results are obtained in mixed characteristic via reduction to characteristic p.

Abstract

We prove a relative decidability result for perfectoid fields. This applies to show that the fields $\mathbb{Q}_p(p^{1/p^{\infty}})$ and $\mathbb{Q}_p(\zeta_{p^{\infty}})$ are (existentially) decidable relative to the perfect hull of $ \mathbb{F}_p(\!(t)\!)$ and $\mathbb{Q}_p^{ab}$ is (existentially) decidable relative to the perfect hull of $\overline{ \mathbb{F}}_p(\!(t)\!)$. We also prove some unconditional decidability results in mixed characteristic via reduction to characteristic $p$.