Decidability via the tilting correspondence

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 $p$.

Final version. Local improvements following suggestions from a study group that took place at Fields Institute in Fall 2021 and two referee reports

References in corpus (1)

• An undecidability result for the asymptotic theory of $p$-adic fields

Cited by in corpus (1)

• Two examples concerning existential undecidability in fields