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Decidability via the tilting correspondence

Konstantinos Kartas

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 $\mathbb{Q}_p(ζ_{p^{\infty}})$ 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 $p$.

Decidability via the tilting correspondence

Abstract

We prove a relative decidability result for perfectoid fields. This applies to show that the fields and are (existentially) decidable relative to the perfect hull of and is (existentially) decidable relative to the perfect hull of . We also prove some unconditional decidability results in mixed characteristic via reduction to characteristic .

Paper Structure

This paper contains 73 sections, 47 theorems, 34 equations.

Table of Contents

  1. Preliminaries
  2. Decidability
  3. Introduction
  4. Uniform decidability
  5. Interpretability
  6. Interpretations
  7. Complexity of interpretations
  8. Recursive interpretations
  9. Uniformly recursive interpretations
  10. Ax-Kochen/Ershov in mixed characteristic
  11. A result by van den Dries
  12. Introduction
  13. Inverse systems
  14. Statement and proof
  15. Existential AKE in mixed characteristic
  16. ...and 58 more sections

Key Result

Corollary A

$(a)$ Assume $\mathbb{F}_p(\!(t)\!)^{1/p^{\infty}}$ is decidable (resp. $\exists$-decidable) in $L_{\text{val}}(t)$. Then $\mathbb{Q}_p(p^{1/p^{\infty}})$ and $\mathbb{Q}_p(\zeta_{p^{\infty}})$ are decidable (resp. $\exists$-decidable) in $L_{\text{val}}$. $(b)$ Assume $\overline{ \mathbb{F}}_p(\!(t

Theorems & Definitions (138)

  • Corollary A
  • Theorem A: Perfectoid transfer
  • Corollary B
  • Theorem B
  • Definition 1.1.3
  • Example 1.1.4
  • Definition 1.1.5
  • Remark 1.1.6
  • Definition 1.2.2
  • Proposition 1.2.3: Reduction Theorem 5.3.2 Hod
  • ...and 128 more