# 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
$\mathbb{Q}_{p}(\zeta_{p^{\infty}})$ 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
$p$.

During this research, the author was funded by EPSRC grant EP/20998761 and was
also supported by the Onassis Foundation - Scholarship ID: F ZP
020-1/2019-2020.

###### Contents

  1. 1 Preliminaries
  2. 2 Ax-Kochen-Ershov in mixed characteristic
  3. 3 Perfectoid fields
  4. 4 Relative decidability for perfectoid fields
  5. 5 Applications: Tame fields of mixed characteristic
  6. 6 Applications: Congruences modulo $p$
  7. 7 Final remarks

## Introduction

After the decidability of the $p$-adic numbers $\mathbb{Q}_{p}$ was
established by Ax-Kochen [AK65] and independently by Ershov [Ers65], several
decidability questions about local fields and their extensions have been
raised and answered:

  * •

In mixed characteristic, Kochen [Koc74] showed that $\mathbb{Q}_{p}^{ur}$, the
maximal unramified extension of $\mathbb{Q}_{p}$, is decidable in the language
$L_{\text{val}}=\\{+,\cdot,{\mathcal{O}},0,1\\}$ (see notation). More
generally, by work of [Zie72], [Ers65], [Bas78], [Bél99] and more recently
[AJ19], [Lee20] and [LL21], we have a good understanding of the model theory
of unramified and finitely ramified mixed characteristic henselian fields.

  * •

In positive characteristic, our understanding is much more limited.
Nevertheless, by work of Denef-Schoutens [DS03], we know that
$\mathbb{F}_{p}((t))$ is existentially decidable in
$L_{t}=\\{+,\cdot,{\mathcal{O}},0,1,t\\}$ (see notation), modulo resolution of
singularities.111The formalism of [DS03] does not include a unary predicate
${\mathcal{O}}$ for the valuation ring. This does not make a difference
because of the equivalences $x\in{\mathcal{O}}\leftrightarrow\exists
y(y^{2}=1+t\cdot x^{2})$ (for $p>2$; replace squares with cubes for $p=2$) and
$x\notin{\mathcal{O}}\leftrightarrow x^{-1}\in t\cdot{\mathcal{O}}$. In fact,
Theorem 4.3 [DS03] applies to show that (assuming resolution) any finitely
ramified extension of $\mathbb{F}_{p}((t))$ is existentially decidable
relative to its residue field. Anscombe-Fehm [AF16] showed unconditionally
that $\mathbb{F}_{p}((t))$ is existentially decidable but in the language
$L_{\text{val}}$, which does not include a constant symbol for $t$.

Our understanding is less clear for infinitely ramified fields and there are
many extensions of $\mathbb{Q}_{p}$ (resp. $\mathbb{F}_{p}((t))$) of great
arithmetic interest, whose decidability problem is still open:

  * •

In mixed characteristic, these include $\mathbb{Q}_{p}^{ab}$, the maximal
abelian extension of $\mathbb{Q}_{p}$ and the totally ramified extension
$\mathbb{Q}_{p}(\zeta_{p^{\infty}})$, obtained by adjoining all $p^{n}$-th
roots of unity. These extensions had already been discussed in Macintyre’s
survey on pg.140 [Mac86] and a conjectural axiomatization of
$\mathbb{Q}_{p}^{ab}$ was given by Koenigsmann on pg.55 [Koe18]. Another
interesting extension is $\mathbb{Q}_{p}(p^{1/p^{\infty}})$, a totally
ramified extension of $\mathbb{Q}_{p}$ obtained by adjoining a compatible
system of $p$-power roots of $p$.

  * •

In positive characteristic, two very natural infinitely ramified fields are
the perfect hulls of $\mathbb{F}_{p}((t))$ and
$\overline{\mathbb{F}}_{p}((t))$. These are semitame fields, using the
terminology of Kuhlmann-Rzepka [KR21], and are both expected to be decidable
(see pg. 4 [KR21]). The ultimate goal of [KR21] is to extend Kuhlmann’s
decidability result for tame fields of equal characteristic (Theorem 1.5
[Kuh16]) in the more general context of semitame fields.

The above fields will be the main objects of interest throughout the paper.
Their $p$-adic (resp. $t$-adic) completions are typical examples of perfectoid
fields in the sense of Scholze [Sch12] (see §3). For any such field $K$, one
can define its tilt $K^{\flat}$ (see §3.2), which intuitively is its local
function field analogue and serves as a good characteristic $p$ approximation
of $K$. In practice, this means that one can often reduce arithmetic problems
about $K$ to arithmetic problems about $K^{\flat}$. This kind of transfer
principle, which works for a fixed $p$, should be contrasted with the Ax-
Kochen principle which achieves such a reduction only asymptotically, i.e.
with the residue characteristic $p\to\infty$. This is explained in detail in
Scholze’s ICM report (see pg.2 [Sch14]).

Our main goal is to set the stage for incorporating ideas from perfectoid
geometry in the model theory of henselian fields. In the present paper we
focus on decidability, although it is conceivable that our methods can be used
in different model-theoretic contexts. We will prove the following relative
decidability result for the fields discussed above:

######  Corollary A.

$(a)$ Assume $\mathbb{F}_{p}((t))^{1/p^{\infty}}$ is decidable (resp.
$\exists$-decidable) in $L_{t}$. 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 $\overline{\mathbb{F}}_{p}((t))^{1/p^{\infty}}$ is decidable
(resp. $\exists$-decidable) in $L_{t}$. Then $\mathbb{Q}_{p}^{ab}$ is
decidable (resp. $\exists$-decidable) in $L_{\text{val}}$.

As usual, we write $\mathbb{F}_{p}((t))^{1/p^{\infty}}$ (resp.
$\overline{\mathbb{F}}_{p}((t))^{1/p^{\infty}}$) for the perfect hull of
$\mathbb{F}_{p}((t))$ (resp. $\overline{\mathbb{F}}_{p}((t))$),
$L_{\text{val}}$ for the language of valued fields and $L_{t}$ for the
language $L_{\text{val}}$ enriched with a constant symbol for $t$ (see
notation). Corollary A is essentially a special case of a general relative
decidability result for perfectoid fields, which is discussed next.222Strictly
speaking, the fields in Corollary A are not perfectoid but one can still
derive Corollary A from Theorem A directly.

Let $F$ be a perfectoid field of characteristic $p$. An untilt of $F$ is a
mixed characteristic perfectoid field $K$ such that $K^{\flat}\cong F$. In
general, there will be many non-isomorphic untilts of $F$. In fact, there will
be too many untilts even up to elementary equivalence (see Proposition
3.6.7$(b),(c)$), thus shattering the naive hope of $K$ being decidable
relative to its tilt $K^{\flat}$ that one might initially have (see Remark
4.3.7). Such a relative decidability result can be salvaged by asking that $K$
be a computable untilt of $F$, a notion which we now explain (see §4.2 for
details).

Write $W({\mathcal{O}}_{F})$ for the ring of Witt vectors over
${\mathcal{O}}_{F}$ (see §3.3). An element $\xi\in W({\mathcal{O}}_{F})$ is
identified with its associated Witt vector, which is an infinite sequence
$(\xi_{0},\xi_{1},...)$ with $\xi_{n}\in{\mathcal{O}}_{F}$. Whenever there is
a computable subring $R_{0}\subseteq{\mathcal{O}}_{F}$ such that $\xi_{n}\in
R_{0}$ and the function $\mathbb{N}\to R_{0}:n\mapsto\xi_{n}$ is recursive, we
say that $\xi$ is $R_{0}$-computable.333As usual, a computable ring $R_{0}$ is
one such that there exists a bijection $R_{0}\simeq\mathbb{N}$, via which the
ring operations are identified with recursive functions (e.g.
$\mathbb{F}_{p}[t]$ is a computable ring). An important result by Fargues-
Fontaine gives a 1-1 correspondence between untilts of $F$ and certain
principal ideals of $W({\mathcal{O}}_{F})$ (Theorem 3.5.10). An untilt $K$ of
$F$ is said to be an $R_{0}$-computable untilt, whenever an $R_{0}$-computable
generator $\xi\in W({\mathcal{O}}_{F})$ of the ideal corresponding to $K$
exists. We then have the following relative decidability result:

######  Theorem A.

Suppose $K$ is an $R_{0}$-computable untilt of $F$. If $F$ is decidable in
$L_{\text{val}}(R_{0})$, then $K$ is decidable in $L_{\text{val}}$.

Here $L_{\text{val}}(R_{0})$ is the language $L_{\text{val}}$ enriched with
constant symbols for the elements of $R_{0}$. We also obtain an existential
version of Theorem A in §4.3. The condition on $K$ being an $R_{0}$-computable
untilt of $F$ is virtually true in all cases of interest (although generically
false), even for a natural choice of $R_{0}$. Corollary A is an immediate
consequence of Theorem A, taking $F$ as the $t$-adic completion of
$\mathbb{F}_{p}((t))^{1/p^{\infty}}$ (resp.
$\overline{\mathbb{F}}_{p}((t))^{1/p^{\infty}}$ for part $(b)$) and
$R_{0}=\mathbb{F}_{p}[t]$. The two key ingredients in the proof of Theorem A
are:

  1. (1)

An (unpublished) Ax-Kochen-Ershov style result by van den Dries, presented in
§2.

  2. (2)

The Fargues-Fontaine correspondence described above, presented in §3.5.

Theorem A allows us to reduce decidability problems from the mixed
characteristic to the positive characteristic world. Although our model-
theoretic understanding of the latter is notoriously limited, this method
still yields several applications. In sections §5 and §6 we obtain new
unconditional decidability results in mixed characteristic via reduction to
characteristic $p$. In §5, we prove the following:

######  Corollary B.

The valued field $(\mathbb{Q}_{p}(p^{1/p^{\infty}}),v_{p})$ (resp.
$(\mathbb{Q}_{p}(\zeta_{p^{\infty}}),v_{p})$) admits a maximal immediate
extension which is decidable in $L_{\text{val}}$.

Note that the fields $\mathbb{Q}_{p}(p^{1/p^{\infty}})$ and
$\mathbb{Q}_{p}(\zeta_{p^{\infty}})$ are not Kaplansky and have many non-
isomorphic maximal immediate extensions, all of which are tame in the sense of
Kuhlmann [Kuh16] (see Example 5.1.6). Although the work of Kuhlmann yields
several decidability results for equal characteristic tame fields, Corollary B
is, to my knowledge, the first decidability result for tame fields of mixed
characteristic (see §5). The proof uses a recent decidability result of
Lisinski [Lis21] for the Hahn field $\mathbb{F}_{p}((t^{\Gamma}))$ with
$\Gamma=\frac{1}{p^{\infty}}\mathbb{Z}$ in the language $L_{t}$, strengthening
Kuhlmann’s earlier result for $\mathbb{F}_{p}((t^{\Gamma}))$ in the language
$L_{\text{val}}$ (see Theorem 1.6 [Kuh16]). Corollary B is then an easy
application of Theorem A.

Without making essential use of the perfectoid machinery but only the
philosophy thereof, we show in §6 the following:

######  Theorem B.

Let $K$ be any of the valued fields
$\mathbb{Q}_{p}(p^{1/p^{\infty}}),\mathbb{Q}_{p}(\zeta_{p^{\infty}})$ or
$\mathbb{Q}_{p}^{ab}$. The existential theory of
${\mathcal{O}}_{K}/p{\mathcal{O}}_{K}$ in the language of rings $L_{\text{r}}$
is decidable.

The proof is again via reduction to characteristic $p$, using a recent
existential decidability result for equal characteristic henselian valued
fields in the language $L_{\text{val}}$, due to Anscombe-Fehm [AF16]. However,
as Corollary A suggests, if we aim to understand the theories of
$\mathbb{Q}_{p}(p^{1/p^{\infty}}),\mathbb{Q}_{p}(\zeta_{p^{\infty}})$ and
$\mathbb{Q}_{p}^{ab}$ via reduction to positive characteristic, we will need
stronger results in the language $L_{t}$ on the characteristic $p$ side. This
is also supported by Proposition 7.2.3 which shows that the Diophantine
problem for $\mathbb{F}_{p}((t))^{1/p^{\infty}}$ in $L_{t}$ is Turing
reducible to the $\forall^{1}\exists$-theory of
$\mathbb{Q}_{p}(p^{1/p^{\infty}})$.

An application of a different flavor, which yields an undecidability result in
mixed characteristic via reduction to characteristic $p$, was recently found
in [Kar21]. Theorem A [Kar21] shows that the asymptotic theory of
$\\{K:[K:\mathbb{Q}_{p}]<\infty\\}$ in the language $L_{\text{val}}$ with a
cross-section is undecidable.

## Notation

  * •

If $(K,v)$ is a valued field, we denote by $\mathcal{O}_{v}$ the valuation
ring. If the valuation is clear from the context, we shall also write
$\mathcal{O}_{K}$. We write $vK$ for the value group and $Kv$ for the residue
field. When $(K,v)$ is of mixed characteristic, we write $p$ for the number
$\text{char}(Kv)$. The notation $p^{n}{\mathcal{O}}_{v}$ stands for the ideal
of ${\mathcal{O}}_{v}$ generated by the element $p^{n}$. If both the field in
question and the valuation $v$ are clear from the context, we shall write
$(p^{n})$ for $p^{n}{\mathcal{O}}_{v}$.

  * •

$\mathbb{Z}_{p}^{ur}$: The valuation ring of $(\mathbb{Q}_{p}^{ur},v_{p})$,
the maximal unramified extension of $\mathbb{Q}_{p}$ equipped with the unique
extension of the $p$-adic valuation.
$\mathbb{Z}_{p}^{ab}$: The valuation ring of $(\mathbb{Q}_{p}^{ab},v_{p})$,
the maximal Galois extension of $\mathbb{Q}_{p}$ whose Galois group over
$\mathbb{Q}_{p}$ is abelian.
We write $\mathbb{F}_{p}((t))^{1/p^{\infty}}$ (resp.
$\overline{\mathbb{F}}_{p}((t))^{1/p^{\infty}}$) for the perfect hull of
$\mathbb{F}_{p}((t))$ (resp. $\overline{\mathbb{F}}_{p}((t))$).

  * •

For a given language $L$, we denote by $\text{Sent}_{L}$ the set of
$L$-sentences and by $\text{Form}_{L}$ the set of $L$-formulas. If $M$ is an
$L$-structure and $A\subseteq M$ is an arbitrary subset, we write $L(A)$ for
the language $L$ enriched with a constant symbol $c_{a}$ for each element
$a\in A$. The $L$-structure $M$ can be updated into an $L(A)$-structure in the
obvious way.

We write $L_{\text{r}}=\\{+,\cdot,0,1\\}$ for the language of rings,
$L_{\text{oag}}=\\{+,<,0\\}$ for the language of ordered abelian groups,
$L_{\text{val}}=L_{r}\cup\\{{\mathcal{O}}\\}$ for the language of valued
fields (where ${\mathcal{O}}$ is a unary predicate ${\mathcal{O}}$ for the
valuation ring), $L_{t}=L_{\text{val}}\cup\\{t\\}$, where $t$ is a constant
symbol whose intended interpretation will always be clear from the context. A
local ring $(R,\mathfrak{m}_{R})$ may be viewed as an
$L_{\text{lcr}}$-structure, where $L_{\text{lcr}}$ is the language of local
rings consisting of the language of rings $L_{\text{r}}$ together with a unary
predicate $\mathfrak{m}$, whose intended interpretation is the maximal ideal
$\mathfrak{m}_{R}\subseteq R$.

##  1\. Preliminaries

###  1.1. Decidability

####  1.1.1. Introduction

Fix a countable language $L$ and let $\text{Sent}_{L}$ be the set of well-
formed $L$-sentences, identified with $\mathbb{N}$ via some Gödel numbering.
Let $M$ be an $L$-structure. Recall that $M$ is decidable if we have an
algorithm to decide whether $M\models\phi$, for a given
$\phi\in\text{Sent}_{L}$. More formally, let
$\chi_{M}:\text{Sent}_{L}\to\\{0,1\\}$ be the characteristic function of
$\text{Th}(M)\subseteq\text{Sent}_{L}$. We say that $M$ is decidable if
$\chi_{M}$ is recursive.

####  1.1.2. Uniform decidability

######  Definition 1.1.3.

For each $n\in\mathbb{N}$, let $f_{n}:\mathbb{N}\to\mathbb{N}$ be a function.
The sequence $(f_{n})_{n\in\omega}$ is uniformly recursive if the function
$\mathbb{N}\times\mathbb{N}\to\mathbb{N}:(n,m)\mapsto f_{n}(m)$ is recursive.

This concept is best illustrated by a non-example:

######  Example 1.1.4.

Let $A\subseteq\mathbb{N}$ be non-recursive. For each $n\in\mathbb{N}$, let
$\delta_{n}:\mathbb{N}\to\mathbb{N}:m\mapsto\delta_{nm}$, where $\delta_{nm}$
is the Kronecker symbol. For each $n\in\mathbb{N}$, define
$f_{n}:\mathbb{N}\to\mathbb{N}$ to be $(1)$ $\delta_{n}$ if $n\in A$ and $(2)$
identically $0$ if $n\notin A$. One readily verifies that each individual
$f_{n}$ is recursive. On the other hand, the sequence $(f_{n})_{n\in\omega}$
is not uniformly recursive. Indeed, if $(n,m)\mapsto f_{n}(m)$ was recursive
then we could solve the membership problem for $A$, using that $n\in A\iff
f_{n}(n)=1$.

Using some Gödel numbering, we can also state a version of Definition 1.1.3
for sequences of functions $(f_{n})_{n\in\omega}$, where
${\mathop{\operatorname{\rm dom}}}(f_{n})=\text{Sent}_{L}$ (or
$\text{cdm}(f_{n})=\text{Sent}_{L}$). One can then define a notion of uniform
decidability for sequences of $L$-structures:

######  Definition 1.1.5.

A sequence $(M_{n})_{n\in\omega}$ of $L$-structures is uniformly decidable if
the sequence of functions $(\chi_{M_{n}})_{n\in\omega}$ is uniformly
recursive, i.e. if the function
$\chi:\mathbb{N}\times\text{Sent}_{L}\to\\{0,1\\}:(n,\phi)\mapsto\chi_{M_{n}}(\phi)$
is recursive.

######  Remark 1.1.6.

If the sequence
$\chi:\mathbb{N}\times\text{Sent}_{L}\to\\{0,1\\}:(n,\phi)\mapsto\chi_{M_{n}}(\phi)$
is recursive when restricted to existential sentences, we naturally say that
the sequence $(M_{n})_{n\in\omega}$ is uniformly existentially decidable.
Other syntactic variants may be defined analogously.

###  1.2. Interpretability

Our formalism follows closely §5.3 [Hod93], where details and proofs may be
found.

####  1.2.1. Definition

Given a language $L$, an unnested atomic $L$-formula is one of the form $x=y$
or $x=c$ or $F(\overline{x})=y$ or $R\overline{x}$, where $x,y$ are variables,
$c$ is a constant symbol, $F$ is a function symbol and $R$ is a relation
symbol of the language $L$.

######  Definition 1.2.2.

An $n$-dimensional interpretation of an $L$-structure $M$ in the
$L^{\prime}$-structure $N$ is a triple consisting of:

  1. (1)

An $L^{\prime}$-formula $\partial_{\Gamma}(x_{1},...,x_{n})$.

  2. (2)

A map $\phi\mapsto\phi_{\Gamma}$, that takes an unnested atomic $L$-formula
$\phi(x_{1},...,x_{m})$ and sends it to an $L^{\prime}$-formula
$\phi_{\Gamma}(\overline{y}_{1},...,\overline{y}_{m})$, where each
$\overline{y}_{i}$ is an $n$-tuple of variables.

  3. (3)

A surjective map $f_{\Gamma}:\partial_{\Gamma}(N^{n})\twoheadrightarrow M$.

such that for all unnested atomic $L$-formulas $\phi(x_{1},...,x_{m})$ and all
$\overline{a}_{i}\in\partial_{\Gamma}(N^{n})$, we have

$M\models\phi(f_{\Gamma}(\overline{a}_{1}),...,f_{\Gamma}(\overline{a}_{m}))\iff
N\models\phi_{\Gamma}(\overline{a}_{1},...,\overline{a}_{m})$

An interpretation of an $L$-structure $M$ in the $L^{\prime}$-structure $N$ is
an $n$-dimensional interpretation, for some $n\in\mathbb{N}$. In that case, we
also say that $M$ is interpretable in $N$. Interpretability is a transitive
relation on the class of structures, i.e. if the $L$-structure $M$ is
interpretable in the $L^{\prime}$-structure $N$ and $N$ is interpretable in
the $L^{\prime\prime}$-structure $P$, then there exists a composite
interpretation of $M$ in $P$ (Exercise 2, pg. 218 [Hod93]).

The formulas $\partial_{\Gamma}$ and $\phi_{\Gamma}$ (for all unnested atomic
$\phi$) are the defining formulas of $\Gamma$. Their complexity defines a
measure of complexity of the interpretation $\Gamma$:

######  Definition 1.2.3 (§5.4$(a)$ [Hod93]).

An interpretation $\Gamma$ of an $L$-structure $M$ in an
$L^{\prime}$-structure $N$ is quantifier-free if the defining formulas of
$\Gamma$ are quantifier-free. Other syntactic variants are defined analogously
(e.g. existential interpretation or positive existential interpretation).

######  Lemma 1.2.4.

If the $L$-structure $M$ is $\exists^{+}$-interpretable in the
$L^{\prime}$-structure $N$ and $N$ is $\exists^{+}$-interpretable in the
$L^{\prime\prime}$-structure $P$, then the composite interpretation of $M$ in
$P$ is also an $\exists^{+}$-interpretation.

###### Proof.

Clear. ∎

If $N$ is an $L^{\prime}$-structure and $\overline{a}=(a_{1},...,a_{m})\in
N^{m}$, we write $(N,\overline{a})$ for the expansion of $N$ in the language
$L(\overline{c})$, which is $L$ together with an $m$-tuple of constant symbols
$(c_{1},...,c_{m})$ with $c_{i}^{(N,\overline{a})}=a_{i}$. If $M$ is
interpretable in $(N,\overline{a})$, for some $\overline{a}\in N^{m}$, we say
that $M$ is interpretable in $N$ with parameters.

######  Proposition 1.2.5 (Reduction Theorem 5.3.2 [Hod93]).

Let $\Gamma$ be an $n$-dimensional interpretation of an $L$-structure $M$ in
the $L^{\prime}$-structure $N$. There exists a map $\phi\mapsto\phi_{\Gamma}$,
extending the map of Definition 1.2.2$(2)$, such that for every $L$-formula
$\phi(x_{1},...,x_{m})$ and all $\overline{a}_{i}\in\partial_{\Gamma}(N^{n})$,
we have that

$M\models\phi(f_{\Gamma}(\overline{a}_{1}),...,f_{\Gamma}(\overline{a}_{m}))\iff
N\models\phi_{\Gamma}(\overline{a}_{1},...,\overline{a}_{m})$

###### Proof.

We describe how $\phi\mapsto\phi_{\Gamma}$ is built, for completeness
(omitting details). By Corollary 2.6.2 [Hod93], every $L$-formula is
equivalent to one in which all atomic subformulas are unnested. One can then
construct $\phi\mapsto\phi_{\Gamma}$ by induction on the complexity of
formulas. The base case is handled by Definition 1.2.2$(2)$. This definition
extends inductively according to the following rules:

  1. (1)

$(\lnot\phi)_{\Gamma}=\lnot(\phi)_{\Gamma}$.

  2. (2)

$(\bigwedge_{i=1}^{n}\phi_{i})_{\Gamma}=\bigwedge(\phi_{i})_{\Gamma}$.

  3. (3)

$(\forall\phi)_{\Gamma}=\forall
x_{1},...,x_{n}(\partial_{\Gamma}(x_{1},...,x_{n})\rightarrow\phi_{\Gamma})$

  4. (4)

$(\exists\phi)_{\Gamma}=\exists
x_{1},...,x_{n}(\partial_{\Gamma}(x_{1},...,x_{n})\land\phi_{\Gamma})$

The resulting map satisfies the desired conditions of the Proposition. ∎

######  Definition 1.2.6.

The map $\text{Form}_{L}\to\text{Form}_{L^{\prime}}:\phi\mapsto\phi_{\Gamma}$
constructed in the proof of Proposition 1.2.5 is called the reduction map of
the interpretation $\Gamma$.

####  1.2.7. Recursive interpretations

######  Definition 1.2.8 (Remark 4, pg. 215 [Hod93] ).

Suppose $L$ is a recursive language. Let $\Gamma$ be an interpretation of an
$L$-structure $M$ in the $L^{\prime}$-structure $N$. We say that the
interpretation $\Gamma$ is recursive if the the map $\phi\mapsto\phi_{\Gamma}$
on unnested atomic formulas is recursive.

######  Remark 1.2.9 (Remark 4, pg. 215 [Hod93]).

If $\Gamma$ is a recursive interpretation of an $L$-structure $M$ in the
$L^{\prime}$-structure $N$, then the reduction map of $\Gamma$ is also
recursive.

####  1.2.10. Uniformly recursive interpretations

######  Definition 1.2.11.

Suppose $L$ is a recursive language, $(M_{n})_{n\in\omega}$ be a sequence of
$L$-structures and $(N_{n})_{n\in\omega}$ be a sequence of
$L^{\prime}$-structure. For each $n\in\mathbb{N}$, let $\Gamma_{n}$ be an
interpretation of $M_{n}$ in $N_{n}$. We say that the sequence of
interpretations $(\Gamma_{n})_{n\in\omega}$ is uniformly recursive if the
sequence of reduction maps $(\phi\mapsto\phi_{\Gamma_{n}})_{n\in\omega}$ on
unnested atomic formulas is uniformly recursive, i.e. if the map
$(n,\phi)\mapsto\phi_{\Gamma_{n}}$ is recursive.

If an $L$-structure $M$ is interpretable in the $L^{\prime}$-structure $N$ and
the latter is decidable, then so is the former. It is not hard to prove the
following uniform version:

######  Proposition 1.2.12.

Suppose $L$ is a recursive language, $(M_{n})_{n\in\omega}$ a sequence of
$L$-structures and $N$ is an $L^{\prime}$-structure. Suppose $N$ is decidable,
$\Gamma_{n}$ is an interpretation of $M_{n}$ in $N$ and the sequence of
interpretations $(\Gamma_{n})_{n\in\omega}$ is uniformly recursive. Then the
sequence $(M_{n})_{n\in\omega}$ is uniformly decidable.

###### Proof.

Rephrasing Proposition 1.2.5 for sentences, yields
$\chi_{M_{n}}(\phi)=\chi_{N}(\phi_{\Gamma_{n}})$ for every
$\phi\in\text{Sent}_{L}$. It follows that the map
$(n,\phi)\mapsto\chi_{M_{n}}(\phi)$ is equal to the map
$(n,\phi)\mapsto\phi_{\Gamma_{n}}\mapsto\chi_{N}(\phi_{\Gamma_{n}})$ and the
latter is recursive as a composition of recursive functions. ∎

If the interpretation of $M$ in $N$ is recursive and so is the interpretation
of $N$ in $P$, then the composite interpretation of $M$ in $P$ is recursive as
well. Indeed, recursive functions are closed under composition. One also has a
uniform version:

######  Lemma 1.2.13.

Let $(M_{n})_{n\in\omega}$ be a sequence of $L$-structures,
$(N_{n})_{n\in\omega}$ be a sequence of $L^{\prime}$-structures and
$(P_{n})_{n\in\omega}$ be a sequence of $L^{\prime\prime}$-structures. For
each $n\in\mathbb{N}$, let $\Gamma_{n}$ be an interpretation of $M_{n}$ in
$N_{n}$ and $\Delta_{n}$ be an interpretation of $N_{n}$ in $P_{n}$ and
suppose that the sequences of interpretations $(\Gamma_{n})_{n\in\omega}$ and
$(\Delta_{n})_{n\in\omega}$ are uniformly recursive. Let $E_{n}$ be the
composite interpretation of $M_{n}$ in $P_{n}$. Then the sequence of
interpretations $(P_{n})_{n\in\omega}$ is uniformly recursive.

###### Proof.

Clear. ∎

##  2\. Ax-Kochen-Ershov in mixed characteristic

###  2.1. A result by van den Dries

####  2.1.1. Introduction

We start with an Ax-Kochen-Ershov style result due to van den Dries
(unpublished) which is briefly discussed on pg. 144 in [vdD12]. We shall
sketch the proof (due to van den Dries), which does not seem to appear
anywhere in the published literature. Some references, which use a similar
coarsening argument, include pg. 2 [vdD99], the proof of Corollary 12.3 [AJ19]
and the proof of Proposition 9.6 [Sca99]. For background material on
coarsenings of valuations, see §7.4 [vdD12].

####  2.1.2. Inverse systems

The formalism of multi-sorted structures, used in this section, is spelled out
in §3 [Sca99]. The proof of van den Dries’ Theorem 2.1.5 requires a technical
lemma for inverse systems, which we now discuss.

Let $\mathcal{R}=(R_{n})_{n\in\omega}$ be a sequence of rings, viewed as a
multi-sorted structure with sorts $(\mathbf{R}_{n})_{n\in\omega}$, each
equipped with the language of rings $L_{\text{r}}$, and for each
$n\in\mathbb{N}$ we have a map $f_{n}:\mathbf{R}_{n+1}\to\mathbf{R}_{n}$. Let
$T$ be the theory that requires of $\mathcal{R}=(R_{n})_{n\in\omega}$ that the
map $f_{n}:R_{n+1}\to R_{n}$ be surjective with
$\text{Ker}(f_{n})=p^{n}R_{n+1}$ , i.e. $R_{n}\cong R_{n+1}/p^{n}R_{n+1}$. If
$\mathcal{R}=(R_{n})_{n\in\omega}$ and $\mathcal{S}=(S_{n})_{n\in\omega}$ are
two models of $T$, they are isomorphic precisely when there is a compatible
system $(\phi_{n})_{n\in\omega}$ of isomorphisms
$\phi_{n}:R_{n}\xrightarrow{\cong}S_{n}$, i.e. such that the diagram commutes

${R_{n+1}}$${S_{n+1}}$${R_{n}}$${S_{n}}$$\scriptstyle{\phi_{n+1}}$$\scriptstyle{f_{n}}$$\scriptstyle{g_{n}}$$\scriptstyle{\phi_{n}}$

for each $n\in\mathbb{N}$. Compatibility of the $\phi_{n}$’s is essential as
there are examples where $R_{n}\cong S_{n}$ for each $n\in\mathbb{N}$ but
$\mathcal{R}\not\cong\mathcal{S}$ (see e.g. Remark 3.6.8). Somewhat
surprisingly, compatibility comes for free in a saturated setting:

######  Lemma 2.1.3.

Assume CH. Let $\mathcal{R}=(R_{n})_{n\in\omega}$ and
$\mathcal{S}=(S_{n})_{n\in\omega}$ be two models of $T$. Suppose that for each
$n\in\mathbb{N}$, we have that $R_{n}\cong S_{n}$ and the rings $R_{n}$,
$S_{n}$ are saturated with $|R_{n}|=|S_{n}|\leq\aleph_{1}$. Then
$\mathcal{R}\cong\mathcal{S}$.

###### Proof.

Let $U$ be a non-principal ultrafilter on $\mathcal{P}(\mathbb{N})$ and
consider the ultraproducts $R_{U}=\prod_{n\in\omega}R_{n}/U$ and
$S_{U}=\prod_{n\in\omega}S_{n}/U$.
Claim 1: $R_{U}\cong S_{U}$.

###### Proof.

Since $R_{n}\equiv S_{n}$ for each $n\in\mathbb{N}$, we get that $R_{U}\equiv
S_{U}$ by Łoś’ s Theorem. For each $n\in\mathbb{N}$, let
$\mathcal{F}_{m}=\\{n\in\mathbb{N}:|R_{n}|\geq m\\}$. If there exists
$m\in\mathbb{N}$ with $\mathcal{F}_{m}\notin U$, then $R_{U}$ and $S_{U}$ are
both finite and thus $R_{U}\cong S_{U}$. If on the other hand
$\mathcal{F}_{m}\in U$ for all $m\in\mathbb{N}$, then $R_{U}$ and $S_{U}$ are
both infinite. Since $|R_{n}|\leq\aleph_{1}$, we get that
$|R_{U}|\leq\aleph_{1}^{\aleph_{0}}=2^{\aleph_{0}^{2}}=2^{\aleph_{0}}=\aleph_{1}$,
using the continuum hypothesis (similarly $|S_{U}|\leq\aleph_{1}$). Moreover,
the ultraproducts $R_{U}$ and $S_{U}$ are $\aleph_{1}$-saturated (Exercise
4.5.37 [Mar02]) and thus saturated of size $\aleph_{1}$. Since $R_{U}$ and
$S_{U}$ are elementary equivalent and both saturated of size $\aleph_{1}$, we
conclude that $R_{U}\cong S_{U}$ (Theorem 4.3.20 [Mar02]). ∎${}_{\textit{Claim
1}}$

Next we prove:
Claim 2: For each $n\in\mathbb{N}$, we have $R_{U}/p^{n}R_{U}\cong R_{n}$.

###### Proof.

Fix $n\in\mathbb{N}$. Since $R_{m}/p^{n}R_{m}\cong R_{n}$ for $m>n$, we get
that $R_{U}/p^{n}R_{U}\equiv R_{n}$ in $L_{\text{r}}$ by Łoś Theorem. If
$R_{n}$ is finite, then $R_{U}/p^{n}R_{U}\cong R_{n}$ and we are done.
Otherwise, we will have that $R_{n}$ is saturated of size $\aleph_{1}$. The
same is true for $R_{U}/p^{n}R_{U}$, being interpretable in the structure
$R_{U}$, which is saturated of size $\aleph_{1}$ by CH (see the proof of Claim
1). We conclude that $R_{U}/p^{n}R_{U}\cong R_{n}$ (Theorem 4.3.20 [Mar02]).
∎${}_{\textit{Claim 2}}$

Similarly, for each $n\in\mathbb{N}$, we have $S_{U}/p^{n}S_{U}\cong S_{n}$.
By Claim 1, we obtain $\phi:R_{U}\xrightarrow{\cong}S_{U}$. Note that
$\phi(p^{n}R_{U})=p^{n}S_{U}$, for each $n\in\mathbb{N}$. By Claim 2, this
gives rise to a compatible system $(\phi_{n})_{n\in\omega}$ of isomorphisms
$\phi_{n}:R_{n}\xrightarrow{\cong}S_{n}$, which yields
$\mathcal{R}\xrightarrow{\cong}\mathcal{S}$. ∎

####  2.1.4. Statement and proof

The following result will be of fundamental importance for the rest of the
paper:

######  Theorem 2.1.5 (van den Dries).

Let $(K,v),(K^{\prime},v^{\prime})$ be two henselian valued fields of mixed
characteristic. Then $(K,v)\equiv(K^{\prime},v^{\prime})$ in $L_{\text{val}}$
if and only if
$\mathcal{O}_{v}/p^{n}{\mathcal{O}}_{v}\equiv{\mathcal{O}}_{v^{\prime}}/p^{n}{\mathcal{O}}_{v^{\prime}}$
in $L_{\text{r}}$ for all $n\in\mathbb{N}$ and
$(vK,vp)\equiv(v^{\prime}K^{\prime},v^{\prime}p)$ in $L_{oag}$ together with a
constant for $vp$.

###### Proof.

"$\Rightarrow$": Clear.
"$\Leftarrow$": As the statement at hand is absolute, we may assume the
continuum hypothesis.444This is explained in great detail in §8 [Sca99] and
also on pg. 122 [vdD12]. We may therefore assume that both $(K,v)$ and
$(K^{\prime},v^{\prime})$ are saturated of size $\aleph_{1}$ (Corolllary
4.3.13 [Mar02]). By our assumption, we have an isomorphism of ordered abelian
groups $(vK,vp)\cong(v^{\prime}K^{\prime},v^{\prime}p)$ and a ring isomorphism
${\mathcal{O}}_{v^{\prime}}/p^{n}{\mathcal{O}}_{v^{\prime}}\cong\mathcal{O}_{v}/p^{n}{\mathcal{O}}_{v}$
for each $n\in\mathbb{N}$. We shall argue that
$(K,v)\cong(K^{\prime},v^{\prime})$.

Consider the finest coarsening $w$ of $v$ for which the associated residue
field $Kw$ has characteristic $0$. The corresponding valuation ring is
$\mathcal{O}_{w}=\mathcal{O}_{v}[\frac{1}{p}]$ and the corresponding value
group is $wK=vK/\text{Conv}(\mathbb{Z}vp)$. Let $\bar{v}$ be the induced
valuation from $v$ on the residue field $Kw$. We then have that
$\mathcal{O}_{\bar{v}}=\mathcal{O}_{v}/\bigcap_{n\in\omega}p^{n}\mathcal{O}_{v}$.
We also consider the analogous objects for $K^{\prime}$.
Claim 1: We have a ring isomorphism
$\mathcal{O}_{\bar{v}}\cong\varprojlim\mathcal{O}_{v}/p^{n}\mathcal{O}_{v}$.

###### Proof.

Consider the ring homomorphism
$f:\mathcal{O}_{\bar{v}}\to\varprojlim\mathcal{O}_{v}/p^{n}\mathcal{O}_{v}:x+\bigcap_{n\in\omega}p^{n}\mathcal{O}_{v}\mapsto(x+p^{n}{\mathcal{O}}_{v})_{n\in\omega}$,
which is clearly injective. We shall argue that it is also surjective. For a
given
$(x_{n}+p^{n}{\mathcal{O}}_{v})_{n\in\omega}\in\varprojlim\mathcal{O}_{v}/p^{n}\mathcal{O}_{v}$,
we may find $x\in{\mathcal{O}}_{v}$ with $x\equiv x_{n}\mod
p^{n}{\mathcal{O}}_{v}$, using that ${\mathcal{O}}_{v}$ is
$\aleph_{1}$-saturated. It follows that
$f(x)=(x_{n}+p^{n}{\mathcal{O}}_{v})_{n\in\omega}$. ∎${}_{\textit{Claim 1}}$

Similarly, one obtains an isomorphism
$\mathcal{O}_{\overline{v^{\prime}}}\cong\varprojlim\mathcal{O}_{v^{\prime}}/p^{n}\mathcal{O}_{v^{\prime}}$.
Claim 2: We have an isomorphism of valued fields
$\phi:(Kw,\overline{v})\cong(K^{\prime}w^{\prime},\overline{v^{\prime}})$.

###### Proof.

Let $\mathcal{R}=({\mathcal{O}}_{v}/p^{n}{\mathcal{O}}_{v})_{n\in\omega}$ and
$\mathcal{S}=({\mathcal{O}}_{v^{\prime}}/p^{n}{\mathcal{O}}_{v^{\prime}})_{n\in\omega}$.
Then $\mathcal{R}\cong\mathcal{S}$ by Lemma 2.1.3. This yields
$\varprojlim\mathcal{O}_{v}/p^{n}\mathcal{O}_{v}\cong\varprojlim\mathcal{O}_{v^{\prime}}/p^{n}\mathcal{O}_{v^{\prime}}$
and thus $\mathcal{O}_{\bar{v}}\cong\mathcal{O}_{\overline{v^{\prime}}}$ by
Claim 1. ∎${}_{\textit{Claim 2}}$

Let $\phi:(Kw,\overline{v})\cong(K^{\prime}w^{\prime},\overline{v^{\prime}})$
be the isomorphism provided by Claim 2. We also have that $wK\cong
w^{\prime}K^{\prime}$, since the isomorphism
$(vK,vp)\cong(v^{\prime}K^{\prime},v^{\prime}p)$ descends to the quotients
$v^{\prime}K^{\prime}/\text{Conv}(\mathbb{Z}v^{\prime}p)\cong
vK/\text{Conv}(\mathbb{Z}vp)$. By Lemma 7.13 [vdD12], the coarsened valued
fields $(K,w)$ and $(K^{\prime},w^{\prime})$ are henselian too. By the Ax-
Kochen-Ershov principle in pure characteristic $0$ (see e.g. Corollary 5.22
[vdD12]), we get that $(K,w)\equiv(K^{\prime},w^{\prime})$ in
$L_{\text{val}}$.

Passing once again to elementary extensions,555 In a suitable language that
includes unary predicates for both ${\mathcal{O}}_{v}$ and
${\mathcal{O}}_{w}$. we may even assume that
$(K,w)\cong(K^{\prime},w^{\prime})$ to begin with. By stable embeddedness of
residue fields for henselian valued fields of pure characteristic $0$ (see
Corollary 5.25 [vdD12]), there is even an isomorphism
$\Phi:(K,w)\xrightarrow{\cong}(K^{\prime},w^{\prime})$ inducing
$\phi:Kw\xrightarrow{\cong}K^{\prime}w^{\prime}$.
Claim 3: The map $\Phi$ is an isomorphism of the valued fields $(K,v)$ and
$(K^{\prime},v^{\prime})$.

###### Proof.

Given $x\in K$, we need to show that $vx\geq 0\iff v^{\prime}(\Phi(x))\geq 0$.
If $vx\geq 0$, then either $(i)$ $wx>0$ or $(ii)$ $wx=0$ and
$\overline{v}\bar{x}\geq 0$, where $\bar{x}\in Kv$ is the image of $x$ via
$\text{res}_{w}:{\mathcal{O}}_{w}\to Kw$. In the first case, we get that
$w^{\prime}(\Phi(x))>0$ as $\Phi:(K,w)\to(K^{\prime},w^{\prime})$ is a valued
field homomorphism and therefore $v^{\prime}(\Phi(x))>0$ as
$\mathfrak{m}_{w^{\prime}}\subset\mathfrak{m}_{v^{\prime}}$. Suppose now that
$wx=0$ and $\overline{v}\bar{x}\geq 0$. Then we also get that $wx=0$ and
$\overline{v^{\prime}}(\phi(\overline{x}))\geq 0$ as
$\phi:(Kw,\overline{v})\to(K^{\prime}w^{\prime},\overline{v^{\prime}})$ is a
valued field homomorphism. Since $\Phi$ induces $\phi$, we get that
$\overline{v^{\prime}}(\overline{\Phi(x)})=\overline{v^{\prime}}(\phi(\overline{x}))\geq
0$ and conclude that $v^{\prime}(\Phi(x))\geq 0$. ∎${}_{\textit{Claim 3}}$

Claim 3 finishes the proof. ∎

###  2.2. Existential AKE in mixed characteristic

In this section we prove an existential version of Theorem 2.1.5. We first
review some known AKE results in the equal characteristic setting.

####  2.2.1. Comparison with the equal characteristic case

For equal characteristic henselian valued fields one has the following simple
Ax-Kochen-Ershov principles due to Anscombe-Fehm:

######  Theorem 2.2.2 (Corollary 1.2 [AF16]).

Let $(K,v),(K^{\prime},v^{\prime})$ be two equal characteristic non-trivially
valued henselian fields. Then $(K,v)\equiv_{\exists}(K^{\prime},v^{\prime})$
in $L_{\text{val}}$ if and only if $Kv\equiv_{\exists}K^{\prime}v^{\prime}$ in
$L_{\text{r}}$.

######  Theorem 2.2.3 (Corollary 7.5 [AF16]).

Let $(K,v)$ be an equal characteristic henselian valued field. Then
$Th_{\exists}(K,v)$ is decidable in $L_{\text{val}}$ if and only if
$Th_{\exists}(Kv)$ is decidable in $L_{\text{r}}$.

######  Remark 2.2.4.

In residue characteristic $0$, Theorems 2.2.2 and 2.2.3 were essentially known
by work of Ax-Kochen-Ershov prior to the work of Anscombe-Fehm (see Remark 7.3
[AF16]).

####  2.2.5. Existential AKE in mixed characteristic

In mixed characteristic, one can easily construct counterexamples of Theorems
2.2.2 and 2.2.3 (see Remark 7.6 [AF16]). It is then natural to ask what an
existential AKE principle in mixed characteristic would look like. This will
be Theorem 2.2.6 below, whose proof follows closely the proof of Theorem
2.1.5.

We write $\mathfrak{m}_{n}$ for the maximal ideal of
${\mathcal{O}}_{v}/p^{n}{\mathcal{O}}_{v}$ and
$({\mathcal{O}}_{v}/p^{n}{\mathcal{O}}_{v},\mathfrak{m}_{n})$ for the local
ring, viewed as an $L_{\text{lcr}}$-structure (see notation).

######  Theorem 2.2.6.

Let $(K,v),(K^{\prime},v^{\prime})$ be two henselian valued fields of mixed
characteristic. Then we have that $(a)$
$(K,v)\equiv_{\exists}(K^{\prime},v^{\prime})$ in $L_{\text{val}}$ if and only
if
$\mathcal{O}_{v}/p^{n}{\mathcal{O}}_{v}\equiv_{\exists}{\mathcal{O}}_{v^{\prime}}/p^{n}{\mathcal{O}}_{v^{\prime}}$
in $L_{\text{r}}$ for all $n\in\mathbb{N}$, and $(b)$
$(K,v)\equiv_{\exists}(K^{\prime},v^{\prime})$ in $L_{\text{val}}$ if and only
if
$(\mathcal{O}_{v}/p^{n}{\mathcal{O}}_{v},\mathfrak{m}_{n})\equiv_{\exists^{+}}({\mathcal{O}}_{v^{\prime}}/p^{n}{\mathcal{O}}_{v^{\prime}},\mathfrak{m}_{n}^{\prime})$
in $L_{\text{lcr}}$ for all $n\in\mathbb{N}$.

###### Proof.

$(a)$ "$\Rightarrow$": Clear.
"$\Leftarrow$": By symmetry, it will suffice to show that
$(K,v)\models\text{Th}_{\exists}(K^{\prime},v^{\prime})$. We may further
assume $(K^{\prime},v^{\prime})$ is countable by downward Löwenheim-Skolem and
that $(K,v)$ is $\aleph_{1}$-saturated.

We again consider the valuations $w,\overline{v}$ (resp.
$w^{\prime},\overline{v^{\prime}}$) that were introduced in the proof of
Theorem 2.1.5. By our assumption, we have for each $n\in\mathbb{N}$ an
embedding of rings
${\mathcal{O}}_{v^{\prime}}/p^{n}{\mathcal{O}}_{v^{\prime}}\hookrightarrow\mathcal{O}_{v}/p^{n}{\mathcal{O}}_{v}$.
Claim: There is an injective ring embedding
$\phi:\mathcal{O}_{\bar{v}^{\prime}}\hookrightarrow\mathcal{O}_{\bar{v}}$.

###### Proof.

Let us fix an enumeration of $\mathcal{O}_{v^{\prime}}$, say
$\mathcal{O}_{v^{\prime}}=(a_{i})_{i\in\mathbb{N}}$. Consider the set of
formulas in countably many variables $x=(x_{n})_{n\in\omega}$ of the form

$\Sigma(x)=\\{x_{i}\diamond x_{j}=x_{v}(p^{n}),x_{m}\Box
x_{\rho}(p^{n}):a_{i}\in\mathcal{O}_{v^{\prime}},a_{i}\diamond
a_{j}=a_{v}(p^{n}),a_{m}\Box a_{\rho}(p^{n})\\}$

where $\diamond$ is either $+$ or $\cdot$ and $\Box$ is either $=$ or $\neq$.
Since for each $n\in\mathbb{N}$ we have an embedding of rings
$\mathcal{O}_{v^{\prime}}/p^{n}\mathcal{O}_{v^{\prime}}\hookrightarrow\mathcal{O}_{v}/p^{n}\mathcal{O}_{v}$,
we get that $\Sigma(x)$ is finitely satisfiable. The ring $\mathcal{O}_{v}$ is
$\aleph_{1}$-saturated and we thus have $b=(b_{n})_{n\in\omega}$ with
$b\models\Sigma(x)$. Using that
$\mathcal{O}_{\bar{v}}=\mathcal{O}_{v}/\bigcap_{n\in\omega}p^{n}\mathcal{O}_{v}$
(resp.
$\mathcal{O}_{\overline{v^{\prime}}}=\mathcal{O}_{v^{\prime}}/\bigcap_{n\in\omega}p^{n}\mathcal{O}_{v^{\prime}}$),
one readily checks that the map
$\mathcal{O}_{v^{\prime}}\to\mathcal{O}_{v}:a_{i}\mapsto b_{i}$ descends to a
ring embedding
$\mathcal{O}_{\bar{v}^{\prime}}\hookrightarrow\mathcal{O}_{\bar{v}}$.
∎${}_{\textit{Claim}}$

Claim 1 provides us with a valued field embedding
$\phi:(K^{\prime}w^{\prime},\bar{v}^{\prime})\hookrightarrow(Kw,\bar{v})$. By
the existential Ax-Kochen-Ershov principle in pure characteristic $0$ (see
Theorem 2.2.2 and Remark 2.2.4), we get that
$(K,w)\models\text{Th}_{\exists}(K^{\prime},w^{\prime})$. Replacing $K$ with
an $\aleph_{1}$-saturated extension in a suitable language (see footnote 5),
we will also have an embedding $(K^{\prime},w^{\prime})\hookrightarrow(K,w)$.

By the relative embedding property for equal characteristic $0$ henselian
valued fields (see Theorem 7.1 in [Kuh16] for a more general statement), we
can even find $\Phi:(K^{\prime},w^{\prime})\hookrightarrow(K,w)$ that induces
$\phi:(K^{\prime}w^{\prime},\bar{v}^{\prime})\hookrightarrow(Kw,\bar{v})$.
Finally, we get that the map
$\Phi:(K^{\prime},v^{\prime})\hookrightarrow(K,v)$ is an embedding of valued
fields, as in the proof of Claim 2, Theorem 2.1.5.
$(b)$ For $f(x_{1},...,x_{m})\in\mathbb{Z}[x_{1},...,x_{m}]$ and
$(a_{1},...,a_{m})\in{\mathcal{O}}_{v}^{m}$, note that $f(a_{1},...,a_{m})\neq
0\mod p^{n}{\mathcal{O}}_{v}$ if and only if there exists
$y\in\mathfrak{m}_{v}$ such that $f(a_{1},...,a_{m})\cdot y=p^{n}\mod
p^{n+1}{\mathcal{O}}_{v}$ (similarly for ${\mathcal{O}}_{v^{\prime}}$).
Consequently, for each $n\in\mathbb{N}$, we see that if
${\mathcal{O}}_{v}/(p^{n+1}{\mathcal{O}}_{v},\mathfrak{m}_{n+1})\equiv_{\exists^{+}}({\mathcal{O}}_{v^{\prime}}/p^{n+1}{\mathcal{O}}_{v^{\prime}},\mathfrak{m}^{\prime}_{n+1})$
in $L_{\text{lcr}}$, then
${\mathcal{O}}_{v}/p^{n}{\mathcal{O}}_{v}\equiv_{\exists}{\mathcal{O}}_{v^{\prime}}/p^{n}{\mathcal{O}}_{v^{\prime}}$
in $L_{\text{r}}$. The conclusion follows from part $(a)$. ∎

###  2.3. Decidability

We now harvest the consequences of Theorem 2.1.5 and 2.2.6 in relation to
decidability.

######  Remark 2.3.1 (see Proposition 7.1.1).

We stress right away that there exist examples of henselian valued fields
$(K,v)$ with $(vK,vp)$ decidable in $L_{\text{oag}}$ and also
${\mathcal{O}}_{K}/(p^{n})$ decidable in $L_{\text{r}}$ for each
$n\in\mathbb{N}$, but with $(K,v)$ undecidable in $L_{\text{val}}$.

We need to ask not only for each individual ${\mathcal{O}}_{K}/(p^{n})$ to be
decidable in $L_{\text{r}}$ but for the sequence
$({\mathcal{O}}_{K}/(p^{n}))_{n\in\omega}$ to be uniformly decidable in
$L_{\text{r}}$:

######  Corollary 2.3.2.

Let $(K,v)$ be a henselian valued field of mixed characteristic. Suppose that
$(vK,vp)$ is decidable in $L_{\text{oag}}$ with a constant for $vp$ and that
the sequence $({\mathcal{O}}_{K}/(p^{n}))_{n\in\omega}$ is uniformly decidable
in $L_{\text{r}}$. Then $(K,v)$ is decidable in $L_{\text{val}}$.

###### Proof.

The identification $vK=K^{\times}/{\mathcal{O}}^{\times}$ furnish us with a
recursive interpretation $\Gamma$ of $(vK,vp)$ in the valued field $(K,v)$.
Let also $\Gamma_{n}$ be the natural interpretation of
${\mathcal{O}}_{K}/(p^{n})$ in the $L_{\text{val}}$-structure $(K,v)$, for
each $n\in\mathbb{N}$. If
$g_{n}:\text{Sent}_{L_{r}}\to\text{Sent}_{L_{\text{val}}}$ denotes the
reduction map of $\Gamma_{n}$, then one can see that the sequence
$(g_{n})_{n\in\omega}$ is uniformly recursive (using that $\Gamma_{n}$ is
uniform in $n\in\mathbb{N}$). Let

$\Sigma:=\text{Hen}_{(0,p)}\cup(\bigcup_{n\in\omega}\\{\phi_{\Gamma_{n}}:{\mathcal{O}}_{K}/(p^{n})\models\phi\\})\cup\\{\phi_{\Gamma}:(vK,vp)\models\phi\\}$

where $\text{Hen}_{(0,p)}$ is a first-order axiom schema capturing Hensel’s
lemma (see e.g. pg.21 [Kuh16]), together with a set of sentences capturing
that the valued field has mixed characteristic $(0,p)$.
Claim: The axiomatization $\Sigma$ is r.e.

###### Proof.

The set $\text{Hen}_{(0,p)}$ is clearly r.e. The set
$\\{\phi_{\Gamma}:(vK,vp)\models\phi\\}\subseteq L_{\text{val}}$ is r.e.,
being the image of a recursive set via the recursive reduction map of
$\Gamma$. Since recursively enumerable sets are closed under finite unions, it
remains to show that
$\bigcup_{n\in\omega}\\{\phi_{\Gamma_{n}}:{\mathcal{O}}_{K}/(p^{n})\models\phi\\}$
is r.e.

Let $\chi:\mathbb{N}\times\text{Sent}_{L_{r}}\to\mathbb{N}$ be the recursive
function associated to the uniformly decidable sequence
$({\mathcal{O}}_{K}/(p^{n}))_{n\in\omega}$. We construct the partial recursive
function
$\chi^{\prime}:\mathbb{N}\times\text{Sent}_{L_{r}}\to\mathbb{N}\times\text{Sent}_{L_{r}}$
which maps $(n,\phi)\mapsto(n,\phi)$ if $\chi(n,\phi)=1$ and is undefined if
$\chi(n,\phi)=0$. Consider also the recursive function
$g:\mathbb{N}\times\text{Sent}_{L_{r}}\to\text{Sent}_{L_{r}}:(n,m)\mapsto
g_{n}(m)$ associated to the uniformly recursive sequence
$(g_{n})_{n\in\omega}$. Observe that
$\bigcup_{n\in\omega}\\{\phi_{\Gamma_{n}}:{\mathcal{O}}_{K}/(p^{n})\models\phi\\}=\text{Im}(F)$
where $F$ is the (partial) recursive function $F=g\circ\chi^{\prime}$. It
follows that
$\bigcup_{n\in\omega}\\{\phi_{\Gamma_{n}}:{\mathcal{O}}_{K}/(p^{n})\models\phi\\}$
is r.e. ∎${}_{\textit{Claim}}$

If $(K^{\prime},v^{\prime})\models\Sigma$, then
$(K^{\prime},v^{\prime})\equiv(K,v)$ in $L_{\text{val}}$ by Theorem 2.1.5. We
therefore get that $\Sigma$ is a complete axiomatization of $(K,v)$. We
conclude that the $L_{\text{val}}$-theory of $(K,v)$ admits a r.e. and
complete axiomatization, whence $(K,v)$ is decidable. ∎

######  Corollary 2.3.3.

Let $(K,v)$ be a henselian valued field of mixed characteristic. Suppose that
the sequence $(({\mathcal{O}}_{K}/(p^{n}),\mathfrak{m}_{n})_{n\in\omega}$ is
uniformly $\exists^{+}$-decidable in $L_{\text{lcr}}$. Then $(K,v)$ is
$\exists$-decidable in $L_{\text{val}}$.

###### Proof.

Similar to Corollary 2.3.2, ultimately using Theorem 2.2.6$(b)$. ∎

##  3\. Perfectoid fields

###  3.1. Introduction

####  3.1.1. Motivation

The theory of perfectoid fields (and spaces), introduced by Scholze in
[Sch12], was initially designed as a means of transferring results available
in positive characteristic to mixed characteristic (see §1 [Sch14]). It
formalizes the earlier Krasner-Kazhdan-Deligne philosophy (due to [Kra56],
[Kaz86] and [Del84]), of approximating a highly ramified mixed characteristic
field with a positive characteristic field. Within the framework of perfectoid
fields, this kind of approximation becomes precise and robust with the use of
the tilting functor (see §3.2).

All this is substantially different from the Ax-Kochen method (see e.g. 2.20
[vdD12]), which achieves a model-theoretic transfer principle asymptotically,
i.e. with the residue characteristic $p\to\infty$. The theory of perfectoid
fields will allow us to transport decidability information for a fixed residue
characteristic (but with high ramification), setting the stage for a different
type of model-theoretic transfer principle.

####  3.1.2. Definition

######  Definition 3.1.3.

A perfectoid field is a complete valued field $(K,v)$ of residue
characteristic $p>0$ such that $vK$ is a dense subgroup of ${\mathbb{R}}$ and
the Frobenius map $\Phi:\mathcal{O}_{K}/(p)\to\mathcal{O}_{K}/(p)$ is
surjective.

######  Example 3.1.4.

$(a)$ The $p$-adic completions of $\mathbb{Q}_{p}(p^{1/p^{\infty}})$,
$\mathbb{Q}_{p}(\zeta_{p^{\infty}})$ and $\mathbb{Q}_{p}^{ab}$ are mixed
characteristic perfectoid fields.
$(b)$ The $t$-adic completions of $\mathbb{F}_{p}((t))^{1/p^{\infty}}$ and
$\overline{\mathbb{F}}_{p}((t))^{1/p^{\infty}}$ are perfectoid fields of
characteristic $p$.

######  Remark 3.1.5.

In characteristic $p$, a perfectoid field is simply a perfect, complete non-
archimedean valued field.

###  3.2. Tilting

####  3.2.1. Introduction

A construction, originally due to Fontaine, provides us with a tilting functor
that takes any perfectoid field $K$ and transforms it into a perfectoid field
$K^{\flat}$ of characteristic $p$. We shall now describe this tilting functor.
For more details, see §3 [Sch12].

####  3.2.2. Definition

Given a perfectoid field $(K,v)$, we shall now define its tilt
$(K^{\flat},v^{\flat})$. Let $\varprojlim_{x\mapsto x^{p}}K$ be the limit of
the inverse system

$...\xrightarrow{x\mapsto x^{p}}K\xrightarrow{x\mapsto
x^{p}}K\xrightarrow{x\mapsto x^{p}}K$

which is identified as $\varprojlim_{x\mapsto
x^{p}}K=\\{(x_{n})_{n\in\omega}:x_{n+1}^{p}=x_{n}\\}$, viewed as a
multiplicative monoid via
$(x_{n})_{n\in\omega}\cdot(y_{n})_{n\in\omega}=(x_{n}\cdot
y_{n})_{n\in\omega}$. Let $\varpi\in{\mathcal{O}}_{K}$ be such that
$0<v\varpi\leq vp$ (e.g. $\varpi=p$ when $\text{char}(K)=0$ and $\varpi=0$
when $\text{char}(K)=p$) and consider the ring
$\varprojlim_{\Phi}{\mathcal{O}}_{K}/(\varpi)$ which is the limit of the
inverse system of rings

$...\xrightarrow{\Phi}{\mathcal{O}}_{K}/(\varpi)\xrightarrow{\Phi}{\mathcal{O}}_{K}/(\varpi)\xrightarrow{\Phi}{\mathcal{O}}_{K}/(\varpi)$

where $\Phi:{\mathcal{O}}_{K}/(\varpi)\to{\mathcal{O}}_{K}/(\varpi):x\mapsto
x^{p}$ is the Frobenius homomorphism.

######  Lemma 3.2.3 (Lemma 3.4 $(i),(ii)$ [Sch12]).

$(a)$ The ring $\varprojlim_{\Phi}{\mathcal{O}}_{K}/(\varpi)$ is independent
of the choice of $\varpi$ and there is a multiplicative isomorphism
$\varprojlim_{x\mapsto
x^{p}}K\xrightarrow{\cong}\varprojlim_{\Phi}{\mathcal{O}}_{K}/(\varpi)$.
Moreover, we get a multiplicative morphism
$\sharp:\varprojlim_{\Phi}{\mathcal{O}}_{K}/(\varpi)\to{\mathcal{O}}_{K}:x\mapsto
x^{\sharp}$ such that if $x=(x_{n}+(\varpi))_{n\in\omega}$, then
$x^{\sharp}\equiv x_{0}\mod(\varpi)$.
$(b)$ There is an element
$\varpi^{\flat}\in\varprojlim_{\Phi}{\mathcal{O}}_{K}/(\varpi)$ with
$v(\varpi^{\flat})^{\sharp}=v\varpi$.

The definition of $\sharp$ goes as follows: Let
$x=(x_{n})_{n\in\omega}\in\varprojlim_{\Phi}{\mathcal{O}}_{K}/(\varpi)$, i.e.
$x_{n}\in{\mathcal{O}}_{K}/(\varpi)$ and $x_{n+1}^{p}=x_{n}$. Let
$\tilde{x}_{n}\in{\mathcal{O}}_{K}$ be an arbitrary lift of
$x_{n}\in{\mathcal{O}}_{K}/(\varpi)$. Then the limit
$\lim_{n\to\infty}\tilde{x}_{n}^{p^{n}}$ exists and is independent of the
choice of the $\tilde{x}_{n}$’s (see Lemma 3.4$(i)$ [Sch12]). We define
$x^{\sharp}:=\lim_{n\to\infty}\tilde{x}_{n}^{p^{n}}$.

We now introduce
$K^{\flat}:=\varprojlim_{\Phi}{\mathcal{O}}_{K}/(\varpi)[(\varpi)^{-1}]$,
which for now is merely a ring. It can be made into a valued field according
to the following:

######  Lemma 3.2.4 (Lemma 3.4 $(iii)$ [Sch12]).

$(a)$ There is a morphism of multiplicative monoids $K^{\flat}\to K:x\mapsto
x^{\sharp}$ (extending the one of Lemma 3.2.3$(a)$) inducing an isomorphism of
multiplicative monoids $K^{\flat}\xrightarrow{\cong}\varprojlim_{x\mapsto
x^{p}}K:x\mapsto(x^{\sharp},(x^{\sharp})^{1/p},...)$. The map
$v^{\flat}:K^{\flat}\to vK$ sending $x\mapsto vx^{\sharp}$ is a valuation on
$K^{\flat}$ that makes $(K^{\flat},v^{\flat})$ into a perfectoid field of
characteristic $p$. If ${\mathcal{O}}_{K^{\flat}}$ is the valuation ring of
$K^{\flat}$, then we have a ring isomorphism
${\mathcal{O}}_{K^{\flat}}\cong\varprojlim_{\Phi}{\mathcal{O}}_{K}/(\varpi)$.
$(b)$ We have an isomorphism of ordered abelian groups
$(vK,v\varpi)\cong(v^{\flat}K^{\flat},v^{\flat}\varpi^{\flat})$ and a field
isomorphism $Kv\cong K^{\flat}v^{\flat}$. Moreover, we have a ring isomorphism
${\mathcal{O}}_{K}/(\varpi)\cong{\mathcal{O}}_{K^{\flat}}/(\varpi^{\flat})$.

######  Remark 3.2.5.

Lemma 3.2.4$(a)$ allows us to identify the multiplicative underlying monoid of
$K^{\flat}$ with $\varprojlim_{x\mapsto x^{p}}K$. It is not hard to see that,
via this identification, addition is described by
$(x_{n})_{n\in\omega}+(y_{n})_{n\in\omega}=(z_{n})_{n\in\omega}$, where
$z_{n}=\lim_{m\to\infty}(x_{n+m}+y_{n+m})^{p^{m}}$.

######  Definition 3.2.6.

We say that the valued field $(K^{\flat},v^{\flat})$ constructed in Lemma
3.2.4$(a)$ is the tilt of the perfectoid field $(K,v)$.

######  Remark 3.2.7 (Lemma 3.4 $(iv)$ [Sch12]).

If $(K,v)$ is a perfectoid field of characteristic $p$, then
$(K^{\flat},v^{\flat})\cong(K,v)$.

######  Example 3.2.8 (see Corollary 4.4.3).

In the examples below, $\widehat{K}$ stands for the $p$-adic (resp. $t$-adic)
completion of the field $K$ depending on whether its characteristic is $0$ or
$p$.
$(a)$
$\widehat{\mathbb{Q}_{p}(p^{1/p^{\infty}})}^{\flat}\cong\widehat{\mathbb{F}_{p}((t))^{1/p^{\infty}}}$
and $t^{\sharp}=p$.
$(b)$
$\widehat{\mathbb{Q}_{p}(\zeta_{p^{\infty}})}^{\flat}\cong\widehat{\mathbb{F}_{p}((t))^{1/p^{\infty}}}$
and $(t+1)^{\sharp}=\zeta_{p}$.
$(c)$
$\widehat{\mathbb{Q}_{p}^{ab}}^{\flat}\cong\widehat{\overline{\mathbb{F}}_{p}((t))^{1/p^{\infty}}}$
and $(t+1)^{\sharp}=\zeta_{p}$.

######  Remark 3.2.9.

The tilting construction makes sense for non-perfectoid fields as well.
However, in the absence of infinite wild ramification, it is too lossy for it
to be useful (e.g. $\mathbb{Q}_{p}^{\flat}=\mathbb{F}_{p}$).

###  3.3. Witt vectors

We review the basics of Witt vectors. Details and proofs can be found in §5,6
[Ser80] and §6 [vdD12].

####  3.3.1. $p$-rings

######  Definition 3.3.2.

A ring $R$ of characteristic $p$ is called perfect if the Frobenius
homomorphism $\Phi:R\to R:x\mapsto x^{p}$ is surjective.

######  Definition 3.3.3.

$(a)$ A $p$-ring is a commutative ring $A$ provided with a filtration
$\mathfrak{a}_{1}\supset\mathfrak{a}_{2}\supset...$ such that
$\mathfrak{a}_{n}\mathfrak{a}_{m}\subset\mathfrak{a}_{n+m}$ and so that $A$ is
Hausdorff and complete with respect to the topology induced by the filtration.
$(b)$ If in addition $\mathfrak{a}_{n}=p^{n}A$ and $p$ is not a zero-divisor,
then we say that $A$ is a strict $p$-ring.

If $A$ is a $p$-ring, we call the quotient ring $A/pA$ the residue ring of the
$p$-ring $A$ and write $\text{res}:A\to A/pA$ for the quotient map. A system
of multiplicative representatives is a multiplicative homomorphism
$f_{A}:A/pA\to A$ such that $f_{A}(\text{res}(x))=x$. A $p$-ring $A$ always
has a system of representatives $f_{A}:A/pA\to A$ and when $A$ is strict every
element $a\in A$ can be written uniquely in the form
$a=\sum_{i=0}^{\infty}f_{A}(\alpha_{i})\cdot p^{i}$ (pg. 37 [Ser80]).

######  Proposition 3.3.4 (Proposition 10, pg. 38 [Ser80]).

Let $A$ and $A^{\prime}$ be two $p$-rings with residue rings $R$ and
$R^{\prime}$ respectively, multiplicative representatives $f_{A}:R\to A$ and
$f_{A^{\prime}}:R^{\prime}\to A^{\prime}$ respectively and suppose that $A$ is
strict. For every ring homomorphism $\phi:R\to R^{\prime}$ there exists a
unique homomorphism $g:A\to A^{\prime}$ making the diagram below commute

${A}$${A^{\prime}}$${R}$${R^{\prime}}$$\scriptstyle{g}$$\scriptstyle{\phi}$

More precisely, $g(\sum_{i=0}^{\infty}f_{A}(\alpha_{i})\cdot
p^{i})=\sum_{i=0}^{\infty}f_{A^{\prime}}(\phi(\alpha_{i}))\cdot p^{i}$.

######  Theorem 3.3.5 (Corollary pg.39 [Ser80]).

For every perfect ring $R$, there exists a unique strict $p$-ring, denoted by
$W(R)$, with residue ring $W(R)/pW(R)\cong R$.

The ring $W(R)$ is said to be the ring of Witt vectors over the ring $R$.

####  3.3.6. Teichmüller representatives

When $R$ is perfect, one has a system of Teichmüller representatives of $R$ in
$W(R)$. This is the (unique) multiplicative homomorphism $[\ ]:R\to W(R)$ with
the property that $\text{res}([x])=x$ for all $x\in R$. Explicitly, for $x\in
R$ and $n\in\mathbb{N}$, let $x_{n}\in R$ be such that $x_{n}^{p^{n}}=x$ and
$\tilde{x}_{n}\in W(R)$ be an arbitrary lift of $x_{n}$. The sequence
$(\tilde{x}_{n}^{p^{n}})_{n\in\omega}$ is a Cauchy sequence, whose limit is
independent of the chosen lifts. We let
$[x]:=\lim_{n\to\infty}\tilde{x}_{n}^{p^{n}}$ (see Proposition 8 pg. 35
[Ser80]).

It is easy to see that any element $x\in W(R)$ can be written uniquely in the
form $x=\sum_{n=0}^{\infty}[x_{n}]\cdot p^{n}$, for some $x_{i}\in R$. The
vector $(x_{1},x_{2},...)\in R^{\omega}$ is called the Teichmüller vector of
$x$.

####  3.3.7. Witt vectors

Since $R$ is perfect, we may also write $x$ in the form
$x=\sum_{n=0}^{\infty}[x_{n}^{p^{n}}]\cdot p^{n}$, for some $x_{i}\in R$ and
the vector $(x_{1},x_{2},...)\in R^{\omega}$ is called the Witt vector of $x$.
The advantage of Witt vectors over Teichmüller vectors, comes from the fact
that the ring operations in $W(R)$ have nicer coordinatewise descriptions when
using the former.

####  3.3.8. Ring operations

By the discussion above, the ring $W(R)$ can be thought of as the $p$-adic
analogue of formal power series with coefficients in $R$. By identifying $x$
with its Witt vector, we see that $W(R)$ has $R^{\omega}$ as its underlying
set. By Lemma 6.5 [vdD12], the ring operations are given by

$(a_{0},a_{1},...)+(b_{0},b_{1},...)=(S_{0}(a_{0},b_{0}),S_{1}(a_{0},a_{1},b_{0},b_{1}),...)$

and

$(a_{0},a_{1},...)\cdot(b_{0},b_{1},...)=(P_{0}(a_{0},b_{0}),P_{1}(a_{0},a_{1},b_{0},b_{1}),...)$

for suitable polynomials
$S_{i},P_{i}\in\mathbb{Z}[x_{0},...,x_{i},y_{0},...,y_{i}]$ which are
universal, in the sense that they do not depend on $R$.

######  Observation 3.3.9.

The polynomials $S_{i}$ (resp. $P_{i}$) are computable, i.e. the function
$\mathbb{N}\to\mathbb{Z}[x_{0},y_{0},...]:n\mapsto S_{n}$ (resp.
$\mathbb{N}\to\mathbb{Z}[x_{0},y_{0},...]:n\mapsto P_{n}$) is recursive.

###### Proof.

For $n\in\mathbb{N}$, we introduce the $n$-th Witt polynomial
$W_{n}(x_{0},...,x_{n})=x_{0}^{p^{n}}+px_{1}^{p^{n-1}}+...+p^{n}\cdot
x_{n}\in\mathbb{Z}[x_{0},...,x_{n}]$ (pg. 135 [vdD12]). The proof of Lemma 6.5
[vdD12] shows that $S_{0}(x_{0},y_{0})=x_{0}+y_{0}$ and
$W_{n-1}(S_{0}^{p},...,S_{n-1}^{p})+p^{n}\cdot
S_{n}=W_{n-1}(S_{0}(x_{0}^{p},y_{0}^{p}),...,S_{n-1}(x_{0}^{p},...,x_{n-1}^{p},y_{0}^{p},...,y_{n-1}^{p}))+p^{n}(x_{n}+y_{n})$,
whence the polynomial $S_{n}$ may be computed recursively from
$S_{0},...,S_{n-1}$. The proof is similar for the $P_{n}$’s. ∎

###  3.4. Truncated Witt vectors

####  3.4.1. Definition

In this paper, we will mostly be working with truncated Witt vectors. These
can be thought of as $p$-adic analogues of truncated power series, i.e.
elements of the ring $R[[t]]/(t^{n})\cong R[t]/(t^{n})$ (over some base ring
$R$). More formally:

######  Definition 3.4.2.

Let $R$ be a perfect ring. Given $n\in\mathbb{N}$, the ring of $n$-truncated
Witt vectors over $R$ is defined as $W_{n}(R):=W(R)/p^{n}W(R)$.

####  3.4.3. Language

For a perfect ring $R$, the pair $(W(R),R)$ (resp. $(W_{n}(R),R)$) is viewed
as a two-sorted structure with sorts W for the Witt ring $W(R)$ (resp.
$W_{n}(R)$) and R for the residue ring $R$. The sort W is equipped with the
language of rings $L_{\text{r}}$, while the sort R may be equipped with any
$L\supseteq L_{r}$. We also have a function symbol for the Teichmüller map $[\
]:\textbf{R}\to\textbf{W}$. For each choice of a language $L$ for the R-sort,
the resulting language will be denoted by $\langle L_{r},L\rangle$.

####  3.4.4. Interpretability in $R$

######  Lemma 3.4.5.

Let $R$ be a perfect ring, viewed as an $L$-structure with $L\supseteq L_{r}$.
For each $n\in\mathbb{N}$, there exists a quantifier-free interpretation
$\Gamma_{n}$ of the $\langle L_{r},L\rangle$-structure $(W_{n}(R),R)$ in the
$L$-structure $R$ such that the sequence of interpretations
$(\Gamma_{n})_{n\in\omega}$ is uniformly recursive.

###### Proof.

By §3.3.8, for $n\in\mathbb{N}$ the underlying set of $W_{n}(R)$ can be
identified with $R^{n}$, so we take $\partial_{\Gamma_{n}}(x_{1},...,x_{n})$
to be $\bigwedge_{i=1}^{n}x_{i}=x_{i}$ and the coordinate map
$f_{\Gamma_{n}}:R^{n}\to W_{n}(R)$ as the identity map. The ring operations of
$W_{n}(R)$ are given by
$(a_{0},...,a_{n-1})+(b_{0},...,b_{n-1})=(S_{0}(a_{0},b_{0}),...,S_{n-1}(a_{0},b_{0},...,a_{n-1},b_{n-1}))$
and
$(a_{0},...,a_{n-1})\cdot(b_{0},...,b_{n-1})=(P_{0}(a_{0},b_{0}),...,P_{n-1}(a_{0},b_{0},...,a_{n-1},b_{n-1}))$,
for certain polynomials
$S_{i},P_{i}\in\mathbb{Z}[x_{0},...,x_{i},y_{0},...,y_{i}]$, for
$i=0,...,n-1$. We now need to describe the map $\phi\mapsto\phi_{\Gamma_{n}}$
on unnested atomic $L_{\text{r}}$-formulas:

  1. (1)

If $\phi(x,y,z)$ is the formula $x+y=z$ (here $x,y,z\in\textbf{W}$), we may
take $\phi_{\Gamma_{n}}(\overline{x},\overline{y},\overline{z})$ to be the
$L_{\text{r}}$-formula
$\bigwedge_{i=0}^{n-1}z_{i}=S_{i}(x_{0},y_{0},...,x_{n-1},y_{n-1})$.

  2. (2)

If $\phi(x,y,z)$ is the formula $x\cdot y=z$ (here $x,y,z\in\textbf{W}$), we
may take $\phi_{\Gamma_{n}}(\overline{x},\overline{y},\overline{z})$ to be the
$L_{\text{r}}$-formula
$\bigwedge_{i=0}^{n-1}z_{i}=P_{i}(x_{0},y_{0},...,x_{n-1},y_{n-1})$.

  3. (3)

If $\phi(x,y)$ is the formula $\iota(x)=y$ (here $x\in\textbf{R}$ and
$y\in\textbf{W}$), we may take $\phi_{\Gamma_{n}}(x,\overline{y})$ to be the
formula $\bigwedge_{i=1}^{n-1}y_{i}=0\land y_{0}=x$.

  4. (4)

If $\phi(\overline{x})$ is an unnested atomic $L$-formula with variables from
the sort R, then we may take
$\phi_{\Gamma_{n}}(\overline{x}):=\phi(\overline{x})$ (here
$x_{i}\in\textbf{R}$ for the latter formula).

The above data defines a quantifier-free interpretation $\Gamma_{n}$ of the
$\langle L_{r},L\rangle$-structure $(W_{n}(R),R)$ in the $L$-structure $R$.
Moreover, since the polynomials $S_{i},P_{i}$ are computable (Observation
3.3.9), the sequence of interpretations $(\Gamma_{n})_{n\in\omega}$ is
uniformly recursive. ∎

###  3.5. Untilting

Expository notes on the material of this section may be found either in §5 of
the Bourbaki seminar given by Morrow [Mor17] or in the lecture series notes by
Lurie (see Lecture 2 [Lur18]).

####  3.5.1. Overview

The functor $K\mapsto K^{\flat}$ is far from being faithful, i.e. there will
be several non-isomorphic mixed characteristic perfectoid fields $K$ that tilt
to the same perfectoid field of characteristic $p$. For example, the $p$-adic
completions of $\mathbb{Q}_{p}(p^{1/p^{\infty}})$ and
$\mathbb{Q}_{p}(\zeta_{p^{\infty}})$ both tilt to the $t$-adic completion of
$\mathbb{F}_{p}((t))^{1/p^{\infty}}$. For a perfectoid field $F$ of
characteristic $p$, an untilt of $F$ is a perfectoid field $K$ such that
$K^{\flat}\cong F$.666In some places in the literature, one often includes the
isomorphism $\iota:K^{\flat}\xrightarrow{\cong}F$ as part of the data that
defines an untilt. This is especially important in the definition of the
Fargues-Fontaine curve but will not be important here.  Fargues and Fontaine
give a description of all possible untilts of $F$ in an intrinsic way, i.e. in
a way that uses only arithmetic from $F$ itself (see Theorem 3.5.10). This
will be useful later in the formulation of Theorem A.

####  3.5.2. The ring $\textbf{A}_{\text{inf}}$

Fix any perfectoid field $(F,w)$ of positive characteristic. We introduce
$\textbf{A}_{\text{inf}}:=W(\mathcal{O}_{F})$, called the infinitesimal period
ring, which is the ring of Witt vectors over the valuation ring of $F$.

######  Definition 3.5.3.

An element $\xi\in\textbf{A}_{\text{inf}}$ is said to be distinguished if it
is of the form $\xi=\sum_{n=0}^{\infty}[c_{n}]p^{n}$, with
$c_{0}\in\mathfrak{m}_{F}$ and $c_{1}\in\mathcal{O}_{F}^{\times}$.

In other words, distinguished elements are those of the form $\xi=[\pi]-up$
where $w\pi>0$ and $u\in\textbf{A}_{\text{inf}}$ is a unit.

######  Remark 3.5.4.

Let $\text{res}:{\mathcal{O}}_{F}\to{\mathcal{O}}_{F}/\mathfrak{m}_{F}$ be the
residue map and $W(\text{res}):\textbf{A}_{\text{inf}}\to
W({\mathcal{O}}_{F}/\mathfrak{m}_{F})$ be the (unique) induced ring
homomorphism provided by Proposition 3.3.4 that maps
$\sum_{n=0}^{\infty}[c_{n}]p^{n}\mapsto\sum_{n=0}^{\infty}[\text{res}(c_{n})]\cdot
p^{n}$. The element $\xi\in\textbf{A}_{\text{inf}}$ is distinguished precisely
when $W(\text{res})(\xi)$ is a unit multiple of $p$ in
$W({\mathcal{O}}_{F}/\mathfrak{m}_{F})$.

####  3.5.5. Distinguished elements and untilts

We now explain the relationship between distinguished elements and untilts of
$F$. Let $(K,v)$ be an untilt of $(F,w)$, i.e.
$(K^{\flat},v^{\flat})\cong(F,w)$. By Lemma 3.2.3$(a)$, we have a morphism of
multiplicative monoids $\sharp:{\mathcal{O}}_{F}\to{\mathcal{O}}_{K}$ inducing
a ring homomorphism ${\mathcal{O}}_{F}\to{\mathcal{O}}_{K}/(p)$. By
Proposition 3.3.4, the latter lifts uniquely to a ring homomorphism
$\theta:\textbf{A}_{\text{inf}}\to{\mathcal{O}}_{K}$ sending
$\sum_{n=0}^{\infty}[c_{n}]\cdot
p^{n}\mapsto\sum_{n=0}^{\infty}c_{n}^{\sharp}\cdot p^{n}$.

By Lemma 3.2.3$(b)$, there exists an element $\pi\in{\mathcal{O}}_{F}$ such
that $v^{\flat}\pi=vp$. It follows that $v\pi^{\sharp}=vp$ and therefore
$\pi^{\sharp}=u\cdot p$ for some $u\in{\mathcal{O}}_{K}^{\times}$. By Lemma
3.2.4$(b)$, we have ${\mathcal{O}}_{K}/(p)\cong{\mathcal{O}}_{F}/(\pi)$. Let
$\overline{u}\in{\mathcal{O}}_{F}/(\pi)$ be the image of $u$ via
${\mathcal{O}}_{K}\to{\mathcal{O}}_{K}/(p)\xrightarrow{\cong}{\mathcal{O}}_{F}/(\pi)$.
Note that $[\pi]-\overline{u}\cdot p\in\textbf{A}_{\text{inf}}$ is a
distinguished element and that $[\pi]-\overline{u}\cdot
p\in\text{Ker}(\theta)$. In fact, the following is true:

######  Proposition 3.5.6 (Corollary 17 [Lur18]).

Let $(F,w)$ be a perfectoid field of characteristic $p$ and $(K,v)$ be an
untilt. Let $\theta:\textbf{A}_{\text{inf}}\to{\mathcal{O}}_{K}$ be as above.
Then $\text{Ker}(\theta)$ is a principal ideal generated by any distinguished
element $\xi\in\text{Ker}(\theta)$.

Starting with an untilt $(K,v)$, we have thus produced a distinguished element
$\xi_{K}\in\textbf{A}_{\text{inf}}$ (unique only up to a unit in
$\textbf{A}_{\text{inf}}$).

Conversely, starting with a distinguished element
$\xi\in\textbf{A}_{\text{inf}}$ we may produce an untilt $(K,v)$ of $(F,w)$ as
follows. Write
$\theta:\textbf{A}_{\text{inf}}\to\textbf{A}_{\text{inf}}/(\xi)$ for the
quotient map. We then have:

######  Lemma 3.5.7 (Lecture 3, pg. 4, Claim $(a)$ [Lur18])).

For every $y\in\textbf{A}_{\text{inf}}/(\xi)$, there exists
$x\in{\mathcal{O}}_{F}$ such that $(y)=(\theta([x]))$.

######  Proposition 3.5.8 (Proposition 16 [Lur18]).

Let $(F,w)$ be a perfectoid field of characteristic $p$ and
$\xi\in\textbf{A}_{\text{inf}}$ a distinguished element. Then the quotient
ring $\textbf{A}_{\text{inf}}/(\xi)$ can be identified with the valuation ring
${\mathcal{O}}_{K}$ of an untilt $(K,v)$ of $(F,w)$. The valuation $v$ is such
that if $y\in\textbf{A}_{\text{inf}}/(\xi)$, then $vy:=wx$ with
$x\in{\mathcal{O}}_{F}$ so that $(y)=(\theta([x]))$ in
$\textbf{A}_{\text{inf}}/(\xi)$.

######  Remark 3.5.9.

Let $(K,v)$ be an untilt of $(F,w)$ and $(\xi)=([\pi]-up)$ be its associated
ideal. Note that $(p)=(\theta([\pi])$ in $\textbf{A}_{\text{inf}}/(\xi)$ and
therefore $vp=w\pi$.

Let $Y_{F}$ denote the set of characteristic $0$ untilts of $(F,w)$, up to
isomorphism. We write $0$ for the isomorphism class of the unique
characteristic $p$ untilt of $(F,w)$, represented by $(F,w)$ itself (Remark
3.2.7), and set $\overline{Y}_{F}=Y_{F}\cup\\{0\\}$ for the set of all untilts
of $(F,w)$, up to isomorphism. The above discussion is summarized in the
following:

######  Theorem 3.5.10 (Fargues-Fontaine).

Let $(F,w)$ be a perfectoid field of characteristic $p$. The map
$\xi\mapsto\text{Frac}({\mathcal{O}}_{F}/(\xi))$ defines a bijective
correspondence between the set of ideals of $\textbf{A}_{\text{inf}}$
generated by a distinguished element and $\overline{Y}_{F}$.

###### Proof.

See Proposition 5.1 [Mor17] or Lecture 2, Corollary 18 [Lur18]. ∎

###  3.6. Space of untilts

In §3.6.3 we exhibit an appealing model-theoretic property of the space
$Y_{F}$ of untilts of $(F,w)$. This will not be used in the rest of the paper.
We then study in §3.6.5 the size of the space of untilts up to elementary
equivalence. We will see that the cardinality is often too big, so that one
cannot possibly expect $K$ to be decidable simply relative to $K^{\flat}$
(Remark 4.3.7).

####  3.6.1. Metric structure on $Y_{F}$

Fargues and Fontaine equip $\overline{Y}_{F}$ with a metric topology, which
allows us to view the space of untilts geometrically. Suppose
$x=(K_{x},v_{x})$ and $y=(K_{y},v_{y})$ are two "points" of
$\overline{Y}_{F}$, corresponding to the ideals $(\xi_{x})$ and $(\xi_{y})$
respectively, provided by Theorem 3.5.10. We choose an embedding
$wF\hookrightarrow\mathbb{R}$, which determines embeddings
$v_{y}K_{y}\hookrightarrow\mathbb{R}$ for all $y\in Y_{F}$ via the canonical
identification $v_{y}K_{y}\cong wF$. One then defines
$d(x,y):=-\log(v_{y}(\theta_{y}(\xi_{x})))$, where
$\theta_{y}:\textbf{A}_{\text{inf}}\twoheadrightarrow\textbf{A}_{\text{inf}}/(\xi_{y})={\mathcal{O}}_{K_{y}}$
is the quotient map.

######  Proposition 3.6.2 (Proposition 2.3.2$(1)$ [FF19]).

Let $d:\overline{Y}_{F}\times\overline{Y}_{F}\to\mathbb{R}$ be as above. The
pair $(\overline{Y}_{F},d)$ is a complete ultrametric space.

###### Proof.

See Propositions 2.3.2, 2.3.4 [FF19] or Lecture 14, Propositions 6, 7 [Lur18].
∎

Recall that $0\in\overline{Y}_{F}$ is the isomorphism class of the untilt
corresponding to $(F,w)$ itself. We can define a radius function
$r(y):=d(0,y)$ for $y\in\overline{Y}_{F}$, which allows us to think of
$\overline{Y}_{F}$ intuitively as the unit disc with center $0$.

####  3.6.3. A model-theoretic property of $Y_{F}$

We now show that limits in the punctured unit disc $Y_{F}$, with respect to
the Fargues-Fontaine metric, agree with limits in the model-theoretic sense:

######  Proposition 3.6.4.

Let $(x_{n})_{n\in\omega}$ be a sequence in $Y_{F}$ such that
$x_{n}\stackrel{{\scriptstyle d}}{{\rightarrow}}x$ and $x\neq 0$. Let
$(K_{n},v_{n})$ be the untilt associated to $x_{n}$ and $(K,v)$ the untilt
associated to $x$. We set

$(K^{*},v^{*}):=\prod_{n\in\omega}(K_{n},v_{n})/U$

for a non-principal ultrafilter $U$ on $\mathcal{P}(\mathbb{N})$. Then
$(K^{*},v^{*})\equiv(K,v)$ in $L_{\text{val}}$.

###### Proof.

Let $(\xi_{n})=([\pi_{n}]-u_{n}p)$ be the ideal in $\textbf{A}_{\text{inf}}$
corresponding to $(K_{n},v_{n})$ and $(\xi)=([\pi]-up)$ be the ideal
corresponding to $(K,v)$. Note that the set
$\\{v_{n}p:n\in\mathbb{N}\\}\subseteq wF$ is bounded from above because
otherwise there would be a subsequence $(x_{n_{k}})_{k\in\mathbb{N}}$ with
$x_{n_{k}}\to 0$.

Fix $m\in\mathbb{N}$ and let $(\bar{\xi}_{n})$ and $(\bar{\xi})$ be the images
of the ideals $(\xi_{n})$ and $(\xi)$ in $W_{m}(\mathcal{O}_{F})$ via
$\textbf{A}_{\text{inf}}\twoheadrightarrow\textbf{A}_{\text{inf}}/(p^{m})=W_{m}(\mathcal{O}_{F})$.
Claim: We have that $(\bar{\xi}_{n})=(\bar{\xi})$, for sufficiently large $n$.

###### Proof.

We let
$\theta:\textbf{A}_{\text{inf}}\to\textbf{A}_{\text{inf}}/(\xi)={\mathcal{O}}_{K}$
be the quotient map. Since $d(x_{n},x)\to 0$, we get that
$v(\theta(\xi_{n}))\to\infty$. We will thus have that $\xi_{n}\equiv
p^{m}\cdot\alpha_{n}\mod(\xi)$, for some
$\alpha_{n}\in\textbf{A}_{\text{inf}}$ and for all sufficiently large $n$.
Consequently, one gets that $(\bar{\xi}_{n})\subseteq(\bar{\xi})$ for large
enough $n\in\mathbb{N}$. Similarly, since $v_{n}(\theta_{n}(\xi))\to\infty$
and $\\{v_{n}p:n\in\mathbb{N}\\}$ is bounded, we get that
$v_{n}(\theta_{n}(\xi))\geq mv_{n}p$ for large enough $n\in\mathbb{N}$. It
follows that for large enough $n\in\mathbb{N}$, there exists
$\beta_{n}\in\textbf{A}_{\text{inf}}$ such that $\xi\equiv
p^{m}\cdot\beta_{n}\mod(\xi_{n})$. We conclude that
$(\bar{\xi}_{n})=(\bar{\xi})$. ∎${}_{\textit{Claim}}$

We have in particular that $(\xi_{n})$ and $(\xi)$ have the same image in
$\textbf{A}_{\text{inf}}/(p)\cong{\mathcal{O}}_{F}$, for sufficiently large
$n$ and therefore $(\pi_{n})=(\pi)$ in $\mathcal{O}_{F}$. By Lemma 3.2.4$(b)$,
we get that $(v_{n}K_{n},v_{n}p)\cong(wK,w\pi_{n})=(wK,w\pi)\cong(vK,vp)$, for
all sufficiently large $n\in\mathbb{N}$. The conclusion follows from Theorem
2.1.5 and Łoś’ s Theorem. ∎

####  3.6.5. The space $Y_{F}/\equiv$

It is natural to consider $Z_{F}$, which is the set $Y_{F}$ up to elementary
equivalence. More precisely:

######  Definition 3.6.6.

Let $(F,w)$ be a perfectoid field of characteristic $p$. For
$x=(K_{x},v_{x}),y=(K_{y},v_{y})\in Y_{F}$ define the equivalence relation
$x\sim y\iff(K_{x},v_{x})\equiv(K_{y},v_{y})$ in $L_{\text{val}}$. We define
$Z_{F}:=Y_{F}/\sim$.

We determine the size of $Z_{F}$ in a few cases.777The construction in part
$(b)$ is an elaborated version of Scholze’s answer [Sch] to a closely related
mathoverflow question asked by the author.

######  Proposition 3.6.7.

$(a)$ Let $(F,w)=(\widehat{\overline{\mathbb{F}_{p}((t))}},v_{t})$, the
$t$-adic completion of the algebraic closure of $\mathbb{F}_{p}((t))$. Then
$|Z_{F}|=1$.
$(b)$ Set $(F,w)=(\widehat{\mathbb{F}_{p}((t))^{1/p^{\infty}}},v_{t})$. Then
$|Z_{F}|=\mathfrak{c}$.
$(c)$ Set $(F,w)=(\mathbb{F}_{p}((t^{\Gamma})),v_{t})$, where
$\mathbb{F}_{p}((t^{\Gamma}))$ is the Hahn field with value group
$\Gamma=\frac{1}{p^{\infty}}\mathbb{Z}$ and residue field $\mathbb{F}_{p}$.
Then $|Z_{F}|=\mathfrak{c}$.

###### Proof.

$(a)$ Let $(K,v)$ be such that
$(K^{\flat},v^{\flat})\cong(\widehat{\overline{\mathbb{F}_{p}((t))}},v_{t})$.
By Proposition 4.3 [Sch12], we get that $K$ is algebraically closed. By
Robinson’s completeness of $\text{ACVF}_{(0,p)}$ (see Corollary 3.34 [vdD12]),
we get that $\text{Th}(K,v)=\text{ACVF}_{(0,p)}$ and hence
$|Z_{F}|=1$.888Alternatively, one can use the fact that we have a ring
isomorphism
${\mathcal{O}}_{K}/(p^{n})\cong{\mathcal{O}}_{{\mathbb{C}}_{p}}/(p^{n})$ for
each $n\in\mathbb{N}$ (Remark 2.24 [FF14]) together with Theorem 2.1.5. One
also has to check that $(vK,vp)\equiv(v_{p}{\mathbb{C}}_{p},v_{p}p)$ in
$L_{\text{oag}}$ with a constant for $vp$. First note that $vK\cong wF$ is
divisible as $F$ is algebraically closed. The conclusion now is an application
of quantifier elimination for the theory ODAG of ordered divisible abelian
groups (Corollary 3.1.17 [Mar02]).
$(b)$ For each $\alpha\in 2^{\omega}$, we define a algebraic extension
$K_{\alpha}$ of $\mathbb{Q}_{p}$, which tilts to
$\widehat{\mathbb{F}_{p}((t))^{1/p^{\infty}}}$ as follows. Set
$\pi_{\alpha(0)}=\zeta_{p}-1$ and $K_{\alpha(0)}=\mathbb{Q}_{p}(\zeta_{p})$.
By Lemma 9.1 [Was97], there is some $u\in\mathbb{Z}[\zeta_{p}]$ with $u\equiv
1\mod p\mathbb{Z}_{p}[\zeta_{p}]$ such that
$\mathbb{Q}_{p}(\zeta_{p},u^{1/p})$ is an unramified $p$-degree extension of
$\mathbb{Q}_{p}(\zeta_{p})$. We now define inductively:
(1) $K_{\alpha(n)}=K_{\alpha(n-1)}(\pi_{\alpha(n-1)}^{1/p})$ and
$\pi_{\alpha(n)}=\pi_{\alpha(n-1)}^{1/p}$ if $\alpha(n)=0$ or
(2) $K_{\alpha(n)}=K_{\alpha(n-1)}((u\pi_{\alpha(n-1)})^{1/p})$ and
$\pi_{\alpha(n)}=(u\pi_{\alpha(n-1)})^{1/p}$ if $\alpha(n)=1$.
We let $K_{\alpha}=\bigcup_{n\in\omega}K_{\alpha(n)}$.
Claim 1: If $\alpha\neq\beta$, then $K_{\alpha}\not\equiv_{\exists}K_{\beta}$
in $L_{\text{r}}$.

###### Proof.

Let $n\in\omega$ be least such that $\alpha(n)\neq\beta(n)$. Note that for
each $n\in\omega$, the elements $\pi_{\alpha(n)},\pi_{\beta(n)}$ are algebraic
over $\mathbb{Q}$. Now if $K_{\alpha}\equiv_{\exists}K_{\beta}$, we would have
that $\pi_{\alpha(n-1)}^{1/p},(u\pi_{\alpha(n-1)})^{1/p}\in K_{\alpha}$. This
would imply that $u^{1/p}\in K_{\alpha}$, which is contrary to the fact that
$K_{\alpha}/\mathbb{Q}_{p}$ is totally ramified. ∎${}_{\textit{Claim 1}}$

Next we prove:
Claim 2: For every $\alpha\in 2^{\omega}$, we have that
$\widehat{K_{\alpha}}^{\flat}=\widehat{\mathbb{F}_{p}((t))^{1/p^{\infty}}}$.

###### Proof.

For every $n\in\mathbb{N}$, we have that $K_{\alpha(n+1)}/K_{\alpha(n)}$ is
totally ramified and
${\mathcal{O}}_{K_{\alpha(n+1)}}={\mathcal{O}}_{K_{\alpha(n)}}[\pi_{\alpha(n+1)}]$
(Corollary pg.19 [Ser80]). It follows that
${\mathcal{O}}_{K_{\alpha}}\cong\mathbb{Z}_{p}[\\{\pi_{\alpha(n)}:n\in\mathbb{N}\\}]$.
Let $\Phi_{p}(x)$ be the $p$-th cyclotomic polynomial. Using that $u\equiv
1\mod p\mathbb{Z}_{p}[\zeta_{p}]$ and that $\Phi_{p}(x)\equiv(x-1)^{p-1}\mod
p\mathbb{Z}[x]$, we compute

${\mathcal{O}}_{K_{\alpha}}/(p)=\mathbb{Z}_{p}[x_{1},x_{2},...]/(p,\Phi_{p}(x_{1}),x_{2}^{p}-ux_{1},...)\cong$
$\cong\mathbb{F}_{p}[x_{1},x_{2},...]/((x_{1}-1)^{p-1},x_{2}^{p}-x_{1},...)\cong\mathbb{F}_{p}[x_{1}^{1/p^{\infty}}]/(x_{1}-1)^{p-1}\stackrel{{\scriptstyle
t=x_{1}-1}}{{=}}\mathbb{F}_{p}[t^{1/p^{\infty}}]/(t^{p-1})$

We thus get that
$\varprojlim_{\Phi}{\mathcal{O}}_{K_{\alpha}}/(p)\cong\widehat{\mathbb{F}_{p}[[t]]^{1/p^{\infty}}}$
and the conclusion follows. ∎${}_{\textit{Claim 2}}$

Claims 1,2 show that $|Z_{F}|\geq\mathfrak{c}$. On the other hand, the bound
$|Z_{F}|\leq\mathfrak{c}$ is automatic, as any first-order
$L_{\text{val}}$-theory can be identified with a subset of
$\text{Sent}_{L_{\text{val}}}\simeq\mathbb{N}$. This finishes the proof of
$(a)$.
$(c)$ For $\alpha\in 2^{\omega}$, we let $K_{\alpha}$ be as in part $(b)$. Let
also $\xi_{\alpha}\in W(\widehat{\mathbb{F}_{p}[[t]]^{1/p^{\infty}}})$ be a
distinguished element such that ${\mathcal{O}}_{K_{\alpha}}\cong
W(\widehat{\mathbb{F}_{p}[[t]]^{1/p^{\infty}}})/(\xi_{\alpha})$ (Theorem
3.5.10). We now let
$L_{\alpha}=\text{Frac}(W(\mathbb{F}_{p}[[t^{\Gamma}]])/(\xi_{\alpha}))$. This
is an immediate extension of $K_{\alpha}$ such that
$L_{\alpha}^{\flat}=\mathbb{F}_{p}((t^{\Gamma}))$. Arguing as in Claim 1 of
part $(b)$, we see that $L_{\alpha}\not\equiv_{\exists}L_{\beta}$ in
$L_{\text{r}}$ when $\alpha\neq\beta$ and in particular
$|Z_{F}|=\mathfrak{c}$, arguing as in part $(b)$. ∎

######  Remark 3.6.8.

Set $(F,w)=(\widehat{\overline{\mathbb{F}_{p}((t))}},v_{t})$. In Remark 2.24
[FF14], Fargues and Fontaine ask whether every untilt $(K,v)$ of $(F,w)$ is
isomorphic to $({\mathbb{C}}_{p},v_{p})$ or equivalently whether $|Y_{F}|$
equals $1$. They note in Remark 2.24 that for any untilt $(K,v)$ of $(F,w)$
one has
${\mathcal{O}}_{K}/(p^{n})\cong{\mathcal{O}}_{{\mathbb{C}}_{p}}/(p^{n})$ for
each $n\in\mathbb{N}$. However, these isomorphisms are obtained in a non-
canonical way and do not necessarily yield a valued field isomorphism between
$K$ and ${\mathbb{C}}_{p}$. In fact, a counterexample was provided by Kedlaya-
Temkin in Theorem 1.3 [KT18], showing that $|Y_{F}|\neq 1$. This should be
contrasted with Proposition 3.6.7$(a)$, which shows that $|Z_{F}|=1$.

######  Remark 3.6.9.

In Remark 7.6 [AF16], the authors write:
"At present, we do not know of an example of a mixed characteristic henselian
valued field $(K,v)$ for which $\text{Th}_{\exists}(Kv)$ and
$\text{Th}_{\exists}(vK,vp)$ are decidable but $\text{Th}_{\exists}(K,v)$ is
undecidable."
Let us provide such an example. By Proposition 3.6.7$(b)$ and the fact that
there are countably many Turing machines, there must exist an undecidable
perfectoid (thus henselian) field $(K,v)$ with
$K^{\flat}\cong\widehat{\mathbb{F}_{p}((t))^{1/p^{\infty}}}$. By Lemma
3.2.4$(b)$, we have that $Kv\cong\mathbb{F}_{p}$ and
$(vK,vp)\cong(\frac{1}{p^{\infty}}\mathbb{Z},\gamma)$ (for some
$\gamma\in\frac{1}{p^{\infty}}\mathbb{Z}$), both of which are decidable.999The
latter is an easy application of Robinson-Zakon [RZ60].

######  Remark 3.6.10.

For future use in Proposition 7.1.1, we record here that Claim 2 gives
$t+1=(\pi_{\alpha(0)}+1+(p),\pi_{\alpha(1)}+1+(p),...)$, via the
identifications
$\widehat{\mathbb{F}_{p}[[t]]^{1/p^{\infty}}}\cong\varprojlim_{\Phi}{\mathcal{O}}_{K_{\alpha}}/(p)$
(resp. via
$\mathbb{F}_{p}[[t^{\Gamma}]]\cong\varprojlim_{\Phi}{\mathcal{O}}_{L_{\alpha}}/(p)$).

######  Question 3.6.11.

Is there a perfectoid field $(F,w)$ of characteristic $p$ with
$1<|Z_{F}|<\mathfrak{c}$?

##  4\. Relative decidability for perfectoid fields

###  4.1. Introduction

We shall now use the results of §3 to prove Theorem A and Corollary A.

####  4.1.1. Work of Rideau-Scanlon

It is my understanding that there is ongoing work by Rideau-Scanlon, which
aims at giving a model-theoretic account of many of the concepts and facts
that were discussed in §3, in the context of continuous logic. In particular,
they obtain a bi-interpretability result between ${\mathcal{O}}_{K}$ and
${\mathcal{O}}_{K^{\flat}}$ in the sense of continous logic. It should be
noted that their bi-interpretability result was conceived prior to the present
paper and in fact influenced the material that is presented here.

####  4.1.2. Plan of action

Without any adjustments, the interpretation of Rideau-Scanlon does not quite
yield an interpretation in the sense of ordinary first-order logic. The
problem is that when one converts statements about ${\mathcal{O}}_{K}$ to
statements about ${\mathcal{O}}_{K^{\flat}}$ via ${\mathcal{O}}_{K}\cong
W({\mathcal{O}}_{K^{\flat}})/(\xi)$, one ends up with infinitely many
variables, coming from the coefficients of the Witt vectors. This problem
disappears by interpreting one residue ring ${\mathcal{O}}_{K}/(p^{n})$ at a
time, via ${\mathcal{O}}_{K}/(p^{n})\cong
W_{n}({\mathcal{O}}_{K^{\flat}})/(\xi\mod(p^{n}))$. This approach is
facilitated by Corollary 2.3.2. In §2.3, we emphasized that uniform
decidability of the residue rings is key, which will eventually come down to
the computability of the distinguished element $\xi$ associated to $K$.

###  4.2. Computable untilts

####  4.2.1. Computable Witt vectors

In analogy with a computable real number, a computable $p$-adic integer is one
for which there is an algorithm which outputs the sequence of its $p$-adic
digits. More precisely, a $p$-adic integer $a\in\mathbb{Z}_{p}$ of the form
$a=\sum_{n=0}^{\infty}[\alpha_{n}]\cdot p^{n}$ is said to be computable
precisely when the function
$\mathbb{N}\to\mathbb{Z}/p\mathbb{Z}:n\mapsto\alpha_{n}$ is recursive. The
notion of a computable $p$-adic integer extends naturally to the more general
concept of a computable Witt vector.

First recall that a computable ring $R_{0}$ is one for which there exists a
bijection $R_{0}\simeq\mathbb{N}$, via which the operations $+:R_{0}\times
R_{0}\to R_{0}$ and $\cdot:R_{0}\times R_{0}\to R_{0}$ are identified with
recursive functions.

######  Definition 4.2.2.

Fix a perfect ring $R$ and let $R_{0}\subseteq R$ a computable subring.
Consider the ring $W(R)$, the ring of Witt vectors over $\xi\in R$. An element
$\xi\in W(R)$ with Witt vector coordinates $(\xi_{0},\xi_{1},...)\in
R^{\omega}$ is said to be $R_{0}$-computable if:

  1. (1)

For each $n\in\mathbb{N}$, we have that $\xi_{n}\in R_{0}$.

  2. (2)

The function $\mathbb{N}\to R_{0}:n\mapsto\xi_{n}$ is recursive.

######  Example 4.2.3.

$(a)$ Let $R_{0}=R=\mathbb{F}_{p}$. Then the computable elements of
$\mathbb{Z}_{p}=W(\mathbb{F}_{p})$ are the usual computable $p$-adic integers.
$(b)$ A non-example: Let $R$ be any perfect ring, $R_{0}\subseteq R$ any
computable subring and $S\subseteq\mathbb{N}$ be a non-recursive set. The
element $\xi=\sum_{n\in S}p^{n}\in W(R)$ is not computable.

####  4.2.4. Computable untilts

Let $(F,w)$ be a perfectoid field of characteristic $p$ and
$\textbf{A}_{\text{inf}}=W({\mathcal{O}}_{F})$. Let also
$R_{0}\subseteq{\mathcal{O}}_{F}$ be a computable subring.

######  Remark 4.2.5.

The reader is welcome to assume that
$F=\widehat{\mathbb{F}_{p}((t))^{1/p^{\infty}}}$ or
$\widehat{\overline{\mathbb{F}}_{p}((t))^{1/p^{\infty}}}$ and
$R_{0}=\mathbb{F}_{p}[t]$. This case is enough for the applications presented
here, i.e. Corollaries A and B.

We shall now define what it means for an untilt $K$ of $F$ to be
$R_{0}$-computable:

######  Definition 4.2.6.

Let $(K,v)$ be an untilt of $(F,w)$ and $R_{0}\subseteq{\mathcal{O}}_{F}$ a
computable subring. We say that $(K,v)$ is an $R_{0}$-computable untilt of
$(F,w)$ if there is an $R_{0}$-computable distinguished element
$\xi_{K}\in\textbf{A}_{\text{inf}}$ with
$\mathcal{O}_{K}\cong\textbf{A}_{\text{inf}}/(\xi_{K})$.

######  Example 4.2.7.

In Corollary 4.4.6 we will see that
$\widehat{\mathbb{Q}_{p}(p^{1/p^{\infty}})}$ and
$\widehat{\mathbb{Q}_{p}(\zeta_{p^{\infty}})}$ are computable untilts of
$\widehat{\mathbb{F}_{p}((t))^{1/p^{\infty}}}$. Also, that
$\widehat{\mathbb{Q}_{p}^{ab}}$ is a computable untilt of
$\widehat{\overline{\mathbb{F}}_{p}((t))^{1/p^{\infty}}}$.

###  4.3. Relative decidability

Let $(K,v)$ be an untilt of $(F,w)$ and $\xi_{K}\in\textbf{A}_{\text{inf}}$ be
such that ${\mathcal{O}}_{K}=\textbf{A}_{\text{inf}}/(\xi_{K})$ (Theorem
3.5.10). We write $\overline{\xi}_{K,n}$ for the image of $\xi_{K}$ in
$W_{n}({\mathcal{O}}_{F})$ via
$\textbf{A}_{\text{inf}}\twoheadrightarrow\textbf{A}_{\text{inf}}/(p^{n})\cong
W_{n}({\mathcal{O}}_{F})$.

####  4.3.1. Interpretability

All the necessary background material related to interpretability is presented
in §1.2.

######  Lemma 4.3.2.

For each $n\in\mathbb{N}$, there is a $\exists^{+}$-interpretation
$\text{A}_{n}$ of the local ring
$({\mathcal{O}}_{K}/(p^{n}),\mathfrak{m}_{n})$ in the $\langle
L_{r},L_{\text{lcr}}\rangle$-structure
$((W_{n}({\mathcal{O}}_{F}),{\mathcal{O}}_{F}),\overline{\xi}_{K,n})$ such
that the sequence of interpretations $(\text{A}_{n})_{n\in\omega}$ is
uniformly recursive.

###### Proof.

Fix $n\in\mathbb{N}$. We have that
${\mathcal{O}}_{K}/(p^{n})\cong\textbf{A}_{\text{inf}}/(p^{n},\xi_{K})\cong
W_{n}(\mathcal{O}_{F})/(\overline{\xi}_{K,n})$. By Proposition 3.5.8, we get
that $\mathfrak{m}_{n}=\\{\theta_{n}([x]):x\in\mathfrak{m}_{F}\\}$, where
$\theta_{n}$ is the composite map
$\textbf{A}_{\text{inf}}\stackrel{{\scriptstyle\theta}}{{\twoheadrightarrow}}\textbf{A}_{\text{inf}}/(\xi_{K})\stackrel{{\scriptstyle\mod
p^{n}}}{{\twoheadrightarrow}}W_{n}({\mathcal{O}}_{F})/(\overline{\xi}_{K,n})\xrightarrow{\cong}{\mathcal{O}}_{K}/(p^{n})$.
We take $\partial_{\text{A}_{n}}(x)$ to be $x=x$. Let $c$ be a constant symbol
with $c^{(W_{n}({\mathcal{O}}_{F}),{\mathcal{O}}_{F})}=\overline{\xi}_{K,n}$.
The reduction map $\text{Form}_{L_{\text{lcr}}}\to\langle
L_{r},L_{\text{lcr}}\rangle\cup\\{c\\}:\phi\mapsto\phi_{\text{A}_{n}}$ on
unnested atomic formulas is described as follows:

  1. (1)

If $\phi(x,y)$ is $x=y$, then we take $\phi_{\text{A}_{n}}$ to be the formula
$x,y\in\textbf{W}\land\exists z\in\textbf{W}(x=y+z\cdot c)$.101010As usual,
the conjunct $x,y\in W$ is an informal way of saying that the variables $x,y$
are taken from the sort W. The cases of $x=0$ and $x=1$ are similar.

  2. (2)

If $\phi(x,y,z)$ is $x\diamond y=z$, then we take $\phi_{\text{A}_{n}}$ to be
the formula $x,y,z\in\textbf{W}\land\exists w\in\textbf{W}(x\diamond
y=z+w\cdot c)$, where $\diamond$ is either $\cdot$ or $+$.

  3. (3)

If $\phi(x)$ is $x\in\mathfrak{m}$, then we take $\phi_{\text{A}_{n}}(x)$ to
be the formula $x\in\textbf{W}\land\exists
z,w\in\textbf{W},y\in\textbf{R}(y\in\mathfrak{m}\land x=[y]\cdot z+w\cdot c)$.

The coordinate map
$f_{\text{A}_{n}}:W_{n}({\mathcal{O}}_{F})\twoheadrightarrow{\mathcal{O}}_{K}/(p^{n})$
is the one induced by the map $\theta_{n}$ above. The above data defines a
recursive $\exists^{+}$-interpretation $\text{A}_{n}$ of
$({\mathcal{O}}_{K}/(p^{n}),\mathfrak{m}_{n})$ in
$((W_{n}({\mathcal{O}}_{F}),{\mathcal{O}}_{F}),\overline{\xi}_{K,n})$. The
sequence of interpretations $(\text{A}_{n})_{n\in\omega}$ is uniform in
$\mathbb{N}$ and thus also (trivially) uniformly recursive. ∎

######  Proposition 4.3.3.

Suppose $(K,v)$ is an untilt of $(F,w)$ and that
$\xi_{K}\in\textbf{A}_{\text{inf}}$ is such that
${\mathcal{O}}_{K}=\textbf{A}_{\text{inf}}/(\xi_{K})$. Let
$(\xi_{0},\xi_{1},...)\in{\mathcal{O}}_{F}^{\omega}$ be the Witt vector
coordinates of $\xi_{K}$. Then:
$(a)$ The value group $(vK,vp)$ is interpretable in the valued field
$((F,w),\xi_{0})$.
$(b)$ For each $n\in\mathbb{N}$, there exists a $\exists^{+}$-interpretation
$\Gamma_{n}$ of the local ring $({\mathcal{O}}_{K}/(p^{n}),\mathfrak{m}_{n})$
in the local ring
$(({\mathcal{O}}_{F},\mathfrak{m}_{F}),\xi_{0},...,\xi_{n-1})$ such that the
sequence of interpretations $(\Gamma_{n})_{n\in\omega}$ is uniformly
recursive.

###### Proof.

$(a)$ By Lemma 3.2.4$(b)$, we will have that $(vK,vp)\cong(wF,w\xi_{0})$ and
the value group $(wF,w\xi_{0})$ is interpretable in the
$L_{\text{val}}\cup\\{c\\}$-structure $(F,w)$ with $c^{(F,w)}=\xi_{0}$.
$(b)$ Lemma 4.3.2 provides us with a $\exists^{+}$-interpretation
$\text{A}_{n}$ of the local ring
$({\mathcal{O}}_{K}/(p^{n}),\mathfrak{m}_{n})$ in the $\langle
L_{\text{r}},L_{\text{lcr}}\rangle\cup\\{c\\}$-structure
$((W_{n}({\mathcal{O}}_{F}),{\mathcal{O}}_{F}),\overline{\xi}_{K,n})$ such
that the sequence of interpretations $(\text{A}_{n})_{n\in\omega}$ is
uniformly recursive. Lemma 3.4.5 provides us with a quantifier-free
interpretation $\text{B}_{n}$ of
$((W_{n}({\mathcal{O}}_{F}),{\mathcal{O}}_{F}),\overline{\xi}_{K,n})$ in the
local ring $(({\mathcal{O}}_{F},\mathfrak{m}_{F}),\xi_{0},...,\xi_{n-1})$,
such that the sequence of interpretations $(\text{B}_{n})_{n\in\omega}$ is
uniformly recursive. Let $\Gamma_{n}$ be the composite interpretation of
$({\mathcal{O}}_{K}/(p^{n}),\mathfrak{m}_{n})$ in
$(({\mathcal{O}}_{F},\mathfrak{m}_{F}),\xi_{0},...,\xi_{n-1})$. This is also a
$\exists^{+}$-interpretation by Lemma 1.2.4 and the sequence of
interpretations $(\Gamma_{n})_{n\in\omega}$ is uniformly recursive by Lemma
1.2.13. ∎

####  4.3.4. Proof of Theorem A

Given a computable subring $R_{0}\subseteq{\mathcal{O}}_{F}$, we write
$L_{\text{val}}(R_{0})$ for the language $L_{\text{val}}$ enriched with a
constant with a constant $c_{a}$ for each $a\in R_{0}$ (see notation). The
valued field $(F,w)$ can then be updated to an
$L_{\text{val}}(R_{0})$-structure with $c_{a}^{(F,w)}=a$.

######  Theorem A.

Let $(F,w)$ be a perfectoid field of characteristic $p$. Suppose
$R_{0}\subseteq{\mathcal{O}}_{F}$ is a computable subring and $(K,v)$ is an
$R_{0}$-computable untilt of $(F,w)$.
$(a)$ If $(F,w)$ is decidable in $L_{\text{val}}(R_{0})$, then $(K,v)$ is
decidable in $L_{\text{val}}$.
$(b)$ If $(F,w)$ is $\exists$-decidable in $L_{\text{val}}(R_{0})$, then
$(K,v)$ is $\exists$-decidable in $L_{\text{val}}$.

###### Proof.

$(a)$ By Proposition 4.3.3$(a)$, we get that $(vK,vp)$ is decidable in
$L_{\text{oag}}$ with a constant for $vp$. By Proposition 4.3.3$(b)$, for each
$n\in\mathbb{N}$, there exists an interpretation $\Gamma_{n}$ of the
$L_{\text{lcr}}$-structure $({\mathcal{O}}_{K}/(p^{n}),\mathfrak{m}_{n})$ in
the $L_{\text{lcr}}\cup\\{c_{0},...,c_{n-1}\\}$-structure
$(({\mathcal{O}}_{F},\mathfrak{m}_{F}),\xi_{0},...,\xi_{n-1})$ with
$c_{m}^{({\mathcal{O}}_{F},\mathfrak{m}_{F})}=\xi_{m}$ for $m=0,...,n-1$.
Moreover, the sequence of interpretations $(\Gamma_{n})_{n\in\omega}$ is
uniformly recursive.
Claim: There exists an interpretation $\Delta_{n}$ of the
$L_{\text{lcr}}\cup\\{c_{0},...,c_{n-1}\\}$-structure
$(({\mathcal{O}}_{F},\mathfrak{m}_{F}),\xi_{0},...,\xi_{n-1})$ in the
$L_{\text{val}}(R_{0})$-structure $(F,w)$ such that the sequence of
interpretations $(\Delta_{n})_{n\in\omega}$ is uniformly recursive.

###### Proof.

Fix $n\in\mathbb{N}$. We take $\partial_{\Delta_{n}}(x)$ to be the formula
$x\in{\mathcal{O}}$. The reduction map on unnested atomic formulas is
described as follows:

  1. (1)

If $\phi(x)$ is the formula $x=c_{m}$, for some $m=0,...,n-1$, we take
$\phi_{\Delta_{n}}(x)$ to be the formula $x=c_{\xi_{m}}$.

  2. (2)

If $\phi(x)$ is the formula $x\in\mathfrak{m}$, we take $\phi_{\Delta_{n}}(x)$
to be the formula $\exists y(xy=1\land y\notin{\mathcal{O}})$.

  3. (3)

If $\phi(x,y,z)$ is $x\diamond y=z$, then we take $\phi_{\Delta_{n}}$ to be
the formula $x,y,z\in{\mathcal{O}}\land\phi(x,y,z)$, where $\diamond$ is
either $\cdot$ or $+$.

The coordinate map
$f_{\Delta_{n}}:\partial_{\Delta_{n}}(F)\to{\mathcal{O}}_{F}$ is the identity.
Since $\mathbb{N}\to R_{0}:m\mapsto\xi_{m}$ is recursive, the reduction map
$\mathbb{N}\times\text{Form}_{L_{\text{lcr}}\cup\\{c_{0},...,c_{n-1}\\}}\to\text{Form}_{L_{\text{val}}(R_{0})}:(n,\phi)\mapsto\phi_{\Delta_{n}}$
restricted to unnested atomic formulas is recursive. By definition, this means
that the sequence of interpretations $(\Delta_{n})_{n\in\omega}$ is uniformly
recursive. ∎${}_{\textit{Claim}}$

By Proposition 1.2.12, we get that the sequence of rings
$({\mathcal{O}}_{K}/(p^{n}))_{n\in\omega}$ is uniformly decidable in
$L_{\text{r}}$. The conclusion follows from Corollary 2.3.2. The proof of
$(b)$ goes along the same lines, using that the interpretations $\Gamma_{n}$
and $\Delta_{n}$ are positive existential. ∎

The assumption of completeness in Theorem A can be easily relaxed to
henselianity:

######  Corollary 4.3.5.

Let $(F,w)$ be a perfect non-trivially valued field with a rank $1$ value
group and $R_{0}\subseteq{\mathcal{O}}_{F}$ be a computable subring. Suppose
$(K,v)$ is a mixed characteristic henselian valued field such that
$(\widehat{K},\widehat{v})$ is an $R_{0}$-computable untilt of
$(\widehat{F},\widehat{w})$. Then:
$(a)$ If $(F,w)$ is decidable in $L_{\text{val}}(R_{0})$, then $(K,v)$ is
decidable in $L_{\text{val}}$.
$(b)$ If $(F,w)$ is $\exists$-decidable in $L_{\text{val}}(R_{0})$, then
$(K,v)$ is $\exists$-decidable in $L_{\text{val}}$.

###### Proof.

Suppose $\widehat{\mathcal{O}}_{K}=W(\widehat{\mathcal{O}}_{F})/(\xi)$, where
$\xi=[\pi]-up$, with $\pi\in\mathfrak{m}_{F}\cap R_{0}$ and $u\in
W(\mathcal{O}_{F})^{\times}$. Note that
$W(\widehat{\mathcal{O}}_{F})/([\pi]-up)\cong W(\mathcal{O}_{F})/([\pi]-up)$.
For each $n\in\mathbb{N}$, one has that

${\mathcal{O}}_{K}/(p^{n})\cong\widehat{\mathcal{O}}_{K}/(p^{n})\cong
W_{n}(\widehat{\mathcal{O}}_{F})/([\pi]-up)\cong
W_{n}(\mathcal{O}_{F})/([\pi]-up)$

We now proceed as in the proof of Theorem A. ∎

####  4.3.6. Remarks on Theorem A

We note that decidability of $K$ relative to its tilt $K^{\flat}$ is false,
without further assumptions:

######  Remark 4.3.7.

Set $(F,w)=(\mathbb{F}_{p}((t^{\Gamma})),v_{t})$ for the Hahn field with value
group $\Gamma=\frac{1}{p^{\infty}}\mathbb{Z}$ and residue field
$\mathbb{F}_{p}$. By Proposition 3.6.7$(c)$, we have that
$|Z_{F}|=\mathfrak{c}$. Since there are countably many Turing machines, there
must exist a valued field $(K,v)$ with
$(K^{\flat},v^{\flat})\cong(\mathbb{F}_{p}((t^{\Gamma})),v_{t})$ and $(K,v)$
undecidable in $L_{\text{val}}$. On the other hand, the valued field
$(\mathbb{F}_{p}((t^{\Gamma})),v_{t})$ is decidable in $L_{\text{val}}$ by
Theorem 1.6 [Kuh16] and even decidable in $L_{t}$ by Lisinski [Lis21].

######  Remark 4.3.8 (see Proposition 7.1.1).

More strikingly, for a generic $\alpha\in 2^{\omega}$, the untilt $L_{\alpha}$
of $\mathbb{F}_{p}((t^{\Gamma}))$ constructed in the proof of Proposition
3.6.7$(c)$, is undecidable and yet each individual residue ring
${\mathcal{O}}_{L_{\alpha}}/(p^{n})$ is decidable.

######  Remark 4.3.9.

Theorem A holds also for untilts $K$ of $F$, which have an associated
distinguished element $\xi_{K}$ satisfying a more relaxed condition than the
one of Definition 4.2.2. Namely, suppose that $\xi_{K}$ has Witt vector
$(\xi_{0},\xi_{1},...)\in{\mathcal{O}}_{F}^{\omega}$ and each $\xi_{n}$ is
definable in the valued field $(F,w)$ by a formula $\phi_{n}(x)\in
L_{\text{val}}(R_{0})$ with parameters from $R_{0}$. Suppose furthermore that
the function
$\mathbb{N}\to\text{Form}_{L_{\text{val}}(R_{0})}:n\mapsto\phi_{n}(x)$ is
recursive. Then the conclusion of Theorem A is still valid, i.e. $(K,v)$ is
decidable in $L_{\text{val}}$ relative to $(F,w)$ in $L_{\text{val}}(R_{0})$.
The existential version also goes through with the caveat that each
$\phi_{n}(x)$ must be an existential formula.

It is clear that when each $\phi_{n}(x)$ is a quantifier-free formula, the
configuration described in Remark 4.3.9 specializes to the notion of a
computable untilt. We have no particular application in mind that requires the
level of generality described in Remark 4.3.9, which is one reason we
restricted ourselves to the quantifier-free case (the other being clarity of
exposition).

###  4.4. Corollary A

In order to prove Corollary A, we need to calculate the tilts of our fields of
interest and compute the associated distinguished elements.

####  4.4.1. Computing the distinguished elements

All computations below are well-known to experts (see e.g. Example 2.22
[FF14]). For lack of a detailed reference, we shall spell out the details.

######  Lemma 4.4.2.

There exist ring isomorphisms:
$(a)$
$\mathbb{Z}_{p}[p^{1/p^{\infty}}]/(p)\cong\mathbb{F}_{p}[t^{1/p^{\infty}}]/(t)$
mapping $p^{1/p^{n}}+(p)\mapsto t^{1/p^{n}}+(t)$.
$(b)$
$\mathbb{Z}_{p}[\zeta_{p^{\infty}}]/(p)\cong\mathbb{F}_{p}[t^{1/p^{\infty}}]/(t^{p-1})$
mapping $\zeta_{p}+(p)\mapsto t+1+(t^{p-1})$.
$(c)$
$\mathbb{Z}_{p}^{ab}/(p)\cong\overline{\mathbb{F}}_{p}[t^{1/p^{\infty}}]/(t^{p-1})$
mapping $\zeta_{p}+(p)\mapsto t+1+(t^{p-1})$.

###### Proof.

$(a)$ For each $n\in\mathbb{N}$, the irreducible polynomial of $p^{1/p^{n+1}}$
over $\mathbb{Q}_{p}(p^{1/p^{n}})$ is the Eisenstein polynomial
$x^{p}-p^{1/p^{n}}\in\mathbb{Z}_{p}[p^{1/p^{n}}][x]$. We therefore compute

$\mathbb{Z}_{p}[p^{1/p^{\infty}}]/(p)\cong\mathbb{Z}_{p}[x_{1},x_{2},...]/(p,x_{1}^{p}-p,x_{2}^{p}-x_{1},...)$
$\cong\mathbb{F}_{p}[x_{1},x_{2},...]/(x_{1}^{p},x_{2}^{p}-x_{1},...)\stackrel{{\scriptstyle
x_{n}\mapsto t^{1/p^{n}}}}{{\cong}}\mathbb{F}_{p}[t^{1/p^{\infty}}]/(t)$

$(b)$ The irreducible polynomial of $\zeta_{p}$ over $\mathbb{Q}_{p}$ is the
cyclotomic polynomial $\Phi_{p}(x)=x^{p-1}+...+1$. Moreover, given $n>1$, the
irreducible polynomial of $\zeta_{p^{n}}$ over
$\mathbb{Q}_{p}(\zeta_{p^{n-1}})$ is
$x^{p}-\zeta_{p^{n-1}}\in\mathbb{Z}[\zeta_{p^{n}}][x]$. We now proceed as in
$(a)$ and compute

$\mathbb{Z}_{p}[\zeta_{p^{\infty}}]/(p)\cong\mathbb{Z}_{p}[x_{1},x_{2},...]/(p,\Phi_{p}(x_{1}),x_{2}^{p}-x_{1},...)\stackrel{{\scriptstyle
x_{n+1}\mapsto
x^{1/p^{n}}}}{{\cong}}\mathbb{F}_{p}[x^{1/p^{\infty}}]/(\overline{\Phi}_{p}(x))$

Note that
$\overline{\Phi}_{p}(x)=\frac{x^{p}-1}{x-1}=\frac{(x-1)^{p}}{x-1}=(x-1)^{p-1}$
and thus

$\mathbb{Z}_{p}[\zeta_{p^{\infty}}]/(p)\cong\mathbb{F}_{p}[x^{1/p^{\infty}}]/(x-1)^{p-1}\stackrel{{\scriptstyle
t=x-1}}{{=}}\mathbb{F}_{p}[t^{1/p^{\infty}}]/(t^{p-1})$

$(c)$ By local Kronecker-Weber (see Theorem 14.2 in [Was97]) and Proposition
17 in [Ser80], we get that
$\mathbb{Z}_{p}^{ab}=\mathbb{Z}_{p}^{ur}[\zeta_{p^{\infty}}]$. We now proceed
as in $(b)$. ∎

######  Corollary 4.4.3.

$(a)$
$\widehat{\mathbb{Q}_{p}(p^{1/p^{\infty}})}^{\flat}\cong\widehat{\mathbb{F}_{p}((t))^{1/p^{\infty}}}$
and $t^{\sharp}=p$.
$(b)$
$\widehat{\mathbb{Q}_{p}(\zeta_{p^{\infty}})}^{\flat}\cong\widehat{\mathbb{F}_{p}((t))^{1/p^{\infty}}}$
and $(t+1)^{\sharp}=\zeta_{p}$.
$(c)$
$\widehat{\mathbb{Q}_{p}^{ab}}^{\flat}\cong\widehat{\overline{\mathbb{F}}_{p}((t))^{1/p^{\infty}}}$
and $(t+1)^{\sharp}=\zeta_{p}$.

###### Proof.

$(a)$ By Lemma 4.4.2, we have that
$\mathbb{Z}_{p}[p^{1/p^{\infty}}]/(p)\cong\mathbb{F}_{p}[t^{1/p^{\infty}}]/(t)$
via an isomorphism mapping $p^{1/p^{n}}+(p)\mapsto t^{1/p^{n}}+(t)$. One can
verify directly that
$\widehat{\mathbb{F}_{p}[[t]]^{1/p^{\infty}}}\to\varprojlim_{\Phi}\mathbb{F}_{p}[t^{1/p^{\infty}}]/(t):x\mapsto(x\mod(t),x^{1/p}\mod(t),...)$
is a ring isomorphism. It follows that
$\widehat{\mathbb{Q}_{p}(p^{1/p^{\infty}})}^{\flat}\cong\widehat{\mathbb{F}_{p}((t))^{1/p^{\infty}}}$
and by definition $t^{\sharp}=\lim_{n\to\infty}(p^{1/p^{n}})^{p^{n}}=p$. The
proofs of $(b)$ and $(c)$ are similar. ∎

######  Proposition 4.4.4 (cf. Example 2.22 [FF14]).

We have the following isomorphisms:
$(a)$ $\widehat{\mathbb{Z}_{p}[p^{1/p^{\infty}}]}\cong
W(\widehat{\mathbb{F}_{p}[[t]]^{1/p^{\infty}}})/([t]-p)$.
$(b)$ $\widehat{\mathbb{Z}_{p}[\zeta_{p^{\infty}}]}\cong
W(\widehat{\mathbb{F}_{p}[[t]]^{1/p^{\infty}}})/([t+1]^{p-1}+[t+1]^{p-2}+...+1)$.
$(c)$ $\widehat{\mathbb{Z}_{p}^{ab}}\cong
W(\widehat{\overline{\mathbb{F}}_{p}[[t]]^{1/p^{\infty}}})/([t+1]^{p-1}+[t+1]^{p-2}+...+1)$.

###### Proof.

$(a)$ By Corollary 4.4.3$(a)$, we have that
$\widehat{\mathbb{Q}_{p}(p^{1/p^{\infty}})}^{\flat}\cong\widehat{\mathbb{F}_{p}((t))^{1/p^{\infty}}}$.
Consider the ring homomorphism
$\theta:\textbf{A}_{\text{inf}}\twoheadrightarrow\widehat{\mathbb{Z}_{p}[p^{1/p^{\infty}}]}$
inducing
$\sharp:\widehat{\mathbb{F}_{p}[[t]]^{1/p^{\infty}}}\to\widehat{\mathbb{Z}_{p}[p^{1/p^{\infty}}]}$.
Since $t^{\sharp}=p$, we get that $\theta([t])=p$ and therefore
$[t]-p\in\text{Ker}(\theta)$. By Proposition 3.5.6, it follows that
$\text{Ker}(\theta)=([t]-p)$.
$(b)$ By Corollary 4.4.3$(b)$, we have that
$\widehat{\mathbb{Q}_{p}(\zeta_{p^{\infty}})}^{\flat}\cong\widehat{\mathbb{F}_{p}((t))^{1/p^{\infty}}}$.
mapping $1+t\mapsto\zeta_{p}$. Consider the ring homomorphism
$\theta:\textbf{A}_{\text{inf}}\twoheadrightarrow\widehat{\mathbb{Z}_{p}[\zeta_{p^{\infty}}]}$
inducing
$\sharp:\widehat{\mathbb{F}_{p}[[t]]^{1/p^{\infty}}}\to\widehat{\mathbb{Z}_{p}[\zeta_{p^{\infty}}]}$.
Since $(t+1)^{\sharp}=\zeta_{p}$, we get that $\theta([t+1])=\zeta_{p}$ and
therefore $[t+1]^{p-1}+....+[t+1]+1\in\text{Ker}(\theta)$. By Proposition
3.5.6, it suffices to show that $\xi=[t+1]^{p-1}+....+[t+1]+1$ is a
distinguished element of $\textbf{A}_{\text{inf}}$. Set
$\text{res}:{\mathcal{O}}_{F}\twoheadrightarrow{\mathcal{O}}_{F}/\mathfrak{m}_{F}$
for the residue map and $W(\text{res})$ for the unique induced homomorphism
$W(\text{res}):W(R)\to W(R/\mathfrak{m}_{R})$ of Proposition 3.3.4. We compute
that $W(\text{res})(\xi)=1+1+...+1=p$, therefore $\xi$ is a distinguished
element of $\textbf{A}_{\text{inf}}$ (Remark 3.5.4).
$(c)$ Similar to $(b)$. ∎

####  4.4.5. Proof of Corollary A

######  Corollary 4.4.6.

We have the following:
$(a)$ $\widehat{\mathbb{Q}_{p}(p^{1/p^{\infty}})}$ and
$\widehat{\mathbb{Q}_{p}(\zeta_{p^{\infty}})}$ are
$\mathbb{F}_{p}[t]$-computable untilts of
$\widehat{\mathbb{F}_{p}((t))^{1/p^{\infty}}}$.
$(b)$ $\widehat{\mathbb{Q}_{p}^{ab}}$ is an $\mathbb{F}_{p}[t]$-computable
untilt of $\widehat{\mathbb{F}_{p}((t))^{1/p^{\infty}}}$.

###### Proof.

The case of $\widehat{\mathbb{Q}_{p}(p^{1/p^{\infty}})}$ is clear. For the
other two cases, note that $\xi:=[t+1]^{p-1}+[t+1]^{p-2}+...+1$ is
$\mathbb{F}_{p}[t]$-computable, using the computable polynomials
$S_{0},...,S_{n}$ for Witt vector addition (Observation 3.3.9). ∎

######  Corollary A.

$(a)$ Assume $\mathbb{F}_{p}((t))^{1/p^{\infty}}$ is (existentially) decidable
in $L_{t}$. Then $\mathbb{Q}_{p}(p^{1/p^{\infty}})$ and
$\mathbb{Q}_{p}(\zeta_{p^{\infty}})$ are (existentially) decidable in
$L_{\text{val}}$.
$(b)$ Assume $\overline{\mathbb{F}}_{p}((t))^{1/p^{\infty}}$ is
(existentially) decidable in $L_{t}$. Then $\mathbb{Q}_{p}^{ab}$ is
(existentially) decidable in $L_{\text{val}}$.

###### Proof.

$(a)$ By Corollary 4.4.6 and Corollary 4.3.5.
$(b)$ Similar to $(a)$. ∎

##  5\. Applications: Tame fields of mixed characteristic

###  5.1. Introduction

####  5.1.1. Motivation

In this section we prove Corollary B, which to my knowledge is the first
decidability result for tame fields of mixed characteristic. Corollary B is
proved by transposing a decidability result for tame fields in positive
characteristic and ultimately using Theorem A.

####  5.1.2. Preliminaries

The algebra and model theory of tame fields was introduced and studied by
Kuhlmann in [Kuh16]. Recall the definition:

######  Definition 5.1.3.

Let $(K,v)$ be a henselian valued field. A finite valued field extension
$(L,w)/(K,v)$ is said to be tame if

  1. (1)

$p\nmid[wL:vK]$, where $p$ is the characteristic exponent of $Kv$, i.e.
$p=\text{char}(Kv)$ if this is positive and $p=1$ if $\text{char}(Kv)=0$.

  2. (2)

The residue field extension $Lw/Kv$ is separable.

  3. (3)

The extension $(L,w)/(K,v)$ is defectless, meaning that we have the
fundamental equality $[L:K]=[wL:vK]\cdot[Lw:Kv]$.

An algebraic valued field extension is said to be tame if every finite
subextension is tame.

######  Definition 5.1.4.

A henselian valued field $(K,v)$ is said to be tame if every finite valued
field extension $(L,w)/(K,v)$ is tame.

In practice, one often needs a more intrinsic description of tame fields. This
is provided by the following:

######  Proposition 5.1.5 (Theorem 3.2 [Kuh16]).

Let $(K,v)$ be henselian valued field. Then the following are equivalent:

  1. (1)

$(K,v)$ is a tame field.

  2. (2)

$(K,v)$ is algebraically maximal, $vK$ is $p$-divisible and $Kv$ is is
perfect.

######  Example 5.1.6.

Using Proposition 5.1.5, one can verify the following:
$(a)$ Any (algebraically) maximal immediate extension of
$\mathbb{Q}_{p}(p^{1/p^{\infty}})$ or $\mathbb{Q}_{p}(\zeta_{p^{\infty}})$ is
tame.
$(b)$ Any (algebraically) maximal immediate extension of
$\mathbb{F}_{p}((t))^{1/p^{\infty}}$ is tame. In particular, the Hahn field
$(\mathbb{F}_{p}((t^{\Gamma})),v_{t})$ with value group
$\Gamma=\frac{1}{p^{\infty}}\mathbb{Z}$ and residue field $\mathbb{F}_{p}$ is
tame.

####  5.1.7. Equal characteristic

Kuhlmann obtained the following Ax-Kochen-Ershov principle for tame fields of
equal characteristic:

######  Theorem 5.1.8 (Theorem 1.4 [Kuh16]).

Let $(K,v)$ and $(K^{\prime},v^{\prime})$ be two equal characteristic tame
fields. Then $(K,v)\equiv(K^{\prime},v^{\prime})$ in $L_{\text{val}}$ if and
only if $Kv\equiv K^{\prime}v^{\prime}$ in $L_{\text{r}}$ and $vK\equiv
v^{\prime}K^{\prime}$ in $L_{\text{oag}}$.

Kuhlmann then deduces the following decidability result (among others):

######  Corollary 5.1.9 (Theorem 1.6 [Kuh16]).

Set $\Gamma=\frac{1}{p^{\infty}}\mathbb{Z}$, i.e. for the $p$-divisible hull
of $\mathbb{Z}$. The Hahn field $(\mathbb{F}_{p}((t^{\Gamma})),v_{t})$ is
decidable in $L_{\text{val}}$.

In recent work, Lisinski combined results of Kuhlmann [Kuh16] with work of
Kedlaya [Ked17] and obtained the following strengthening of Corollary 5.1.9:

######  Theorem 5.1.10 (Theorem 1 [Lis21]).

Set $\Gamma=\frac{1}{p^{\infty}}\mathbb{Z}$. The Hahn field
$(\mathbb{F}_{p}((t^{\Gamma})),v_{t})$ is decidable in $L_{t}$.

####  5.1.11. Mixed characteristic

Kuhlmann’s Theorem 5.1.8 fails as such for mixed characteristic tame fields. A
counterexample is given in Theorem 1.5$(c)$ [AK16]. The lack of such a
principle has been a fundamental obstacle in obtaining decidability results
for such fields.

###  5.2. Mixed characteristic tame fields

While we do not know whether $\mathbb{Q}_{p}(p^{1/p^{\infty}})$ and
$\mathbb{Q}_{p}(\zeta_{p^{\infty}})$ are (existentially) decidable, we will
show that they admit decidable maximal immediate extensions. These are tame
fields by Example 5.1.6$(a)$. First we give a self-contained proof of the fact
that tilting preserves maximality:

######  Lemma 5.2.1 (Remark 2.23 [FF14]).

Let $(K,v)$ be a perfectoid field. Then $(K,v)$ is maximal if and only if
$(K^{\flat},v^{\flat})$ is maximal.

###### Proof.

"$\Rightarrow$": Suppose $(K,v)$ is maximal and $(F,w)/(K^{\flat},v^{\flat})$
be a proper immediate extension. Let $\xi\in W({\mathcal{O}}_{K^{\flat}})$ be
a distinguished element such that ${\mathcal{O}}_{K}\cong
W({\mathcal{O}}_{K^{\flat}})/(\xi)$ (Theorem 3.5.10). The proper inclusion
$\iota:{\mathcal{O}}_{K^{\flat}}\hookrightarrow{\mathcal{O}}_{F}$ induces a
ring homomorphism $W(\iota):W({\mathcal{O}}_{K^{\flat}})\to
W({\mathcal{O}}_{F})$ by Proposition 3.3.4. By Theorem 3.5.10, there is a
mixed characteristic perfectoid field $(L,v^{\prime})$ with
${\mathcal{O}}_{L}=W({\mathcal{O}}_{F})/(\xi)$. Since the map $W(\iota)$
induces $\iota$, it descends to a non-surjective local ring homomorphism
${\mathcal{O}}_{K}\to{\mathcal{O}}_{L}$ which renders $(L,v^{\prime})/(K,v)$
into a proper valued field extension. By Lemma 3.2.4$(b)$, the extension
$(L,v^{\prime})/(K,v)$ is also immediate, which contradicts our assumption on
$(K,v)$.
"$\Leftarrow$": Suppose $(K^{\flat},v^{\flat})$ is maximal and let
$(L,v^{\prime})/(K,v)$ be a proper immediate extension.
Claim: ${\mathcal{O}}_{K}+p{\mathcal{O}}_{L}\subsetneqq{\mathcal{O}}_{L}$.

###### Proof.

Suppose that ${\mathcal{O}}_{K}+p{\mathcal{O}}_{L}={\mathcal{O}}_{L}$.
Inductively, for each $n\in\mathbb{N}$, one gets that
${\mathcal{O}}_{L}={\mathcal{O}}_{K}+p{\mathcal{O}}_{K}+...+p^{n}{\mathcal{O}}_{K}+p^{n+1}{\mathcal{O}}_{L}$.
This implies that ${\mathcal{O}}_{K}$ is dense in ${\mathcal{O}}_{L}$. Since
$(K,v)$ is complete, it follows that ${\mathcal{O}}_{K}={\mathcal{O}}_{L}$,
which is contrary to our assumption that $K\subsetneqq L$.
∎${}_{\textit{Claim}}$

By the Claim, we see that the natural inclusion
${\mathcal{O}}_{K}/(p)\hookrightarrow{\mathcal{O}}_{L}/(p)$ is not surjective.
It follows that the induced inclusion map
$\varprojlim_{\Phi}{\mathcal{O}}_{K}/(p)\hookrightarrow\varprojlim_{\Phi}{\mathcal{O}}_{L}/(p)$
is also non-surjective, whence
$(L^{\flat},v^{\prime\flat})/(K^{\flat},v^{\flat})$ is a proper valued field
extension. Since $(L,v^{\prime})/(K,v)$ is immediate so is
$(L^{\flat},v^{\prime\flat})/(K^{\flat},v^{\flat})$, using Lemma 3.2.4$(b)$.
This contradicts our assumption on $(K^{\flat},v^{\flat})$. ∎

######  Corollary B.

The valued field $(\mathbb{Q}_{p}(p^{1/p^{\infty}}),v_{p})$ (resp.
$(\mathbb{Q}_{p}(\zeta_{p^{\infty}}),v_{p})$) admits a maximal immediate
extensions which is decidable in $L_{\text{val}}$.

###### Proof.

We only provide a proof for $\mathbb{Q}_{p}(p^{1/p^{\infty}})$ since the
argument for $\mathbb{Q}_{p}(\zeta_{p^{\infty}})$ is similar. Recall from
Corollary 4.4.4 that
$\widehat{\mathbb{Z}_{p}[p^{1/p^{\infty}}]}=W(\widehat{\mathbb{F}_{p}[[t]]^{1/p^{\infty}}})/([t]-p)$.
Let $\mathbb{F}_{p}((t^{\Gamma}))$ be the Hahn field with value group
$\Gamma=\frac{1}{p^{\infty}}\mathbb{Z}$ and residue field $\mathbb{F}_{p}$. It
is a (non-unique) maximal immediate extension of
$\mathbb{F}_{p}((t))^{1/p^{\infty}}$. By Theorem 3.5.10, there exists a
perfectoid field $(K,v)$ with
${\mathcal{O}}_{K}=W(\mathbb{F}_{p}[[t^{\Gamma}]])/([t]-p)$. Lemma 3.2.4$(b)$
shows that $(vK,vp)=(\frac{1}{p^{\infty}}\mathbb{Z},1)$ and
$Kv=\mathbb{F}_{p}$. By Proposition 3.3.4, the natural inclusion
$\iota:\widehat{\mathbb{F}_{p}[[t]]^{1/p^{\infty}}}\hookrightarrow\mathbb{F}_{p}[[t^{\Gamma}]]$
induces uniquely a ring homomorphism
$W(\iota):W(\widehat{\mathbb{F}_{p}[[t]]^{1/p^{\infty}}})\to
W(\mathbb{F}_{p}[[t^{\Gamma}]])$. The map $W(\iota)$ descends to a local ring
homomorphism $\widehat{\mathbb{Z}_{p}[p^{1/p^{\infty}}]}\to{\mathcal{O}}_{K}$
and renders $(K,v)/(\mathbb{Q}_{p}(p^{1/p^{\infty}}),v_{p})$ into an immediate
valued field extension. The valued field $\mathbb{F}_{p}((t^{\Gamma}))$ is
maximal and so is $(K,v)$ by Lemma 5.2.1. By Theorem 5.1.10, the Hahn field
$(\mathbb{F}_{p}((t^{\Gamma})),v_{t})$ is decidable in $L_{t}$. We conclude
from Theorem A. ∎

######  Remark 5.2.2.

It is also true that $\mathbb{Q}_{p}^{ab}$ admits a decidable maximal
immediate extension (in this case unique) but this already follows from well-
known results in the model theory of algebraically maximal Kaplansky fields
(see pg.4 Part (f) [Kuh16]).

##  6\. Applications: Congruences modulo $p$

###  6.1. Introduction

####  6.1.1. Goal

We shall now prove Theorem B, which shows the existence of an algorithm that
decides whether a system of polynomial equations and inequations, defined over
$\mathbb{Z}$, has a solution modulo $p$ over the valuation rings of our fields
of interest.

####  6.1.2. Strategy

The proof is via a local field approximation argument, using the computations
of Lemma 4.4.2. One eventually encodes the above problem in the existential
theory of the tilt in $L_{\text{val}}$, where the Anscombe-Fehm Theorem 2.2.3
applies. This should be contrasted with Corollary A, which requires
decidability in the language $L_{t}$ on the characteristic $p$ side.

###  6.2. From residue rings to valuation rings

The crux of the argument lies in the following:

######  Proposition 6.2.1.

Let $f_{i}(x),g_{j}(x)\in\mathbb{F}_{p}[x]$ be multi-variable polynomials in
$x=(x_{1},...,x_{m})$ for $i,j=1,...,n$. Then

$\mathbb{F}_{p}[t^{1/p^{\infty}}]/(t)\models\exists x\bigwedge_{1\leq i,j\leq
n}(f_{i}(x)=0\land g_{j}(x)\neq 0)\iff$
$\mathbb{F}_{p}[[t]]^{1/p^{\infty}}\models\exists x\bigwedge_{1\leq i,j\leq
n}(v(f_{i}(x))>v(g_{j}(x)))$

###### Proof.

First observe that

$\mathbb{F}_{p}[[t]]^{1/p^{\infty}}/(t)\cong\varinjlim\mathbb{F}_{p}[[t^{1/p^{n}}]]/(t)\cong\varinjlim\mathbb{F}_{p}[t^{1/p^{n}}]/(t)\cong\mathbb{F}_{p}[t^{1/p^{\infty}}]/(t)\
\ (\dagger)$

"$\Rightarrow$": Let $a\in(\mathbb{F}_{p}[t^{1/p^{\infty}}]/t)^{m}$ be such
that $f_{i}(a)=0\land g_{j}(a)\neq 0$, for $1\leq i,j\leq n$ and let
$\tilde{a}$ be any lift of $a$ in $\mathbb{F}_{p}[[t]]^{1/p^{\infty}}$ via the
isomorphism $(\dagger)$. We see that $v(f_{i}(\tilde{a}))\geq
1>v(g_{j}(\tilde{a}))$, for all $1\leq i,j\leq n$.
"$\Leftarrow$": Let $b\in(\mathbb{F}_{p}[[t]]^{1/p^{\infty}})^{m}$ be such
that $v(g_{j}(b))<v(f_{i}(b))$ for all $1\leq i,j\leq n$. Set
$\gamma_{1}=\max\\{v(g_{j}(b)):j=1,...,n\\}$ and
$\gamma_{2}=\min\\{v(f_{i}(b)):i=1,...,n\\}$ and consider the open interval
$I=(\gamma_{1},\gamma_{2})\subseteq\frac{1}{p^{\infty}}\mathbb{Z}^{\geq 0}$.
Since $\frac{1}{p^{\infty}}\mathbb{Z}$ is dense in $\mathbb{R}$, we can find
$q\in\frac{1}{p^{\infty}}\mathbb{Z}$ such that $1\in qI$.

We now make use of the fact that for each
$q\in\frac{1}{p^{\infty}}\mathbb{Z}^{>0}$, there is an embedding

$\rho:\mathbb{F}_{p}((t))^{1/p^{\infty}}\to\mathbb{F}_{p}((t))^{1/p^{\infty}}$

which maps $t\mapsto t^{q}$. Indeed, if $q\in\frac{1}{p^{N}}\mathbb{Z}^{>0}$
for some $N\in\mathbb{N}$, then there exists an embedding
$\rho:\mathbb{F}_{p}((t))^{1/p^{N}}\to\mathbb{F}_{p}((t))^{1/p^{N}}$ mapping
$t\mapsto t^{q}$, exactly as in Remark 7.9 [AF16]. Such a map can also be
extended uniquely to the perfect hull $\mathbb{F}_{p}((t))^{1/p^{\infty}}$.
Now let
$\rho:\mathbb{F}_{p}((t))^{1/p^{\infty}}\to\mathbb{F}_{p}((t))^{1/p^{\infty}}$
be as above. Then, since $f_{i}(x),g_{j}(x)\in\mathbb{F}_{p}[x]$, we get

$v(g_{j}(\rho(b)))=v(\rho(g_{j}(b)))=qv(g_{j}(b))<qv(f_{i}(b))=v(\rho(f_{i}(b)))=v(f_{i}(\rho(b)))$

for all $1\leq i,j\leq n$. We may thus replace our witness $b$ with
$a=\rho(b)$.

Since $1\in qI$, we get $f_{i}(a)=0\mod(t)\land g_{j}(a)\neq 0\mod(t)$, for
all $i,j=1,...,n$. The reduction of $a$ modulo $(t)$, seen as a tuple in
$\mathbb{F}_{p}[t^{1/p^{\infty}}]/t$ via $(\dagger)$, is the desired witness.
∎

######  Remark 6.2.2.

The same argument used in Lemma 6.2.1 shows that

$k[t^{1/p^{\infty}}]/(t^{p-1})\models\exists x\bigwedge_{1\leq i,j\leq
n}(f_{i}(x)=0\land g_{j}(x)\neq 0)\iff$ $k[[t]]^{1/p^{\infty}}\models\exists
x\bigwedge_{1\leq i,j\leq n}(v(f_{i}(x))>v(g_{j}(x)))\iff$

where $k=\mathbb{F}_{p}$ or $\overline{\mathbb{F}}_{p}$. The argument of
Proposition 6.2.1 needs only a slight modification for the converse direction;
one needs to take $q\in\frac{1}{p^{\infty}}\mathbb{Z}$ such that $(p-1)\in qI$
instead of $1\in qI$ and the same proof goes through.

###  6.3. Proof of Theorem B

We may now prove the following:

######  Theorem B.

Let $K$ be any of the fields
$\mathbb{Q}_{p}(p^{1/p^{\infty}}),\mathbb{Q}_{p}(\zeta_{p^{\infty}})$ or
$\mathbb{Q}_{p}^{ab}$. Then the existential theory of $\mathcal{O}_{K}/(p)$ is
decidable in $L_{\text{r}}$.

###### Proof.

By the Anscombe-Fehm Theorem 2.2.3, the valued fields
$(\mathbb{F}_{p}((t))^{1/p^{\infty}},v_{t})$ and
$(\overline{\mathbb{F}}_{p}((t))^{1/p^{\infty}},v_{t})$ are
$\exists$-decidable in $L_{\text{val}}$. For
$\mathbb{Q}_{p}(p^{1/p^{\infty}})$ the conclusion now follows from Lemma
4.4.2$(a)$ and Proposition 6.2.1. For the other two fields, one has to use
Lemma 4.4.2$(b),(c)$ and Remark 6.2.2. ∎

######  Remark 6.3.1.

Note that Corollary A requires decidability in the language $L_{t}$ on the
characteristic $p$ side. However, for the purposes of Theorem B, the Anscombe-
Fehm results in $L_{\text{val}}$ turned out to be sufficient. This became
possible because of Proposition 6.2.1, which "eliminates" any references to
$t$.

##  7\. Final remarks

###  7.1. An almost decidable field

In §2.3, we emphasized that uniform decidability of
$({\mathcal{O}}_{K}/(p^{n}))_{n\in\omega}$ is essential for the decidability
of $(K,v)$. Indeed, there are examples where each individual ring
${\mathcal{O}}_{K}/(p^{n})$ is decidable but $(K,v)$ is undecidable:

######  Proposition 7.1.1.

There exists a mixed characteristic henselian valued field $(K,v)$ such that:

  1. (1)

The valued field $(K,v)$ is undecidable in $L_{\text{val}}$.

  2. (2)

For each $n\in\mathbb{N}$, the ring ${\mathcal{O}}_{K}/(p^{n})$ is decidable
in $L_{\text{r}}$.

  3. (3)

The value group $(vK,vp)$ decidable in $L_{\text{oag}}$.

###### Proof.

For the convenience of the reader, we first review the construction in the
proof of Proposition 3.6.7$(b),(c)$. Set $\pi_{\alpha(0)}=\zeta_{p}-1$ and
$K_{\alpha(0)}=\mathbb{Q}_{p}(\zeta_{p})$. By Lemma 9.1 [Was97], there is some
$u\in\mathbb{Z}[\zeta_{p}]$ with $u\equiv 1\mod p\mathbb{Z}_{p}[\zeta_{p}]$
such that $\mathbb{Q}_{p}(\zeta_{p},u^{1/p})$ is an unramified $p$-degree
extension of $\mathbb{Q}_{p}(\zeta_{p})$. For each $\alpha\in 2^{\omega}$,
define inductively
$K_{\alpha(n)}=K_{\alpha(n-1)}((u^{\alpha(n)}\cdot\pi_{\alpha(n-1)})^{1/p})$
and $\pi_{\alpha(n)}=(u^{\alpha(n)}\cdot\pi_{\alpha(n-1)})^{1/p}$ for
$n\in\mathbb{N}$. Let $K_{\alpha}=\bigcup_{n\in\omega}K_{\alpha(n)}$. Claim 2
of Proposition 3.6.7 shows that
$\widehat{K_{\alpha}}^{\flat}=\widehat{\mathbb{F}_{p}((t))^{1/p^{\infty}}}$.
Let $\xi_{\alpha}\in W(\widehat{\mathbb{F}_{p}[[t]]^{1/p^{\infty}}})$ be a
distinguished element such that ${\mathcal{O}}_{K_{\alpha}}\cong
W(\widehat{\mathbb{F}_{p}[[t]]^{1/p^{\infty}}})/(\xi_{\alpha})$ (Theorem
3.5.10). We now let
$L_{\alpha}=\text{Frac}(W(\mathbb{F}_{p}[[t^{\Gamma}]])/(\xi_{\alpha}))$,
where $\Gamma=\frac{1}{p^{\infty}}\mathbb{Z}$. This is an immediate extension
of $K_{\alpha}$ such that $L_{\alpha}^{\flat}=\mathbb{F}_{p}((t^{\Gamma}))$.
Since $L_{\alpha}\not\equiv L_{\beta}$ in $L_{\text{val}}$ for
$\alpha\neq\beta$ and there are countably many Turing machines, there will
exist an $\alpha\in 2^{\omega}$ such that $L_{\alpha}$ is undecidable in
$L_{\text{val}}$. On the other hand, we shall argue below that any
$L_{\alpha}$ satisfies conditions (2) and (3).

For $\alpha\in 2^{\omega}$, set $\overline{a}=\sum_{k\geq 0}\alpha(k)\cdot
p^{k}\in\mathbb{Z}_{p}$ and
$\overline{\alpha}_{n}=\sum_{k=0}^{n-1}\alpha(k)\cdot p^{k}\in\mathbb{Z}$ for
all $n\in\mathbb{N}$ ($\overline{\alpha}_{0}=0$ by convention). Since $u\equiv
1\mod p{\mathcal{O}}_{L_{\alpha}}$, one gets by induction that
$u^{p^{n}}\equiv 1\mod p^{n+1}{\mathcal{O}}_{L_{\alpha}}$ for all
$n\in\mathbb{N}$. It follows that the limit $\lim_{n\to\infty}u^{\alpha_{n}}$
exists in ${\mathcal{O}}_{L_{\alpha}}$ and we denote it by $u^{\overline{a}}$.
Claim 1:  We have $(t+1)^{\sharp}=u^{\overline{a}}\cdot\pi+1$.

###### Proof.

We have that $t+1=(\pi_{\alpha(0)}+1+(p),\pi_{\alpha(1)}+1+(p),...)$, via the
identification
$\mathbb{F}_{p}[[t^{\Gamma}]]\cong\varprojlim_{\Phi}{\mathcal{O}}_{L_{\alpha}}/(p)$
(Remark 3.6.10). By induction, one sees that
$\pi_{\alpha(n)}^{p^{n}}=u^{\overline{\alpha}_{n}}\cdot\pi$ for all
$n\in\mathbb{N}$. We thus compute
$(t+1)^{\sharp}=\lim_{n\to\infty}(\pi_{\alpha(n)}+1)^{p^{n}}=\lim_{n\to\infty}(u^{\overline{a}_{n}}\cdot\pi+1+p^{n}\cdot\epsilon_{n})=u^{\overline{a}}\cdot\pi+1$.
∎${}_{\textit{Claim 1}}$

Fix $n\in\mathbb{N}$ for the rest of the proof. Since $u^{p^{n-1}}\equiv 1\mod
p^{n}{\mathcal{O}}_{L_{\alpha}}$, we get that $(t+1)^{\sharp}\equiv
u^{\overline{\alpha}_{n-1}}\cdot\pi+1\mod p^{n}{\mathcal{O}}_{L_{\alpha}}$.
Set $\beta:=u^{\overline{\alpha}_{n-1}}\cdot\pi+1\in\mathbb{Z}[\zeta_{p}]$ and
let $f(x)=\text{irr}_{(\beta,\mathbb{Q})}(x)\in\mathbb{Z}[x]$. If we set
$g(x):=f(x+1)$, then
$g(x)=\text{irr}_{(u^{\overline{\alpha}_{n-1}}\cdot\pi,\mathbb{Q})}(x)\in\mathbb{Z}[x]$
is an Eisenstein polynomial of degree $p-1$ as
$u^{\overline{\alpha}_{n-1}}\cdot\pi$ is a uniformizer of the totally ramified
extension $\mathbb{Q}_{p}(\zeta_{p})$ of $\mathbb{Q}_{p}$. In particular, we
deduce that $p|f^{(k)}(1)$ for $k=0,1,...,p-2$ and $p^{2}\nmid f(1)$. Set
$\xi^{\prime}:=f([t+1])$.
Claim 2: The element $\xi^{\prime}$ is a distinguished element of
$\textbf{A}_{\text{inf}}$.

###### Proof.

Let $\text{res}:\mathbb{F}_{p}[[t^{\Gamma}]]\to\mathbb{F}_{p}$ be the residue
map and
$W(\text{res}):\textbf{A}_{\text{inf}}\to\mathbb{Z}_{p}:\sum_{k=0}^{\infty}[c_{k}]\cdot
p^{n}\mapsto\sum_{k=0}^{\infty}[\text{res}(c_{k})]\cdot p^{n}$ be the induced
ring homomorphism provided by Proposition 3.3.4. We have that
$W(\text{res})(\xi^{\prime})=W(\text{res})(f([t+1]))=f(W(\text{res})([t+1]))=f(\text{res}(t+1))=f(1)$.
Since $p|f(1)$ and $p^{2}\nmid f(1)$, we get that
$W(\text{res})(\xi^{\prime})$ is an associate of $p$ and thus $\xi^{\prime}$
is a distinguished element (Remark 3.5.4). ∎${}_{\textit{Claim 2}}$

By Theorem 3.5.10, there is a mixed characteristic perfectoid field
$(L^{\prime},v^{\prime})$ with
${\mathcal{O}}_{L^{\prime}}=\textbf{A}_{\text{inf}}/(\xi^{\prime})$.
Claim 3: We have an equality of ideals $(p,\xi_{\alpha})=(p,\xi^{\prime})$ in
$\textbf{A}_{\text{inf}}$.

###### Proof.

Equivalently, we need to show that $(\overline{f}(1+t))=(t^{p-1})$ in
$\mathbb{F}_{p}[[t^{\Gamma}]]$, where $\overline{f}(x)\in\mathbb{F}_{p}[x]$ is
the image of $f(x)$ via $\mathbb{Z}[x]\xrightarrow{\mod p}\mathbb{F}_{p}[x]$.
Consider the formal Taylor expansion
$f(1+t)=\sum_{k=0}^{p-1}\frac{f^{(k)}(1)\cdot t^{k}}{k!}$. Since
$p|f^{(k)}(1)$ for $k=0,1,...,p-2$, we deduce that $\overline{f}(1+t)$ is an
associate of $t^{p-1}$. ∎${}_{\textit{Claim 3}}$

In particular, we have that
${\mathcal{O}}_{L_{\alpha}}/(p)\cong{\mathcal{O}}_{L^{\prime}}/(p)$.
Claim 4: We have an isomorphism of rings
${\mathcal{O}}_{L_{\alpha}}/(p^{n})\cong{\mathcal{O}}_{L^{\prime}}/(p^{n})$.

###### Proof.

We write $\theta_{\alpha}$ for the map
$\theta_{\alpha}:\textbf{A}_{\text{inf}}\xrightarrow{\mod(\xi_{\alpha})}{\mathcal{O}}_{L_{\alpha}}$.
We have that
$\theta_{\alpha}(\xi^{\prime})=\theta_{\alpha}(f([t+1]))=f(\theta_{\alpha}([t+1]))=f((t+1)^{\sharp})$.
By Claim 1, we get that $\theta_{\alpha}(\xi^{\prime})\equiv f(\beta)\equiv
0\mod(p^{n})$. In particular, the map $\theta_{\alpha}$ induces a surjective
ring homomorphism
$\phi:{\mathcal{O}}_{L^{\prime}}/(p^{n})\twoheadrightarrow{\mathcal{O}}_{L_{\alpha}}/(p^{n})$,
which descends to an isomorphism
$\overline{\phi}:{\mathcal{O}}_{L^{\prime}}/(p)\xrightarrow{\cong}{\mathcal{O}}_{L_{\alpha}}/(p)$
by Claim 2. For $x\in{\mathcal{O}}_{L^{\prime}}/(p^{n})$, write $x=p^{m}\cdot
x^{\prime}$ with $m=0,1,...,n-1$ and $x^{\prime}\not\equiv 0\mod(p)$. Then
$\phi(x)\neq 0$ since $\overline{\phi}(x\mod(p))\neq 0$. Therefore
$\text{Ker}(\phi)=(0)$ and we conclude that $\phi$ is an isomorphism.
∎${}_{\textit{Claim 4}}$

By Theorem 5.1.10, the Hahn field $(\mathbb{F}_{p}((t^{\Gamma})),v_{t})$ is
decidable in the language $L_{t}$. Since the ring
${\mathcal{O}}_{L^{\prime}}/(p^{n})$ is interpretable in the $L_{t}$-structure
$(\mathbb{F}_{p}((t^{\Gamma})),v_{t})$, it is also decidable.111111In fact,
the valued field $(L^{\prime},v^{\prime})$ is decidable in $L_{\text{val}}$ by
Theorem A. By Claim 2, we also have that ${\mathcal{O}}_{L_{\alpha}}/(p^{n})$
is decidable in $L_{\text{r}}$. The value group
$(v_{\alpha}L_{\alpha},v_{\alpha}p)\cong(\frac{1}{p^{\infty}}\mathbb{Z},p-1)$
is also decidable in $L_{\text{oag}}$ with a constant symbol for $p-1$. Since
$n$ was arbitrary, this concludes the proof. ∎

######  Remark 7.1.2.

Note that the valued field constructed in the proof of Proposition 7.1.1 is
tame. Indeed, by the characterization of Proposition 5.1.5, it suffices to
observe that it is maximal (Lemma 5.2.1), has value group isomorphic to
$\frac{1}{p^{\infty}}\mathbb{Z}$ and residue field $\mathbb{F}_{p}$ (Lemma
3.2.4$(b)$).

###  7.2. Reversing the direction

#### 7.2.1.

Given that our understanding of decidability problems in characteristic $p$ is
limited, our philosophy of reducing decidability questions from mixed
characteristic to positive characteristic may seem impractical. Nevertheless,
we have already seen two applications in §5 and §6. We also mention another
application in [Kar21], which proves an undecidability result for the
asymptotic theory of $\\{K:[K:\mathbb{Q}_{p}]<\infty\\}$ in the language
$L_{\text{val}}$ with a cross-section, again by transposing a result in
positive characteristic.

####  7.2.2.

We shall now demonstrate that the characteristic $p$ difficulties in the
language $L_{t}$ are already encoded in the mixed characteristic setting, by
showing a relative decidability result in the opposite direction:

######  Proposition 7.2.3.

If $\mathbb{Q}_{p}(p^{1/p^{\infty}})$ is $\forall^{1}\exists$-decidable in
$L_{\text{val}}$, then $\mathbb{F}_{p}[[t]]^{1/p^{\infty}}$ is
$\exists^{+}$-decidable in $L_{t}$.

###### Proof.

It is enough to prove the first statement. Let
$f_{i}(X_{1},...,X_{m},T)\in\mathbb{F}_{p}[X_{1},...,X_{m},T]$ for
$i=1,...,n$. We claim that

$\mathbb{F}_{p}[[t]]^{1/p^{\infty}}\models\exists
x_{1},...,x_{m}(\bigwedge_{1\leq i\leq n}f_{i}(x_{1},...,x_{m},t)=0)\iff$
$\mathbb{F}_{p}[t^{1/p^{\infty}}]/(t)\models\forall y\in\mathfrak{m}\exists
x_{1},...,x_{m}(\bigwedge_{1\leq i\leq n}f_{i}(x_{1},...,x_{m},y)=0)$

It is enough to prove the claim since
$\mathbb{Z}_{p}[p^{1/p^{\infty}}]/(p)=\mathbb{F}_{p}[t^{1/p^{\infty}}]/(t)$ by
Lemma 4.4.2$(a)$.
$"\Rightarrow":$ Let
$(\alpha_{1},...,\alpha_{m})\in(\mathbb{F}_{p}[[t]]^{1/p^{\infty}})^{m}$ be
such that
$f_{1}(\alpha_{1},....,\alpha_{m},t)=f_{2}(\alpha_{1},....,\alpha_{m},t)=...=f_{n}(\alpha_{1},....,\alpha_{m},t)=0$
and let $y\in\mathfrak{m}\subset\mathbb{F}_{p}[[t]]^{1/p^{\infty}}$. Let
$\kappa\in\mathbb{N}$ be such that $y^{p^{\kappa}}\in\mathbb{F}_{p}[[t]]$.
Then there exists a ring endomorphism $\rho$ on $\mathbb{F}_{p}[[t]]$ mapping
$t\mapsto y^{p^{\kappa}}$, which extends uniquely to
$\mathbb{F}_{p}[[t]]^{1/p^{\infty}}$. Let $\beta_{i}:=\rho(\alpha_{i})$ for
$i=1,...,m$. Since $f_{i}(X_{1},...,X_{m},T)\in\mathbb{F}_{p}[X,T]$, we get
$f_{1}(\beta_{1},...,\beta_{m},y^{p^{\kappa}})=f_{2}(\beta_{1},...,\beta_{m},y^{p^{\kappa}})=...=f_{n}(\beta_{1},...,\beta_{m},y^{p^{\kappa}})=0$
and thus
$f_{1}(\beta_{1}^{1/p^{\kappa}},...,\beta_{m}^{1/p^{\kappa}},y)=f_{2}(\beta_{1}^{1/p^{\kappa}},...,\beta_{m}^{1/p^{\kappa}},y)=...=f_{n}(\beta_{1}^{1/p^{\kappa}},...,\beta_{m}^{1/p^{\kappa}},y)=0$.
In particular, the tuple
$(x_{1},...,x_{m}):=(\beta_{1}^{1/p^{\kappa}},...,\beta_{m}^{1/p^{\kappa}})$
is a solution modulo $(t)$ to the above system of equations.
$"\Leftarrow"$ A generalized version of Greenberg’s Theorem due to Moret-
Bailly (see Corollary 1.2.2 [MB11]) shows that

$f_{1}(x_{1},...,x_{m},t)=f_{2}(x_{1},...,x_{m},t)=...=f_{n}(x_{1},...,x_{m},t)=0$

has a solution in $\mathbb{F}_{p}[[t]]^{1/p^{\infty}}$ if and only if it has a
solution modulo $(t^{N})$, for all $N\in\mathbb{N}$. Equivalently, if and only
it has a solution modulo $(t^{p^{N}})$ for all $N\in\mathbb{N}$. Using the
$N$-th iterated Frobenius, we see that for any $N\in\mathbb{N}$ we have

$\mathbb{F}_{p}[[t]]^{1/p^{\infty}}\models\exists
x_{1},...,x_{m}(\bigwedge_{1\leq i\leq n}f_{i}(x_{1},...,x_{m},t)=0\mod
t^{p^{N}})\iff$ $\mathbb{F}_{p}[[t]]^{1/p^{\infty}}\models\exists
x(\bigwedge_{1\leq i\leq n}f_{i}(x_{1},...,x_{m},t^{1/p^{N}})=0\mod t)$

and the latter is true by assumption, for any $N\in\mathbb{N}$. ∎

######  Remark 7.2.4.

By an identical argument, one can prove a similar result for
$\mathbb{Q}_{p}(\zeta_{p^{\infty}})$ or $\mathbb{Q}_{p}^{ab}$. In the case of
the latter, one gets that $\overline{\mathbb{F}}_{p}[[t]]^{1/p^{\infty}}$ is
$\exists^{+}$-decidable in $L_{t}$, provided that $\mathbb{Q}_{p}^{ab}$ is
$\forall^{1}\exists$-decidable in $L_{\text{val}}$.

It would be nice to have a version of Proposition 7.2.3 for the full theories.
Together with Corollary A, this would yield a Turing equivalence between the
theories of $\mathbb{Q}_{p}(p^{1/p^{\infty}})$ and
$\mathbb{F}_{p}((t))^{1/p^{\infty}}$. Nevertheless, Proposition 7.2.3 still
suggests that if we eventually want to understand the theories of
$\mathbb{Q}_{p}(p^{1/p^{\infty}})$, $\mathbb{Q}_{p}(\zeta_{p^{\infty}})$ and
$\mathbb{Q}_{p}^{ab}$ (even modest parts of their theories), we would have to
face certain characteristic $p$ difficulties, posed in Question 7.2.5 below:

######  Question 7.2.5.

$(a)$ Is $\text{Th}_{\exists^{+}}(\mathbb{F}_{p}[[t]]^{1/p^{\infty}})$
decidable in $L_{t}$?
$(b)$ Is
$\text{Th}_{\exists^{+}}(\overline{\mathbb{F}}_{p}[[t]]^{1/p^{\infty}})$
decidable in $L_{t}$?

### Acknowledgements

A great mathematical debt is owed to E. Hrushovski from whom I learned many of
the crucial ideas in this work. I would like to thank J. Koenigsmann for
careful readings, insightful suggestions and for his encouragement throughout.
Finally, many thanks to T. Scanlon for his hospitality while I was a visitor
in UC Berkeley and for many inspiring discussions.

## References

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