!renewcal O
!newfrak m
!newbb U
!newbb Q
!newbb F
!newop acl
!newop dcl
!newop eq
!newop tp
!newop im
!newop dom
!newop ACVF
!newop GL
!newtup x

\newcommand{\acleq}{\acl^\eq}
\newcommand{\dcleq}{\dcl^\eq}

\newcommand{\Ueq}{\U^\eq}

\newcommand{\code}[1]{\ulcorner #1 \urcorner}

{\it Notes from a seminar in Münster on [Hrushovski-Rideau ``Valued Fields, 
Metastable Groups''], Nov 2019.}

Work in $\U |= ACVF$ sufficiently saturated, in geometric language (sorts 
$K,\Gamma,k,S_n,T_n$). Definable means definable over a small subset of $\U$.
We write $\dcl$ resp.\ $\acl$ for $\dcleq$ resp.\ $\acleq$.

# Pure imaginaries

Definition:
  $e (- \Ueq$ is #purely# #imaginary# over $C (= \U$
  if $\acl(Ce) \cap K = \acl(C) \cap K$.
  .

Lemma:
  $e$ is purely imaginary over $C$ iff
  $\dcl(Ce) \cap K (= \acl(C)$.
  .
Proof:
  Symmetric polynomials.

  STS $\acl(Ce) \cap K (= \acl(\dcl(Ce) \cap K)$.

  But if $a (- \acl(Ce) \cap K$,
  say $a=a_1,...,a_n$ are the conjugates over $Ce$,
  then the coefficients of $\prod_i (x-a_i)$
  are in $\dcl(Ce) \cap K$.
  .

Definition:
  An $\infty$-definable set $D$ is #purely# #imaginary# if there is no 
  definable map $D --> K$ with infinite image.

  Equivalently: for any $C$ over which $D$ is defined, any 
  $e (- D$ is purely imaginary over $C$.
  .

Definition:
  A $\infty$-definable set $D$ is #boundedly# #imaginary# if any 
  definable map $D --> \Gamma$ is bounded.

  Equivalently: for any $C$ over which $D$ is defined, any 
  $e (- D$ is boundedly imaginary over $C$, where
  $e (- \Ueq$ is #boundedly# #imaginary# over $C (= \U$
  if for every $\gamma (- \Gamma(Ce) := \Gamma \cap \dcl(Ce)$,
  $\tp(\gamma/C)$ is bounded (i.e. not $+\infty$ or $-\infty$,
  i.e. $\gamma$ is in the convex hull of $\Gamma(C)$).
  .

Lemma{boundedPure}:
  Any boundedly imaginary $D$ is purely imaginary.
  .
Proof:
  If the image of $D --> K$ is infinite, it contains a ball $B$.
  But for $b (- B$,
  $x |-> v(x-b)$ is an unbounded map $B --> \Gamma$.
  .

Define $\alpha\O := \{ x (- K : v(x) >= \alpha \}$
and $\beta\m := \{ x (- K : v(x) > \beta \}$.

Lemma{orth}:
  Let $X$ be the set of closed or open balls of a fixed radius $\alpha$,
  i.e. $X = K / \alpha\O$ or $X = K / \alpha\m$.

  There is no definable finite correspondence $\Gamma --> X$
  with infinite image,
  i.e. for any definable $Z (= \Gamma\times X$ with $\forall \gamma (- 
  \Gamma.\; |\pi_1^{-1}(\gamma) \cap Z| < \aleph_0$, $|\pi_2(Z)| < \aleph_0$.
  .
Proof:
  Else, by Swiss cheese decomposition applied to the union of the balls, 
  $\pi_2(Z)$ contains all but finitely many of the balls within a closed ball 
  $B$ of some radius $\gamma <= \alpha$, with $\gamma < \alpha$ in the case $X 
  = K / \alpha\O$.

  Then $Z$ induces a finite correspondence from $\Gamma$ onto $B / \gamma\m$, 
  and hence onto an infinite subset of $k$.
  By definable Skolem functions for $\Gamma$, this yields a definable partial 
  function $k --> \Gamma$ with infinite image, contradicting strong minimality 
  of $k$.
  .

Lemma{ballsBounded}:
  Let $\alpha <= 0 <= \beta (- \Gamma$ and $n >= 1$.
  Then $(\alpha\O / \beta\m)^n$ is boundedly imaginary, and hence purely 
  imaginary.
  .
Proof:
  \underline{$n=1$}: we may assume $\alpha = 0$. Suppose $f : \O / \beta\m --> 
  \Gamma$ is an unbounded $C$-definable map. Say it is unbounded above. Let 
  $\gamma > \Gamma(C)$.
  Let $g : \O --> \O / \beta\m$ be the quotient map.
  Then $g^{-1}(f^{-1}(\gamma))$ is a boolean combination of balls each of 
  radius in $[0,\beta] \cap \Gamma(C\gamma)$.
  Now $\Gamma$ is a pure divisible ordered abelian group,
  so any definable map from $\Gamma$ to the bounded interval $[0,\beta]$ is 
  eventually constant,
  so $[0,\beta] \cap \Gamma(C\gamma) (= \Gamma(C)$.

  But the balls vary with $\gamma$ since $f$ is a function,
  so we contradict Lemma~{orth}.

  \underline{$n=k+1$}:
  Given $f : (\alpha \O / \beta\m)^{k+1} --> \Gamma$,
  inductively $y |-> \sup_\x f(\x,y)$ defines a map
  $\alpha \O / \beta\m --> \Gamma$, so this map is bounded, and hence so is $f$.
  .

Lemma{pureBalls}:
  $e (- \Ueq$ is purely imaginary over $C$
  iff
  for some $\alpha <= 0 <= \beta$ with $\alpha,\beta (- \Gamma(Ce)$,
    $e (- \dcl(\acl(C), \alpha\O / \beta\m)$.
  .
Proof:
  $<==$:
    Immediate from Lemma~{ballsBounded}.
    %Let $X := \{ \code{\O(1,a) + 2\beta\O(0,1)} : a (- K \} (= S_2$.
    %Then $X$ is $\beta$-definable
    %and intersection with $(1,K)$ gives a ${}$-definable bijection
    %%$X --> K / 2\beta\O$,
    %and the quotient map induces a $\beta$-definable surjection
    %$K / 2\beta\O -->> K / \beta\m )= \alpha\O / \beta\m$.
    %So $\alpha\O / \beta\m (= \dcl(\beta S_2)$.
    %(Thanks to Martin Hils for explaining this trick of representing torsors 
    %as submodules.)

    %Then $e (- \dcl(\acl(C)\beta d)$ for some $d (- S_2^n$,
    %so $\acl(Ce) \cap K (= \acl(C\beta d) \cap K = \acl(C) \cap K$
    %since $\beta d$ is purely imaginary over $C$.

  $==>$: 
    By the EI and the definition of purely imaginary,
    $e$ is interdefinable with a 
    finite tuple from $\bigcup_n S_n \cup \bigcup_n T_n \cup (\acl(C) \cap 
    K)$.

    Note that $K -) x |-> \min (\beta, v(x))$ induces a map
    $\alpha\O / \beta\m -->> [\alpha,\beta]$,
    so $[\alpha,\beta] (= \dcl(\alpha\O / \beta\m)$.
    Hence if $\alpha <= \alpha' < 0 < \beta' <= \beta$
    then $\alpha'\O / \beta'\m (= \dcl(\alpha\O / \beta\m)$.

    So it suffices to consider the case $e (- S_n$ or $e (- T_n$ for some $n$.

    Let $\Lambda <= K^n$ be a rank $n$ free $\O$-submodule,
    say with $\O$-basis $(\lambda_1,...,\lambda_n)$.
    
    Let $\Lambda^- := \{ \alpha : \Lambda (= (\alpha\O)^n \}$
    and $\Lambda^+ := \{ \beta : (\beta\m)^n (= \Lambda \}$.
    Note $\sup\Lambda^- = \min_{i,j} v((\lambda_i)_j) (- \Lambda^-$.

    Let $\alpha = \min \{0, \sup \Lambda^- \} (- \Lambda^-$,
    and $\beta = \max \{0, \inf \Lambda^+ \} (- \Lambda^+$.

    Then $\alpha <= 0 <= \beta$ and $\alpha,\beta (- \Gamma(Ce)$
    and $(\beta\m)^n (= \Lambda (= (\alpha\O)^n$.
    then $S_n -) \code{\Lambda} (- \dcl((\lambda_i / \beta\m)_i) (= \dcl(\alpha\O 
    / \beta\m)$.

    Also $\beta\m^n (= \m\Lambda$ (since $\m(\beta\m) = \beta\m$),
    so $T_n )= \Lambda/\m\Lambda (= \dcl(\alpha\O / \beta\m)$.
    
    Since $\Lambda$ was arbitrary, we conclude.
  .

Remark{SnTnPure}:
  It follows from the proof that each $S_n$ and $T_n$ is purely imaginary.
  .

Lemma{boundedBalls}:
  Let $D$ be a $\infty$-definable set over $C$. TFAE:
  ***(1) $D$ is boundedly imaginary.
  **(2) There exists a definable surjection $g : (\O / \beta\m)^n -->> D$.
  **(3) For some $\alpha <= 0 <= \beta$ with $\alpha,\beta (- \Gamma(C)$,
  $D (= \dcl(\acl(C), \alpha\O / \beta\m)$.

  {\it NOTE: the paper has $C$ rather than $\acl(C)$, but I don't see how to get that.
  This statement is good enough for the application in Corollary~6.4.}
  *ee*
  .
Proof:
  (2) $==>$ (1): by Lemma~{ballsBounded}.

  (3) $==>$ (2): by compactness, we get finitely many $g_i : (\alpha_i \O / \beta_i\m)^{n_i} --> D$
  with $D = \bigcup_i \im g_i$.
  We can assume $\alpha_i = 0$ by multiplying,
  and we can assume $\beta_i = \max_i \beta_i =: \beta$,
  and then combine the $g_i$ into a surjection $g : (\O / \beta\m)^{\sum_i n_i} --> D$.

  (1) $==>$ (3): By compactness, it suffices to show that if $e (- D$,
  then such $\alpha,\beta (- \Gamma(C)$ exist with $e (- \dcl(C, \alpha\O / 
  \beta\m)$.

  Now $D$ is purely imaginary by Lemma~{boundedPure},
  so by Lemma~{pureBalls} we can find such $\alpha,\beta (- \Gamma(Ce)$.

  Say $\alpha = f(e)$, where $f$ is over $C$. Since $D$ is boundedly 
  imaginary, $\alpha >= \inf_{D \cap \dom f} f =: \alpha' (- \Gamma(C)$.
  Similarly we find $\beta <= \beta' (- \Gamma(C)$.
  
  Then $\alpha\O / \beta\m (= \dcl(\alpha'\O / \beta'\m)$,
  so $e (- \dcl(\acl(C), \alpha'\O / \beta'\m)$ as required.
  .

Remark:
  Martin Hils remarks that in Lemma~{pureBalls} we can equivalently ask just 
  for $\alpha,\beta (- \Gamma$ rather than require them in $\Gamma(Ce)$. From 
  this one can conclude that $\infty$-definable $D$ is purely imaginary iff 
  there is a definable surjection $f : (B^{\"open})^n -->> D$ where 
  $B^{\"open}$ is the set of all open balls (of all radii).

  Then it follows from this and Lemma~{boundedBalls}, that if $D$ is 
  $\infty$-definable and purely resp.\ boundedly imaginary, it is contained in 
  a purely resp.\ boundedly imaginary definable set.
  .


# Redundant

Alternative direct proof of Remark~{SnTnPure}:

Lemma:
  Any product of $S_n$'s and $T_n$'s is purely imaginary.

  So by the EI, $e (- \Ueq$ is purely imaginary over $C$ iff it is interdefinable with a 
  finite tuple from $\bigcup_n S_n(\U) \cup \bigcup_n T_n(\U) \cup (\acl(C) \cap K)$.
  .
Proof:
  Let $T$ be a completion of $\ACVF$.

  It suffices to find an uncountable model of $T$ in which each $S_n$ and 
  $T_n$ is countable,
  since the image of a definable map to $K$ with infinite image contains a 
  ball and so has the same cardinality as $K$.

  Thanks to Martin Hils for providing the following example.

  Let $L |= T$ be countable with $\Gamma(L) = \Q$.
  (We can take $L$ to be an algebraic closure of $\Q(t)$, $\F_p(t)$, or $\Q$ 
  with the $p$-adic valuation.) 
  Consider $L$ as a normed field (with $||x|| := 2^{-v(x)}$).
  
  \newcommand{\Lbar}{\bar{L}}

  Let $\Lbar$ be the completion of $L$.
  Fact: $\Lbar |= \ACVF$.
  We have $|\Lbar| = 2^{\aleph_0}$,
  but $S_n(\Lbar) = \GL_n(\Lbar) / \GL_n(\O(\Lbar)) <-- \GL_n(L) / \GL_n(\O(L)) 
  = S_n(L)$ is a bijection
  since $\GL_n(\O(\Lbar))$ is an open neighbourhood of the identity
  and $\GL_n(\O(\Lbar)) \cap \GL_n(L) = \GL_n(\O(L))$.
  Similarly $k(\Lbar) = \O(\Lbar) / \m(\Lbar) <-- \O(L) / \m(L) = k(L)$ is a bijection.

  So $|S_n(\Lbar)| = \aleph_0 = |T_n(\Lbar)|$.
  .
