==== TABLE OF CONTENTS ====
The-Margulis-Platonov-conjecture-over-number-fields//paper.tex
The-Margulis-Platonov-conjecture-over-number-fields//preamble.tex
The-Margulis-Platonov-conjecture-over-number-fields//references.tex
The-Margulis-Platonov-conjecture-over-number-fields//sections/characters.tex
The-Margulis-Platonov-conjecture-over-number-fields//sections/colors.tex
The-Margulis-Platonov-conjecture-over-number-fields//sections/exceptional-types.tex
The-Margulis-Platonov-conjecture-over-number-fields//sections/finite-groups.tex
The-Margulis-Platonov-conjecture-over-number-fields//sections/foundations.tex
The-Margulis-Platonov-conjecture-over-number-fields//sections/introduction.tex
The-Margulis-Platonov-conjecture-over-number-fields//sections/palettes.tex
The-Margulis-Platonov-conjecture-over-number-fields//sections/unitary-induction.tex
===========================

==== BEGIN The-Margulis-Platonov-conjecture-over-number-fields//paper.tex ====
\input{preamble}
\title{The Margulis--Platonov conjecture\\over number fields}
\author{OpenAI}
\date{September 23, 2026}
\begin{document}
\maketitle
\begin{abstract}
We prove the Margulis--Platonov conjecture over number fields.
For an absolutely almost simple simply connected group, every noncentral
abstract normal subgroup of its rational points is the inverse image of an
open normal subgroup of the product of its anisotropic nonarchimedean
local groups.
\end{abstract}
\tableofcontents
\clearpage
\input{sections/introduction}
\input{sections/foundations}
\input{sections/colors}
\input{sections/characters}
\input{sections/palettes}
\input{sections/finite-groups}
\input{sections/unitary-induction}
\input{sections/exceptional-types}
\input{references}
\end{document}
==== END The-Margulis-Platonov-conjecture-over-number-fields//paper.tex ====

==== BEGIN The-Margulis-Platonov-conjecture-over-number-fields//preamble.tex ====
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==== END The-Margulis-Platonov-conjecture-over-number-fields//preamble.tex ====

==== BEGIN The-Margulis-Platonov-conjecture-over-number-fields//references.tex ====
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==== END The-Margulis-Platonov-conjecture-over-number-fields//references.tex ====

==== BEGIN The-Margulis-Platonov-conjecture-over-number-fields//sections/characters.tex ====
\section{Excluding finite abelian characters}\label{sec:characters}

Throughout this section, $L/k$ is quadratic, $D$ is a central division
algebra over $L$ of odd degree $n\geq3$, and $*$ is a unitary involution.
We retain $U=\U(D,*)$ and $S=\SU(D,*)$ from
Section~\ref{sec:colors}. Our aim is the following assertion, which
eliminates the abelian alternative of
Proposition~\ref{prop:color-dichotomy}.

\begin{theorem}\label{thm:no-characters}
Every homomorphism from $U(k)$ to a finite abelian group is trivial on
$S(k)$.
\end{theorem}

Fix such a homomorphism $\chi:U(k)\to B$, and write $B$ additively.
The proof first converts $\chi$ into a difference function on $D$.
Exact elementary transformations make this difference independent of its
arguments. A real-signature argument then converts that algebraic
identity into continuity on $S(k)$, after which local commutators finish
the proof. No continuity of $\chi$ is assumed.

\subsection{A noncommutative difference identity}

Let $\mathcal D$ be the set of nonzero $b\in D$ for which
$b+b^*=a^*a$ for some $a\in D$. Define
\begin{equation}\label{eq:chars-f}
 m_b=1-ab^{-1}a^*,\qquad f(b)=\chi(m_b).
\end{equation}
These definitions also allow $a=0$, in which case $b$ is skew and
$m_b=1$. Multiplication using $a^*a=b+b^*$ gives
$m_bm_b^*=1$. If $a'{}^*a'=a^*a\ne0$, then $a'=va$ with
$v\in U(k)$, and the two resulting elements $m_b$ are conjugate by
$v$. Thus $f$ is well-defined. Moreover,
\begin{equation}\label{eq:chars-congruence}
 f(x^*bx)=f(b)\qquad(x\in D^\times),
\end{equation}
because $ax$ can be used in the definition on the left.

For the moment fix $d\in D^\times$ such that the reduced spectra of
$d$ and $d^*$ are disjoint. For $(a,c)\in D^2\setminus\{(0,0)\}$,
there is a unique $e\in D$ satisfying
\begin{equation}\label{eq:chars-sylvester}
 e+e^*=c^*c,\qquad ed+d^*e^*=a^*a.
\end{equation}
Indeed the equation
\[
 ed-d^*e=a^*a-d^*c^*c
\]
has a unique solution: after splitting the algebra its linear part is
the Sylvester operator, whose eigenvalues are the differences of the
two disjoint spectra. Taking adjoints shows that $c^*c-e^*$ is another
solution of this same equation, giving the first identity in
Equation~\ref{eq:chars-sylvester} and then the second. The solution is
nonzero, for $e=0$ would imply $a=c=0$. Consequently
\[
 F_d(a,c)=f(ed)-f(e)
\]
is defined on all nonzero pairs.

\begin{lemma}\label{lem:chars-pair-invariance}
The function $F_d$ is invariant under independent left multiplications
of its two arguments by elements of $U(k)$, and under
\begin{equation}\label{eq:chars-pair-move}
 (a,c)\longmapsto(a-cd,a-c).
\end{equation}
\end{lemma}

\begin{proof}
The independent unitary multiplications leave
Equation~\ref{eq:chars-sylvester} unchanged. For the second assertion put
$b=ed$ and $z=b^*+e-a^*c$. Expansion, with the order of all factors
unchanged, gives
\[
 z+z^*=(a-c)^*(a-c),\qquad
 zd+d^*z^*=(a-cd)^*(a-cd).
\]
The transformation in Equation~\ref{eq:chars-pair-move} is invertible:
its kernel would give $a=c$ and $c(1-d)=0$, whereas $d\ne1$ by spectral
disjointness. In particular $z\ne0$.

Set $A'=ce^{-1}-ab^{-1}$ and $q=b^{-*}ze^{-1}$, where
$b^{-*}=(b^*)^{-1}$. A direct expansion gives $A'{}^*A'=q+q^*$.
Also
\[
 q=(b^{-1})^*(zd)b^{-1},
\]
so Equation~\ref{eq:chars-congruence} identifies $f(q)$ with $f(zd)$.
For clarity, the multiplicative identity needed here is
\begin{equation}\label{eq:chars-three-unitaries}
 1-(A'e)z^{-1}(b^*A'{}^*)=m_bm_zm_e^*,
\end{equation}
where the defining vectors for $m_b,m_z,m_e$ are respectively
$a,c-a,c$. It follows from the two identities
\begin{align*}
 A'e&=-ab^{-1}z+m_b(c-a),\\
 b^*A'{}^*&=ze^{-*}c^*+(c-a)^*m_e^*.
\end{align*}
To check the remaining terms after multiplying these identities, use
$m_ba=-ab^{-1}b^*$ and $c^*c=e+e^*$. The part of the left side of
Equation~\ref{eq:chars-three-unitaries} not containing
$-(m_b(c-a))z^{-1}((c-a)^*m_e^*)$ is then $m_bm_e^*$.
The left side of Equation~\ref{eq:chars-three-unitaries} is precisely
the unitary element defining $f(q)$, because $q^{-1}=ez^{-1}b^*$.
Applying $\chi$ proves
\[
 f(zd)=f(b)+f(z)-f(e),
\]
which is the required invariance. These computations take place in
$D$; no commutativity of their entries is used.
\end{proof}

\subsection{Producing exact elementary transformations}

Suppose now that $d$ lies in a $*$-stable maximal subfield $E\subset D$
which is Galois over $L$. Write $\bar z=z^*$ for $z\in E$, and suppose
that the $n$ conjugates $t_1,\ldots,t_n$ of $d$ are distinct and disjoint
from $\bar t_1,\ldots,\bar t_n$. Skolem--Noether gives a decomposition
\[
 D=\bigoplus_{j=1}^n E w_j,\qquad w_jd=t_jw_j.
\]
Let $E^1=\{z\in E^\times:z\bar z=1\}$. The pair move in
Equation~\ref{eq:chars-pair-move} and independent left multiplications
by elements of $E^1$ preserve the $n$ two-dimensional $E$-blocks in
$D\oplus D$.

In the invariance group take the subgroup $H$ generated by diagonal
$E^1$-multiplications on either coordinate and by the simultaneous matrices
\begin{equation}\label{eq:chars-boosts}
 B_h(t_j)=\begin{pmatrix}1&t_j\bar h\\h&1\end{pmatrix}
 \qquad(h\in E^1).
\end{equation}
Indeed changing the sign of the second output in
Equation~\ref{eq:chars-pair-move} gives $B_{-1}$, and conjugating by
diagonal phases gives all $B_h$. Each determinant is $1-t_j\ne0$.

\begin{lemma}\label{lem:chars-pure-shears}
For each block $i$, $H$ contains a nonidentity upper elementary
unipotent supported only on block $i$, and a nonidentity lower
elementary unipotent supported only on block $i$.
\end{lemma}

\begin{proof}
We first separate one block from the others. On a block with parameter
$t$, the product $B_h(t)B_{hz}(t)$ sends the first coordinate axis to
the line of slope
\[
 h\frac{1+z}{1+tz}\qquad(z,h\in E^1).
\]
At $t=t_j$ the norm of the slope with $h=1$ is
\begin{equation}\label{eq:chars-slope-norm}
 \ell_j(z)=\frac{(1+z)^2}{(1+t_jz)(z+\bar t_j)}.
\end{equation}
This is a degree-two rational function. Apart from finitely many $z$,
the other solution $z'$ of $\ell_j(z')=\ell_j(z)$ is distinct from
$z$ and lies in $E$. Explicitly, with $a=t_j$ and $b=\bar t_j$,
\[
 z'=\frac{(a-b)z+1+ab-2b}{(1+ab-2a)z+b-a}.
\]
It belongs to $E^1$: conjugate reciprocation
preserves the two roots and fixes the first, so fixes the second.
If $s_t(z)=(1+z)/(1+tz)$, choose
$h'=s_{t_j}(z)/s_{t_j}(z')\in E^1$. Then
\[
 g_j(t)=\bigl(B_{h'}(t)B_{h'z'}(t)\bigr)^{-1}B_1(t)B_z(t)
\]
has lower-left entry zero at $t_j$.

The lower-left numerator of $g_j(t)$ is
\[
 (1+tz')(1+z)-h'(1+z')(1+tz).
\]
This is a nonzero linear polynomial in $t$: the corresponding two
fractional slope functions have distinct poles and nonzero numerators.
Its unique root is $t_j$. Thus the
lower-left entry of $g_j(t)$ is nonzero at every other $t_\ell$ and at
every $\bar t_\ell$. Matrices of the form
\[
 \begin{pmatrix}\alpha(t)&t\overline{\beta}(t)\\
                 \beta(t)&\overline{\alpha}(t)\end{pmatrix},
\]
where the bar acts on coefficients and fixes the indeterminate $t$,
are closed under multiplication and inversion. The boosts have this
form. Consequently the upper-right entry of $g_j(t)$ vanishes exactly
when its lower-left entry vanishes at $\bar t$. It is therefore
nonzero at all $t_\ell$, including $t_j$.

We may also arrange that all diagonal entries at every $t_\ell$ are
nonzero. Here is a check that the excluded conditions are proper.
At $z=-1$, Equation~\ref{eq:chars-slope-norm} has a double zero and its
degree-two interchange sends $z$ to $z'$ with derivative $-1$.
Thus $h'\to-1$ as $z\to-1$, and both boost products tend to
$(1-t)I$. Hence $g_j(t)\to I$. These are algebraic limits on the
norm-one parameter curve over the fixed field of bar; its rational
points $E^1$ are infinite, by Hilbert 90, so avoid the finitely many
proper exceptional conditions.
We have obtained a triangular, nondiagonal matrix at $j$, and matrices
with all four entries nonzero at the other blocks. Repeating the
construction with $\bar t_j$ in place of $t_j$ gives a lower triangular,
nondiagonal matrix at $j$.

Fix a block $i$. For each $j\ne i$, let $H^{(j)}$ be the subgroup
generated by $g_j$ and the diagonal phases. Its projection at $j$ is
upper triangular. Its projection at $i$ is projectively Zariski dense
in $\PGL_2$ over an algebraic closure of $E$: it contains a
Zariski-dense subgroup of the diagonal
torus and a matrix with all entries nonzero. A proper algebraic subgroup
containing the diagonal torus lies in a Borel subgroup or its torus
normalizer, neither of which contains that matrix. Since the second
derived group of a triangular group is trivial,
$(H^{(j)})''$ acts trivially at $j$, but remains projectively Zariski
dense at $i$. The latter assertion follows by taking the Zariski
closures of commutators in the perfect algebraic group $\PGL_2$.

Importantly, use the \emph{full} kernels
\[
 K_j=\ker\bigl(H\longrightarrow\GL_2(E)
                         \text{ on block }j\bigr).
\]
Each $K_j$ is normal in $H$ and contains $(H^{(j)})''$, so its image
at $i$ is projectively dense. A commutator of two such kernels is
trivial on the union of their excluded blocks and remains projectively
dense at $i$: the commutators of two dense images are dense in the
derived algebraic group $\PGL_2$. Iterating gives a projectively dense
image at $i$ of
\[
 K_{\ne i}=\bigcap_{j\ne i}K_j.
\]
Write $H_i$ for the projection of $H$ at $i$, and $P_i$ for the image
of $K_{\ne i}$ there. We have proved that $P_i\triangleleft H_i$ and
that $P_i$ is projectively Zariski dense.

We next pass from this density to actual unipotents. Commuting the
upper and lower triangular, nondiagonal elements already constructed
with diagonal phases shows that $H_i$ contains upper and lower
elementary unipotents with some nonzero coefficients. Phase conjugation
and addition show that their coefficients contain scalar multiples of
the additive group $\mathbb Z E^1$. Its rational span is all of $E$.
Indeed that span is a finite-dimensional subring of a field, hence a
field; the Cayley ratios $(1+x)/(1-x)$ for $\bar x=-x$ show that it
contains the whole skew subspace, which generates $E$. It follows that
the allowable coefficients for both upper and lower unipotents contain
a common nonzero integral ideal $I\subset\mathcal O_E$.

Choose $p\in P_i$ with $p\infty=r\ne\infty$ on $\mathbb P^1(E)$.
Choose an infinite-order unit $u\in\mathcal O_E^\times$ sufficiently
close to $1$ modulo an integral ideal that
\[
 u=1+st\quad(s,t\in I),\qquad (1-u^2)r\in I.
\]
Such units exist by the unit theorem: $[E:\mathbb Q]\ge6$, and a
congruence subgroup of the unit group has finite index. More explicitly,
fix $0\ne s\in I$, require $u-1\in sI$, and impose the additional
congruence clearing the denominator of $r$. With
$U(x)=\left(\begin{smallmatrix}1&x\\0&1\end{smallmatrix}\right)$
and $L(x)=\left(\begin{smallmatrix}1&0\\x&1\end{smallmatrix}\right)$,
the exact identity
\[
 U(s)L(t)U(-s/u)L(-tu)=\diag(u,u^{-1})
\]
shows that $H_i$ contains the affine transformation
\[
 h:x\longmapsto u^2x+(1-u^2)r.
\]
It fixes $r$ and $\infty$. Normality gives
$q=h^{-1}p^{-1}hp\in P_i$. This element fixes $\infty$ and has affine
slope $u^{-4}$: in a local coordinate at $\infty$ the derivative of
$h$ is $u^{-2}$, whereas at $r$ it is $u^2$.
Thus, for any nonzero $x\in I$,
\[
 qU(x)q^{-1}U(-x)=U((u^{-4}-1)x)
\]
is an actual nonidentity upper unipotent in $P_i$. Scalar factors in
a representative of $q$ cancel in this identity. By the definition of
$P_i$ it lifts to an element of $H$ which is the identity on every
other block. Interchanging the axes proves the lower assertion.
\end{proof}

To turn a single nonzero shear coefficient into an arbitrary additive
change in $D$, we need the following exact statement, not merely local
density.

\begin{lemma}\label{lem:chars-additive-span}
The additive subgroup generated by $U(k)$ is all of $D$:
\[
 \mathbb Z U(k)=D.
\]
\end{lemma}

\begin{proof}
Let $M=\mathbb Z U(k)$. It is a subring, since $U(k)$ is a group.
For a rational prime $p$, let $C_p$ be its additive closure in
$D_p=D\otimes_{\mathbb Q}\mathbb Q_p$. Weak approximation for $U$
puts the product of its local groups above $p$ inside $C_p$. The largest
$\mathbb Q_p$-vector subspace of $C_p$ is
\[
 V_p=\bigcap_{j\ge0}p^jC_p.
\]
It is closed; here closed additivity implies stability under
$\mathbb Z_p$. Left and right multiplication by every local unitary
element preserve $V_p$. Differentiating these actions shows stability
under multiplication by every skew element, separately in each local
component. The skew elements generate the local algebra: an invertible
central skew scalar and its inverse, together with skew multiples of
Hermitian elements, generate all Hermitian and skew elements. Hence
$V_p$ is a two-sided ideal of $D_p$.

Every simple factor occurs in this ideal. At a nonsplit finite place of
$k$, the algebra over $L$ splits and the unitary group is isotropic
because $n\ge3$. Choose an unbounded sequence in that local unitary
group, taking the identity in the other local components. Multiplication
by suitable positive powers $p^{r_\nu}$, with $r_\nu\to\infty$, gives
a subsequence tending to a nonzero vector in this factor and zero in
the others. For each fixed $j$, the same sequence multiplied by
$p^{r_\nu-j}$ lies in $C_p$ eventually, so its limit is divisible by
$p^j$ in $C_p$. The nonzero limit belongs to $V_p$.
At a split place the two factors are interchanged by $*$; the unitary
group contains reciprocal central scalars. Taking $(p^{-r},p^r)$ and
scaling by $p^r$ gives a nonzero limit in one factor alone; reversing the
two scalars gives the other. Thus the ideal $V_p$ contains every simple
factor, proving $C_p=D_p$.

This local density implies that the rational span of $M$ is $D$, so
$M$ contains a full $\mathbb Z$-lattice $\Lambda$ in $D$. Fix
$x=p^{-j}\lambda$, with $\lambda\in\Lambda$. Choose $a\in M$ with
$a-x\in\Lambda\otimes\mathbb Z_p$. There is a positive integer $m$
prime to $p$, congruent to $1$ modulo $p^j$, which clears all denominators
of $a$ away from $p$ relative to $\Lambda$. Then
$ma-mx\in\Lambda$ at every rational prime, and hence globally, while
$(m-1)x\in\Lambda$. Therefore $x\in M$. Inverting all rational primes
in $\Lambda$ now gives $M=D$.
\end{proof}

\begin{proposition}\label{prop:chars-difference}
Let $E\subset D$ be a $*$-stable maximal subfield, Galois over $L$.
Let $d\in E^\times$ have $n$ distinct $L$-conjugates
$t_1,\ldots,t_n$ whose set is disjoint from
$\{\bar t_1,\ldots,\bar t_n\}$, where $\bar z=z^*$ on $E$.
Then, whenever $b,bd\in\mathcal D$,
\begin{equation}\label{eq:chars-difference}
 f(bd)-f(b)=\chi(\bar d/d).
\end{equation}
\end{proposition}

\begin{proof}
We show that the invariance group of $F_d$ is transitive on nonzero
pairs. If $c\ne0$, one of its $E$-block coordinates is nonzero. The
corresponding pure upper shear of Lemma~\ref{lem:chars-pure-shears}
changes $a$ by a fixed nonzero element $y\in D$, leaving $c$ unchanged.
Conjugating this transformation by left multiplication of the first
coordinate by $v\in U(k)$ changes $a$ by $vy$. Products and inverses
therefore permit addition of every element of $(\mathbb ZU(k))y=D$,
by Lemma~\ref{lem:chars-additive-span} and division. The lower shears
likewise permit arbitrary changes of $c$ when $a\ne0$. These operations
connect every nonzero pair to a pair with both entries nonzero, and then
connect any two such pairs. Thus $F_d$ is constant.

At $e=(d-\bar d)^{-1}$, Equation~\ref{eq:chars-sylvester} is solved
by $c=0$, $a=1$. The unitary elements defining $f(e)$ and $f(ed)$ are
respectively $1$ and $\bar d/d$. This evaluates the constant and proves
Equation~\ref{eq:chars-difference}.
\end{proof}

\subsection{Scalar normalization and a cyclic maximal field}

Let $\Omega$ be the nonsplit real places of $k$. At $v\in\Omega$
write the involution as $x^*=H_v^{-1}\bar x^{\,t}H_v$ on
$M_n(\mathbb C)$. For a nonzero Hermitian element $h\in D$, call its
signature admissible when the signature of $H_vh$ is that of $H_v$ up
to overall sign, for every $v\in\Omega$. This does not depend on the
matrix realization. Define $\mathcal D^\#$ to consist of the nonzero
skew elements and the $b\in D$ for which $h=b+b^*\ne0$ has admissible
signature. Division makes every such $h$ invertible.

\begin{lemma}\label{lem:chars-normalization}
If $h=h^*\ne0$ has admissible signature and $s=\Nrd(h)\in k^\times$,
there is $a\in D^\times$ with
\begin{equation}\label{eq:chars-normalized-norm}
 a^*a=sh,\qquad \Nrd(a)=s^{(n+1)/2}.
\end{equation}
Consequently
\[
 f^\#(b)=f(sb)\qquad(b\in\mathcal D^\#)
\]
is well-defined using any $s\in k^\times$ for which $sb\in\mathcal D$.
It is invariant under congruence by $D^\times$ and multiplication by
$k^\times$.
\end{lemma}

\begin{proof}
At a nonsplit finite place the algebra splits; classification of
Hermitian forms says that its dimension and determinant modulo norms
determine its isometry class. The determinant ratio for $sh$ is
$s^{n+1}$, which is a norm since $n+1$ is even. At a nonsplit real
place, $\operatorname{sign}(s)$ is positive if $H_vh$ has the signature
of $H_v$, and negative if it has the opposite signature, because $n$ is
odd. Thus $sH_vh$ has precisely the signature of $H_v$. At split finite
places the assertion reduces to surjectivity of the local reduced norm;
at split archimedean places the algebra is split, again by odd degree.
These facts give local solutions of $a^*a=sh$. Their reduced norms can
be made exactly $s^{(n+1)/2}$, since the determinant map of the local
full unitary group is onto its norm-one group. The variety in
Equation~\ref{eq:chars-normalized-norm} is a torsor under the simply
connected group $S$, and so has a rational point by its Hasse principle.
We use here the local Hermitian classification and the simply connected
Hasse principle in their number-field forms
\cite{KMRT,PRbook}.

If both $sb$ and $tb$ are in $\mathcal D$ and $b$ is not skew, taking
reduced norms of the two defining Hermitian equations shows that
$(t/s)^n$ is a norm from $L^\times$. The quotient
$k^\times/N_{L/k}(L^\times)$ has exponent two. Since $n$ is odd,
$t/s$ itself is a norm, say $t/s=\bar z z$. Central congruence by $z$
and Equation~\ref{eq:chars-congruence} give $f(tb)=f(sb)$. For skew
$b$ both values are zero. This proves independence. Congruence and
scalar invariance follow by applying the same argument to any
permissible normalization.
\end{proof}

\begin{lemma}\label{lem:chars-cyclic-field}
There is a $*$-stable maximal subfield $E\subset D$ which is cyclic of
degree $n$ over $L$. Its norm-one torus
$T=\ker(N_{E/L}:\Res_{E/L}\mathbb G_m\to\mathbb G_m)$ has weak
approximation over $L$.
\end{lemma}

\begin{proof}
By the odd-degree case of Grunwald--Wang, choose a cyclic extension
$F/k$ of degree $n$ having full local degree at each place below the
Brauer support of $D$ \cite{Neukirch}; see also
\cite[Chapter~VIII, Theorem~2.4 and Corollary~2.5]{MilneCFT}.
It is disjoint from $L$.
The local invariant criterion shows that $E_0=LF$ splits $D$; since
$[E_0:L]=n$, the embedding criterion for central simple algebras gives
an embedding $\alpha:E_0\hookrightarrow D$ \cite{Reiner}.
Let bar on $E_0$ fix $F$ and restrict to the nonidentity automorphism
of $L/k$. Skolem--Noether supplies $h\in D^\times$ with
\[
 \alpha(\bar z)^*=h\alpha(z)h^{-1}\qquad(z\in E_0).
\]
Replacing $z$ by $\bar z$ and taking adjoints gives
\[
 \alpha(z)=h^{-*}\alpha(\bar z)^*h^*.
\]
Thus $h^*$ also intertwines the two embeddings, and
$c=h^{-1}h^*$ lies in $\alpha(E_0)$. On this field the adjoint satisfies
$x^*=h\bar xh^{-1}$, so
\[
 \bar c=h^{-1}c^*h=h^{-*}h=c^{-1}.
\]
Hilbert 90 gives $t/\bar t=c$. The replacement
$h\mapsto h\alpha(t)$ preserves the intertwining identity and is
Hermitian, because
$(h\alpha(t))^*=h\alpha(\bar t)c=h\alpha(t)$.

At a real place in $\Omega$, the odd cyclic extension $F/k$ splits
completely. For the Hermitian form with matrix $H_vh$, the $n$
eigenlines of $\alpha(E_0)$ are orthogonal. Replacing $h$ by
$h\alpha(z)$, with $z\in F^\times$, scales their Hermitian coefficients
independently by the real embeddings of $z$. Weak approximation in $F$
therefore makes its signature admissible at every such place, even
equal to that of $H_v$ if desired. Apply
Lemma~\ref{lem:chars-normalization} to obtain $a^*a=sh$. The embedding
$z\mapsto a\alpha(z)a^{-1}$ is $*$-stable and induces bar, since
\[
 (a\alpha(\bar z)a^{-1})^*
 =(a^*)^{-1}h\alpha(z)h^{-1}a^*
 =a\alpha(z)a^{-1}.
\]
Its image is the required $E$. Finally, for a cyclic generator $\sigma$
of $\Gal(E/L)$, Hilbert 90 parametrizes $T(L)$ by
$y/\sigma(y)$ with $y\in E^\times$. The same parametrization holds at
each completion, and weak approximation in $E$ proves weak
approximation for $T$.
\end{proof}

Fix this field $E$ and an integer $m\ge1$ annihilating $B$. Call
$d\in T(L)^m$ admissible if its conjugates over $L$ are distinct and
disjoint from their bars. These conditions specify a nonempty $k$-Zariski
open subset of $\Res_{L/k}T$: after extending scalars, the two
determinant-one eigenvalue
lists can be chosen independently, avoiding equality within or between
the lists. In particular admissible elements exist, since the rational
points of $T$ are Zariski dense and the power map is dominant.
Let $\mathcal C$ be their $D^\times$-conjugates. This set is closed
under inversion.

\begin{corollary}\label{cor:chars-step-invariance}
If $t\in\mathcal C$ and $b,bt\in\mathcal D^\#$, then
\begin{equation}\label{eq:chars-step-invariance}
 f^\#(bt)=f^\#(b).
\end{equation}
\end{corollary}

\begin{proof}
Write $t=xdx^{-1}$. Congruence by $x$ gives
$x^*(bt)x=(x^*bx)d$. Choose separate scalar normalizations of these
two arguments. Their ratio only replaces $d$ in
Proposition~\ref{prop:chars-difference} by a nonzero $k$-multiple, which
preserves spectral disjointness and leaves $\bar d/d$ unchanged.
If $d=y^m$ with $y\in T(L)$, then
$\bar d/d=(\bar y/y)^m$, an $m$-th power in $E^1\subset U(k)$.
Thus its character is zero, and
Equation~\ref{eq:chars-step-invariance} follows.
\end{proof}

\subsection{Keeping products in the real signature domain}

The subgroup $J\subset\SL_1(D)(L)$ generated by $\mathcal C$ is
noncentral and normal. The established inner-type case of the
Margulis--Platonov theorem, applied over the number field $L$, gives an
open neighborhood $W$ of $1$ in
\[
 \prod_{w\in A_D}\SL_1(D)(L_w)
\]
whose rational inverse image is contained in $J$. Here $A_D$ is the set
of finite places of $L$ where $D\otimes_LL_w$ is division. This use of
the known inner case is independent of the outer case being proved
\cite[Theorem~2]{SegevSeitz2002}\cite[Sections~3.4--3.5]{PRsurvey}.
Although every element of $J$ is a word in the allowed steps, an
arbitrary word need not preserve the real signature domain at its
intermediate products. The next lemma supplies the required control.

\begin{lemma}\label{lem:chars-local-invariance}
Let $b\in\mathcal D^\#$ have nonzero Hermitian part. For every
$r\in\SL_1(D)(L)$ with $r_{A_D}\in W$ and $r_\Omega$ sufficiently
close to $1$, one has
\begin{equation}\label{eq:chars-local-invariance}
 f^\#(br)=f^\#(b).
\end{equation}
\end{lemma}

\begin{proof}
If $\Omega$ is empty, the signature condition is absent: every nonzero
element of $D$ is in $\mathcal D^\#$, either with nonzero Hermitian
part or skew. A word in $\mathcal C$ therefore proves the assertion
directly by Corollary~\ref{cor:chars-step-invariance}.

Suppose $\Omega\ne\varnothing$ and write
$G_\Omega=\prod_{v\in\Omega}\SL_n(\mathbb C)$. Let $\mathcal O$
be the open set of $g\in G_\Omega$ for which $bg$ has nondegenerate
Hermitian part with the required signatures. It contains $1$. Restrict
$r_\Omega$ to a sufficiently small neighborhood of $1$ that it can be
joined to $1$ by a path $\gamma:[0,1]\to\mathcal O$.
Since $r\in J$, write it as a word in elements of $\mathcal C$.
Join each factor of this word to $1$ in the corresponding conjugate of
$T$ at $\Omega$; these complex tori are connected. The product of the
factor paths is a path $p$ from $1$ to $r_\Omega$.

We explain why the based loop $p^{-1}\gamma$ is a finite product of
based loops in rational conjugates of $T$. The Lie algebras of finitely
many such conjugates span the real Lie algebra of $G_\Omega$.
Indeed the full conjugacy span is the whole Lie algebra by simplicity,
and rational conjugators in $D^\times$ are dense independently at the
finitely many places. The multiplication map from finitely many of
these tori consequently has a continuous local section near $1$, taking
$1$ to the tuple of identities. Applying this section pointwise factors
every based loop sufficiently uniformly close to $1$ into based torus
loops. The subgroup they generate in the based loop group is therefore
open. The based loop group is connected, because every
$\SL_n(\mathbb C)$ is simply connected; hence that open subgroup is
the whole based loop group. This proves the desired finite
factorization of $p^{-1}\gamma$.

Appending these factors to those of $p$ writes $\gamma$ pointwise as
a finite product of paths in rational conjugates of $T$, each starting
at $1$ and with rational prescribed endpoint. Every endpoint belongs
to the corresponding rational $m$-th-power group: the original word
has this property, and the appended loops end at $1$.

Take a fine mesh in $[0,1]$ with at least one interior point. At its
interior nodes approximate each torus factor by rational points in its
$m$-th-power group, leaving the endpoints unchanged. These rational
powers are dense at $\Omega$: use weak approximation for $T$ and
surjectivity of the power map on each complex torus. We can further
require that every consecutive ratio in every factor is admissible,
including the ratios next to the fixed endpoints. To see this precisely,
regard all interior nodes as independent variables in products of
$\Res_{L/k}T$. Every consecutive-ratio map is dominant; beside a fixed
endpoint it is simply translation or inverse translation of the free
variable. The complement of admissibility therefore pulls back to a
proper closed subset in each case. The finite union of these subsets
cannot fill the irreducible parameter variety. One may work in the
parameters before taking $m$-th powers, whose power map is dominant:
weak approximation then supplies rational parameters arbitrarily close
to the chosen local roots while avoiding the finite proper closed
exclusions. Thus both endpoint ratios and all interior ratios can be
made admissible simultaneously.

Pass from one mesh product to the next by changing one factor at a
time. Each change is right multiplication by a rational
$D^\times$-conjugate of an admissible increment. All intermediate
products remain in $\mathcal O$ if the mesh is sufficiently fine and
the approximations sufficiently close. To justify the uniform choice,
the finitely many factor paths are uniformly continuous, their images
are compact, and $\gamma([0,1])$ is a compact subset of the open set
$\mathcal O$; mixed products from two sufficiently nearby mesh nodes
are uniformly close to $\gamma$. At the rational intermediate points
the real nondegeneracy also guarantees nonzero Hermitian part in $D$.
Corollary~\ref{cor:chars-step-invariance} now applies to every move.
The initial and final products are exactly $1$ and $r$, so their
character differences give Equation~\ref{eq:chars-local-invariance}.
\end{proof}

\subsection{Continuity and the final vanishing}

\begin{proof}[Proof of Theorem~\ref{thm:no-characters}]
Choose $u\in U(k)$ with $\Nrd(u)\ne1$. Such an element can be chosen
central, since the norm-one torus of $L/k$ has infinitely many rational
points. Division implies that $1-u$ is invertible. Put
$b=(1-u)^{-1}$. Direct calculation gives
\begin{equation}\label{eq:chars-final-b}
 b+b^*=1,\qquad b^*=-ub,\qquad f^\#(b)=f(b)=\chi(u).
\end{equation}
If $u'\in U(k)$ has the same reduced norm as $u$, define
$b'=(1-u')^{-1}$; it satisfies the same identities. In particular
\[
 q=\frac{\Nrd(b')}{\Nrd(b)}\in k^\times.
\]
Indeed $\overline{\Nrd(b)}=-\Nrd(u)\Nrd(b)$ by oddness, and the same
identity for $b'$ makes the ratio fixed by $L/k$.

Suppose $u'$ is sufficiently close to $u$ at $A_D$ and at $\Omega$,
using both places of $L$ over every split place of $k$. Choose
$a\in D^\times$ with
\[
 \Nrd(a)=q^{(n-1)/2},
\]
and with $a$ close to $1$ at all these places. The global reduced norm
theorem gives one such element without the approximation conditions:
the only possible archimedean sign obstruction occurs at quaternionic
real factors, which cannot occur in odd degree. Locally near $1$ the
reduced norm is a submersion, so there are local solutions near $1$ for
$q$ near $1$. Weak approximation for the simply connected inner group
$\SL_1(D)$ then adjusts the global solution to these local solutions
\cite{PRbook,Reiner}.

Set $b''=q^{-1}a^*b'a$. Congruence and scalar invariance give
$f^\#(b'')=f^\#(b')$, while
\[
 \Nrd(b'')=q^{-n}q^{n-1}\Nrd(b')=\Nrd(b).
\]
Thus $r=b^{-1}b''\in\SL_1(D)(L)$ is close to $1$ at $A_D$ and
$\Omega$. Lemma~\ref{lem:chars-local-invariance} gives
$f^\#(b'')=f^\#(b)$, and hence $\chi(u')=\chi(u)$.
Taking $u'=us$ shows that $\chi|_{S(k)}$ is continuous for the topology
induced by these finitely many local factors.

The set $A_D$ consists precisely of the two places over each place in
$A$ from Section~\ref{sec:colors}. At nonsplit places the algebra
splits, and at a split place its local norm-one group is anisotropic
exactly when the component algebra is division. Weak approximation for
$S$ therefore extends this finite-valued continuous restriction uniquely
to a continuous homomorphism on
\[
 \prod_{v\in A}S(k_v)\ \times\ \prod_{v\in\Omega}S(k_v).
\]
For completeness, write $H$ for the dense rational subgroup and $K$ for
the kernel of its character. Continuity makes $K$ closed in $H$, so
$\overline K\cap H=K$. Finitely many cosets $h_iK$ cover $H$; the
finite closed union of $h_i\overline K$ therefore covers the ambient
local product. The subgroup $\overline K$ consequently has finite index
and is open, since its complement is a finite union of closed cosets.
It is normal by density and continuity of conjugation. The resulting
map $H/K\to G/\overline K$, where $G$ denotes the local product, is an
isomorphism: injectivity is the intersection identity, and surjectivity
is the coset cover. This gives the asserted continuous extension. Each real
factor is connected and so maps trivially to the finite group $B$.

Finally fix a split place $v\in A$, writing
$U(k_v)=\mathcal D_v^\times$ and $S(k_v)=\SL_1(\mathcal D_v)$.
Weak approximation in $U$ approximates any two local elements at $v$
by rational unitary elements, with the identity prescribed at the
other places of $A$ and $\Omega$. Their rational commutators belong to
$S(k)$ and have character zero. Continuity of the extended restriction
therefore kills every local commutator of $\mathcal D_v^\times$.
Lemma~\ref{lem:colors-local-abelianization} says that these commutators
generate a dense subgroup of $\SL_1(\mathcal D_v)$, so this factor also
maps trivially. All factors have been killed, proving
$\chi(S(k))=0$. This includes $A=\varnothing$, when only the connected
real factors occur, and includes the case where both products are
empty.
\end{proof}
==== END The-Margulis-Platonov-conjecture-over-number-fields//sections/characters.tex ====

==== BEGIN The-Margulis-Platonov-conjecture-over-number-fields//sections/colors.tex ====
\section{Finite quotients of an odd-degree unitary group}
\label{sec:colors}

We now consider the first remaining family: special unitary groups defined
by division algebras of odd reduced degree.  The finite defect constructed
in Proposition~\ref{prop:finite-defect} is a quotient of a finite-index
normal subgroup of the special unitary group.  The purpose of this section
is to use a nontrivial simple quotient of that defect to construct a
homomorphism defined on the \emph{full} unitary group.  The
abelian and nonabelian alternatives will then be excluded separately.

Let $L/k$ be a quadratic extension of number fields, let $D$ be a central
simple $L$-algebra of reduced degree $n$, and let $*$ be a unitary involution
on $D$.  Write
\[
 S=\SU(D,*),\qquad U=\U(D,*),\qquad
 L^1=\{a\in L^\times:a\bar a=1\},
\]
where the bar is the nontrivial automorphism of $L/k$.  The determinant
homomorphism on $U$ is the reduced norm:
\[
 \det=\Nrd_{D/L}:U(k)\longrightarrow L^1,
 \qquad S(k)=\ker(\det).
\]
Fix $i\in L^\times$ with $\bar i=-i$.  On a dense open subset of the
$k$-vector space of Hermitian elements, the Cayley map
\begin{equation}\label{eq:colors-cayley}
 x\longmapsto (x+i)(x-i)^{-1},\qquad x=x^*,
\end{equation}
is a birational parametrization of $U$.  In particular, $U(k)$ satisfies
weak approximation: first perturb finitely many local targets into this
open chart and then approximate their inverse Cayley coordinates.

We recall the relevant local descriptions of these groups
\cite{PRbook,Reiner,KMRT}.  At a nonsplit nonarchimedean place of $L/k$,
the algebra $D$ splits.  Indeed, the existence of the unitary involution
makes its Brauer corestriction zero
\cite[Theorem~3.1(2)]{KMRT}, and corestriction for a local field
extension preserves the local Brauer invariant.  A Hermitian form of
dimension at least three over a quadratic extension of nonarchimedean
local fields is isotropic.  Thus, when $n\geq3$, the finite places where
$S$ has rank zero are split places of $L/k$.  At each such place $v$,
choose one of the two components of $D\otimes_k k_v$ and denote it by
$\mathcal D_v$.  It is a central division algebra of degree $n$ over
$k_v$, the involution exchanges the two components, and
\begin{equation}\label{eq:colors-anisotropic-factors}
 U(k_v)=\mathcal D_v^\times,
 \qquad S(k_v)=\SL_1(\mathcal D_v).
\end{equation}
Under this identification, the local determinant is the reduced norm on
$\mathcal D_v$.

For the rest of this section assume that $D$ is a division algebra and
that $n\geq3$ is odd.  Then $S$ is $k$-anisotropic.  Let $A$ be its finite
set of anisotropic nonarchimedean places and use the notation of
Proposition~\ref{prop:finite-defect}:
\begin{equation}\label{eq:colors-defect}
 R=S(k)^e,\qquad
 P_0=\overline{\delta(R)}\subset S_A,
 \qquad V_0=\delta^{-1}(P_0),\qquad V=V_0/R,
\end{equation}
where $e\geq1$, $S_A=\prod_{v\in A}S(k_v)$, and the power subgroup is
abstract.  Finally, put $A_0=\det U(k)\subset L^1$.  An empty product over
$A$ is the trivial group throughout.

\begin{proposition}[Finite quotient dichotomy]\label{prop:color-dichotomy}
Suppose that the finite group $V$ in Equation~\eqref{eq:colors-defect} is
nontrivial.  Then at least one of the following exists:
\begin{enumerate}[label=\textup{(\roman*)}]
 \item a homomorphism $\chi:U(k)\to C$ to a finite abelian group such
 that $\chi(S(k))\ne1$;
 \item a homomorphism $\phi:U(k)\to\mathcal F$ to a finite nonabelian
 simple group such that
 \[
  \phi(V_0)=\mathcal F,\qquad R\subset\ker\phi.
 \]
\end{enumerate}
\end{proposition}

The first step is an approximation statement that retains both
commutativity and exact determinants.  Its second part is also required
to retain a specified coset modulo the abstract subgroup $R$.

\subsection{Commuting approximation}

\begin{proposition}\label{prop:commuting-approximation}
With the preceding notation, the following statements hold.
\begin{enumerate}[label=\textup{(\roman*)}]
 \item Let $a,b\in A_0$.  For every $v\in A$, suppose that
 $x_v,t_v\in U(k_v)$ commute and have determinants $a_v,b_v$.
 There exist commuting $x',t\in U(k)$ with
 $\det x'=a$ and $\det t=b$ whose $A$-components approximate the
 prescribed pair arbitrarily closely.
 \item Let $\gamma\in V_0$ and $h\in U(k)$.  There exist commuting
 $x',t\in U(k)$ with
 \[
  x'\in\gamma R,\qquad \det t=\det h,
 \]
 and with $t_A$ arbitrarily close to $h_A$.
\end{enumerate}
\end{proposition}

\begin{proof}
For part~\textup{(ii)}, set $a=1$ and $b=\det h$, so that $a,b$ denote
the two prescribed determinants in either part.
We construct a regular first element $x'$ whose centralizer has enough
local determinant norms everywhere.  Lemma~\ref{lem:norm-torus} will
then produce the second element in that centralizer with its prescribed
determinant and approximations.

\paragraph{Local choices at the prescribed places.}
At $v\in A$, a commuting pair in $\mathcal D_v^\times$ lies in a common
maximal subfield.  This is because the algebra it generates is a
commutative domain in a division algebra, and hence a field, which can
be extended to a maximal subfield.  In the corresponding maximal torus
of $U(k_v)$, perturb the first element within its fixed-determinant
fiber until it is regular.  Such regular elements are dense in that
fiber: geometrically the fiber is a translate of
\[
 \{(z_1,\ldots,z_n)\in\bbG_m^n:z_1\cdots z_n=1\},
\]
and no equality $z_r=z_s$ holds identically on it.  These fibers are
smooth, so their nonempty Zariski opens are dense in the local topology.
The conjugation map around a regular semisimple element is a local
submersion.  Consequently, sufficiently small changes of this first
element have centralizer tori obtained by conjugation with elements
arbitrarily close to $1$.  Conjugating the second element with the same
elements retains its determinant and its prescribed approximation.

For part~\textup{(ii)}, take the first local element sufficiently close
to $1$ in a maximal torus containing $h_v$, and with determinant $1$.
The choice near $1$ is compatible with the closure of $\gamma R$ because
$\gamma\in V_0$.  The same regular perturbation applies.  Hence in both
parts we have nonempty open conditions for the first element such that
its local centralizer admits the required second element.

Enlarge the finite set of prescribed places to include all archimedean
places, places of bad reduction or small residue characteristic, places
ramified in $L/k$, and places where $a$ or $b$ is not a unit.  At every
additional place choose a maximal torus whose determinant image contains
both required determinants.  At nonsplit places the algebra splits, and
a diagonal unitary torus has determinant image equal to the local
norm-one group.  At a split finite place outside $A$, write the chosen
component of the algebra as $M_l(\Delta)$.  If $\Delta$ is nontrivial,
then $l\geq2$: use an unramified maximal subfield of $\Delta$ in one
diagonal block and a totally ramified maximal subfield in another.
Their product norm image contains all units and an element of valuation
one, and is therefore all of $k_v^\times$.  Complete these two blocks
to a maximal \'etale subalgebra.  If $\Delta=k_v$, the usual diagonal
torus suffices.  At split archimedean places the algebra is split because
its degree is odd.  In each case we may choose the first element regular
with the specified determinant and then take a sufficiently small
opening around it.

We also prescribe finitely many auxiliary split places, outside all
exceptions, at which the algebra splits and $a,b$ are units.  Choose
unramified maximal tori having every possible permutation cycle type in
$S_n$, one type at each such place.  Norms from their unramified field
factors cover the unit group.  These conditions will force the joint
Galois group needed by Lemma~\ref{lem:norm-torus}.

In part~\textup{(i)}, the determinant-$a$ slice is a rational translate
of $S$ because $a\in A_0$.  Weak approximation for $S$, followed by
Proposition~\ref{prop:power-approximation} on that translate, permits all
the local openings just chosen.  In part~\textup{(ii)}, apply that
proposition in the coset $\gamma R$.  The openings near $1$ at $A$ are
allowed by Equation~\eqref{eq:colors-defect} and the closure statement
in Proposition~\ref{prop:finite-defect}; the other local openings are
unrestricted.  We now check the reduction and pole conditions required
by Proposition~\ref{prop:power-approximation}.

\paragraph{Integral reductions and single poles.}
At an integral reduction outside the prescribed set, exclude matrices
whose characteristic polynomial has no simple root over an algebraic
closure of the residue field.  This exclusion has geometric codimension
at least two in a fixed nonzero determinant slice of $\GL_n$.
Indeed, if every eigenvalue has multiplicity at least two, there are at
most $\lfloor n/2\rfloor$ distinct eigenvalues.  After fixing the
determinant, they contribute at most $\lfloor n/2\rfloor-1$ parameters.
For each Jordan type with fixed eigenvalues, the orbit dimension is at
most $n^2-n$.  Thus the excluded set has dimension at most
\[
 n^2-n+\lfloor n/2\rfloor-1
 \leq (n^2-1)-2.
\]
There are only finitely many Jordan types, so the same bound holds for
their union and its closure.  These conditions descend to the unitary
determinant slice and spread out after finitely many model exceptions.

A simple absolute residue root belongs to a residue-field irreducible
factor of multiplicity one.  Hensel lifting gives an unramified field
factor of the algebra generated by the first element.  We do not need
the simple root itself to lie in the ground residue field.

At a single pole in Proposition~\ref{prop:power-approximation}, the place
splits in the fixed splitting field and the first element has the form
\[
 x'=g_L\,\diag(t^{e d_1},\ldots,t^{e d_n})\,g_R,
 \qquad d_1<\cdots<d_n,
\]
where $\ord_v(t)>0$ and the two side factors are integral and invertible.
Require every principal minor on the first $j$ indices of $g_Rg_L$ to
be nonzero modulo the place, for $1\leq j\leq n$.  This is a nonempty
open condition on each pole hyperplane: an undamaged basic parameter
list in Proposition~\ref{prop:power-approximation} is dominant, so its
varying side factor can meet the finitely many principal-minor opens.
If $c_j$ is the $j$th elementary characteristic coefficient, principal
minor expansion gives
\begin{equation}\label{eq:colors-pole-valuations}
 \ord_v(c_j)=e(d_1+\cdots+d_j)\ord_v(t).
\end{equation}
The term from the first $j$ indices is the unique term of least
valuation.  The successive Newton slopes are strictly distinct and all
segments have horizontal length one.  The characteristic polynomial
therefore splits into distinct linear factors over $k_v$.

The additional denominator places in the approximation procedure admit
the same determinant-compatible local tori described above.  At the
later finite model exceptions, integral unit parameters giving regular
reduction provide the required openings.  This checks all hypotheses
of Proposition~\ref{prop:power-approximation}.

\paragraph{The global centralizer and its norm torsor.}
Choose $x'$ by that proposition, also imposing global regularity.  Since
$D$ is division,
\[
 E=L(x')\subset D
\]
is a maximal field of degree $n$ over $L$.  It is stable under $*$ because
$(x')^*=(x')^{-1}$.  Put $F=\{z\in E:z^*=z\}$, the fixed field.
Then $E=LF$ and
$[F:k]=n$.

Let $\Gamma\subset S_n\times C_2$ be the Galois group of the normal
closure of $F$ together with $L$, acting on the embeddings of $F$ and on
$L$.  At the auxiliary places split in $L$, the prescribed cycle types
show that the kernel of $\Gamma\to C_2$ meets every conjugacy class of
$S_n$.  A proper subgroup of a finite group cannot have this property:
the transitive action on its cosets would have a derangement, by
averaging the number of fixed cosets, whereas every conjugacy class
meeting the subgroup would rule out all derangements.  Hence that
kernel is $S_n\times\{1\}$.  The projection to $C_2$ is surjective, so
\[
 \Gamma=S_n\times C_2.
\]

The unitary torus in the centralizer of $x'$ is
\[
 T_E=\Res_{F/k}\bigl(\Res^1_{E/F}\bbG_m\bigr),
\]
and its determinant is $N_{E/L}:T_E\to\Res^1_{L/k}\bbG_m$.
At all prescribed places its image contains $b_v$ by construction.
At every other integral place, take the full $*$-orbit of the unramified
field factor of $E$ found above; all its fields are unramified over $k_v$
since $L/k$ is unramified here.  Its fixed algebra is an unramified field
factor $F_j/k_v$ of $F\otimes_k k_v$: it is the fixed field when the
factor is stable, and the diagonal fixed algebra when two factors are
exchanged.  The corresponding $*$-stable algebra is
$F_j\otimes_{k_v}(L\otimes_k k_v)$, possibly split, so no individual
$E$-factor is required to be $*$-stable.  Restrict the norm
map to this factor.  It is a surjective map of unramified tori with
connected kernel.  To check connectedness, over an algebraic closure
the dual map sends a generator to a vector with entries $1$ or $-1$;
this vector is primitive.  Lang's theorem on the residue-field tori and
smooth lifting show that the norm map is onto integral points.  As $b_v$
is a unit, it is a local determinant norm.  At a single pole the torus
splits by Equation~\eqref{eq:colors-pole-valuations}, and the same
conclusion holds without an integrality restriction.

Thus the fiber $N_{E/L}^{-1}(b)$ has points everywhere locally.  Its
acting kernel is exactly the torus of Lemma~\ref{lem:norm-torus}, with
the joint Galois group just verified.  That lemma gives a rational point
$t$ in the fiber approximating the chosen local second elements at $A$.
It belongs to the centralizer of $x'$, so $x'$ and $t$ commute.  The
construction retained the required determinant and, in part~\textup{(ii)},
the coset $\gamma R$.  This proves both assertions.
\end{proof}

\subsection{A simple quotient preserved by the full unitary group}

The passage from class preservation to a quotient of the ambient group
follows the inner-type argument of Rapinchuk--Potapchik
\cite[Corollary~2.3 and Theorem~2.4]{RapinchukPotapchik1996}.
Here Proposition~\ref{prop:commuting-approximation} supplies the required
unitary input. Conjugation by $U(k)$ preserves $R$, since $R$ is
characteristic in $S(k)$.  It also preserves $P_0$ and $V_0$ by passage to local closures.
In fact the resulting action on $V$ preserves each conjugacy class.
To see this, take $\gamma\in V_0$ and $h\in U(k)$.  By
Proposition~\ref{prop:commuting-approximation}\textup{(ii)}, choose
$x'\in\gamma R$ commuting with $t$, with $\det t=\det h$ and $t_A$
sufficiently close to $h_A$ that $t h^{-1}\in V_0$.  In $V$, conjugation
by $t$ fixes the image of $x'$, whereas $t h^{-1}$ induces an inner
automorphism.  Conjugation by $h$ consequently carries $\gamma R$ to a
conjugate of itself.

If $V\ne1$, choose a maximal proper normal subgroup of $V$ and denote
the resulting simple quotient by $B$.  Its kernel is invariant under
the $U(k)$ action because a normal subgroup is a union of conjugacy
classes.  The induced automorphisms of $B$ are class-preserving.  If $B$
is nonabelian, the theorem of Feit and Seitz
\cite[Theorem~C]{FeitSeitz} says that every such automorphism is inner.
Since a nonabelian simple group has trivial center, the identification
$\Inn(B)=B$ gives
\[
 \phi:U(k)\longrightarrow B.
\]
Its restriction to $V_0$ is the quotient map to $B$, and $R$ acts
trivially.  This is alternative~\textup{(ii)} of
Proposition~\ref{prop:color-dichotomy}.

It remains to treat the case that $B$ is cyclic of prime order.  Let
$R'$ be its kernel upstairs, so that
\begin{equation}\label{eq:colors-central-quotient}
 R\subset R'\subset V_0,\qquad B=V_0/R'.
\end{equation}
Class preservation on this abelian quotient means that the action is
trivial.  Thus $R'$ is normal in $U(k)$ and $B$ is central in $U(k)/R'$.
The rest of the section uses a nonzero character of this central
subgroup to construct a finite character of $U(k)$ that is nonzero on
$S(k)$.

\subsection{Local abelianization and the metaplectic input}

We need two local facts: the continuous characters of the compact groups
$S(k_v)$, and a splitting theorem for their finite product.  The first
fact also identifies the closed subgroup generated by commutators of
the full local multiplicative group.

\begin{lemma}\label{lem:colors-local-abelianization}
Let $F_v$ be a nonarchimedean local field of characteristic zero, with
residue field $\bbF_q$, and let $\mathcal D$ be a central division algebra
over $F_v$ of odd degree $n\geq3$.  Put $S_v=\SL_1(\mathcal D)$.
Reduction induces an isomorphism
\[
 S_v/\overline{[S_v,S_v]}
 \simeq T_v:=\ker\bigl(N_{\bbF_{q^n}/\bbF_q}:
                       \bbF_{q^n}^\times\to\bbF_q^\times\bigr).
\]
Moreover,
\[
 \overline{[\mathcal D^\times,\mathcal D^\times]}=S_v.
\]
If an element of $\mathcal D^\times$ has reduced-norm valuation $r$,
its action on $T_v$ is $\sigma^r$, where $\sigma:z\mapsto z^{q^h}$
for some $h$ relatively prime to $n$.
\end{lemma}

\begin{proof}
Use the cyclic description of a division algebra over a local field
\cite{Reiner}.  There is an unramified maximal subfield $E_v/F_v$ and
a prime element $\Pi$ such that conjugation by $\Pi$ induces a generator
$\sigma$ of $\Gal(E_v/F_v)$.  The residue algebra is $\bbF_{q^n}$ and
$\sigma$ acts there by $q^h$ with $(h,n)=1$.  Normalize valuations so
that $\ord_v(\Nrd\Pi)=1$.  Every element of $S_v$ is integral and its
residue lies in $T_v$.  The Teichm\"uller representatives in $E_v$ lift
$T_v$ to a subgroup of $S_v$, so reduction onto $T_v$ is surjective.

Let $\mathfrak P=(\Pi)$ be the maximal ideal of the maximal order of
$\mathcal D$, and set
\[
 S_r=S_v\cap(1+\mathfrak P^r),\qquad r\geq1.
\]
In leading-coefficient coordinates, the quotients of this filtration
are
\begin{equation}\label{eq:colors-filtration}
 S_r/S_{r+1}\simeq
 \begin{cases}
  (\bbF_{q^n},+),&n\nmid r,\\
  \ker\Tr_{\bbF_{q^n}/\bbF_q},&n\mid r.
 \end{cases}
\end{equation}
For completeness, the norm filtration is
\[
 \Nrd(1+\mathfrak P^r)=1+\mathfrak p^{\lceil r/n\rceil}.
\]
The inclusion follows from the valuations of the characteristic
coefficients; the reverse inclusion follows already from norms of
principal units in the unramified maximal subfield.  If $n\nmid r$,
the norm images for $r$ and $r+1$ agree, so any leading coefficient can
be corrected at the next level to have norm one.  If $n\mid r$, the
first norm coefficient is the residue trace, and the next-level norm
image gives exactly the trace-zero restriction in
Equation~\eqref{eq:colors-filtration}.  The finite-field trace is onto
also when the residue characteristic divides $n$.

For $n\nmid r$, commute an element at level $r$ with a Teichm\"uller
element of $T_v$ not fixed by $\sigma^r$.  Such an element exists:
$T_v$ is cyclic of order $(q^n-1)/(q-1)$ and is not contained in the
multiplicative group of a proper subfield.  On the leading coefficient
this commutator is multiplication by a nonzero residue scalar.
Thus commutators fill $S_r/S_{r+1}$ in this case.

For $n\mid r$, use brackets between levels $1$ and $r-1$.  Both levels
have unrestricted leading coefficients because $n\geq3$.  Taking the
coefficient $1$ in degree $1$ makes the bracket coefficients contain
\[
 (\sigma-1)\bbF_{q^n}=\ker\Tr_{\bbF_{q^n}/\bbF_q}.
\]
Hence commutators again fill the quotient.  Successive approximation
now puts all of $S_1$ in $\overline{[S_v,S_v]}$.  The reverse inclusion
holds because the residue quotient $T_v$ is abelian.  This proves the
first assertion.

Finally, commutators of $\Pi$ with unramified units have residues
$\sigma(z)/z$, which fill $T_v$ by finite-field Hilbert 90.  Together
with the already obtained subgroup $S_1$, they generate a dense subgroup
of $S_v$.  Every commutator in $\mathcal D^\times$ has reduced norm one,
proving the second assertion.  A local unit acts trivially on the
commutative residue field, while $\Pi$ acts by $\sigma$.  Writing an
arbitrary element as a unit times $\Pi^r$ gives the last statement.
\end{proof}

Here is the precise metaplectic consequence we use.  Let
$I=\bbR/\bbZ$, written additively.  If $G$ is a simply connected
absolutely almost simple group over a number field, not of type $A_1$,
and $A'$ is a finite set of nonarchimedean places where $G$ is anisotropic,
then the restriction map
\begin{equation}\label{eq:colors-metaplectic}
 H^2_{\mathrm m}\!\left(\prod_{v\in A'}G(k_v),I\right)
 \longrightarrow H^2\bigl(G(k),I\bigr)
 \quad\text{is injective}.
\end{equation}
This is the specialization $V_2=\varnothing$ of
\cite[Proposition~2.6, p.~114]{PRmetaplectic}. If $A'$ is nonempty,
the local classification forces the absolute type to be $A$, so the
type-$D$ exclusion inherited from Proposition~2.4 there is harmless.
If $A'$ is empty, the source is $H^2_{\mathrm m}(1,I)=0$.
The source uses measurable
cohomology for locally compact groups and abstract cohomology on the
rational group.  In degree two these classify, respectively,
topological central circle extensions with Borel sections and abstract
central extensions; see \cite[Section~3.2]{PRsurvey}.  Thus a topological
central circle extension of the displayed finite local product that
splits over the abstract rational group has a continuous homomorphic
section.  Indeed, the zero measurable class gives a Borel homomorphic
section, and Borel homomorphisms between these locally compact
second-countable groups are continuous.

We apply only Equation~\eqref{eq:colors-metaplectic} with $G=S$ and
$A'=A$.  Here $n\geq3$ excludes type $A_1$, and every support place is
nonarchimedean and anisotropic.  In particular, this invocation involves
neither a comparison with real-place metaplectic kernels nor an
identification of a congruence kernel requiring the normal-subgroup
assertion we are proving.

\subsection{A circle extension of the full local determinant group}

Give $A_0$ the discrete topology and define
\begin{equation}\label{eq:colors-Q}
 Q=\{(y,a)\in U_A\times A_0:\det y=a_A\},
 \qquad U_A=\prod_{v\in A}U(k_v).
\end{equation}
Its determinant kernel is $S_A$.  The group $Q$ is locally compact and
second countable: each of its countably many determinant fibers is a
translate of the compact group $S_A$.  There is a natural homomorphism
\[
 \rho:U(k)\longrightarrow Q,\qquad u\longmapsto(u_A,\det u).
\]
The subgroup $P_0\subset S_A$ is open and normal in $Q$.  Normality
follows first under the dense group $\rho(U(k))$ and then under $Q$.
In fact $\rho(U(k))$ is dense in every determinant fiber by weak
approximation for $S$.  The same argument, applied modulo the open
subgroup $P_0$, gives an isomorphism of discrete groups
\begin{equation}\label{eq:colors-discrete-quotient}
 Q/P_0\simeq U(k)/V_0.
\end{equation}

The central quotient in Equation~\eqref{eq:colors-central-quotient}
provides the discrete central extension
\[
 1\longrightarrow B\longrightarrow U(k)/R'
 \longrightarrow U(k)/V_0\longrightarrow1.
\]
Pull it back along Equation~\eqref{eq:colors-discrete-quotient} and push
it out by an injective character $\iota:B\hookrightarrow I$.  We obtain
a locally compact second-countable central extension
\begin{equation}\label{eq:colors-circle-extension}
 1\longrightarrow I\longrightarrow\widetilde Q
 \xrightarrow{p}Q\longrightarrow1.
\end{equation}
The pullback description gives continuous local sections, and it also
gives a specified abstract lift
\[
 \ell:U(k)\longrightarrow\widetilde Q,
 \qquad p\circ\ell=\rho.
\]
Over $P_0$ there is a canonical continuous homomorphic section
$c:P_0\to\widetilde Q$, obtained by taking the identity element in the
discrete quotient $U(k)/R'$.  For $u\in V_0$, these descriptions give
\begin{equation}\label{eq:colors-canonical-lift}
 \ell(u)=c(u_A)\,\iota(uR'),
\end{equation}
where the last factor denotes the central element of $\widetilde Q$
corresponding to $\iota(uR')\in I$.

\begin{lemma}\label{lem:colors-commuting-lifts}
If two elements of $Q$ commute, any lifts of them to $\widetilde Q$
commute.
\end{lemma}

\begin{proof}
The two determinant coordinates are elements $a,b\in A_0$.  By
Proposition~\ref{prop:commuting-approximation}\textup{(i)}, their local
components are limits of commuting rational pairs with these exact
determinants.  The specified lifts under $\ell$ of each rational pair
commute.  For a commuting pair in the base of a central extension, the
commutator of lifts is independent of the choices of lifts.  Use the
continuous local sections of Equation~\eqref{eq:colors-circle-extension}
to pass these commutators to the limit.  Their limit is $1$, as required.
\end{proof}

Restriction of Equation~\eqref{eq:colors-circle-extension} to $S_A$
splits abstractly over $S(k)$ by $\ell$.  Equation~\eqref{eq:colors-metaplectic}
therefore gives a continuous homomorphic section
\[
 j:S_A\longrightarrow\widetilde Q.
\]
Our remaining task is to modify $j$ so that this splitting extends from
$S_A$ to all of $Q$.  We first make $j(S_A)$ normal and then eliminate
the commutator pairing on the determinant quotient $A_0$.

\subsection{Making the splitting equivariant}

For $q\in Q$, choose a lift $\widetilde q$ and define the discrepancy
character $d_q:S_A\to I$ by
\begin{equation}\label{eq:colors-discrepancy}
 \widetilde q\,j(s)\,\widetilde q^{-1}
   =j(qsq^{-1})\,d_q(s).
\end{equation}
We again regard values in $I$ as central factors.  This is a continuous
character of $s$, independent of the lift of $q$.  It is zero for
$q\in S_A$, since there a lift differs from $j(q)$ by a central element.
Composition of conjugations shows that the $d_q$ form a cocycle for the
conjugation action.  More explicitly,
\[
 d_{q_1q_2}(s)=d_{q_1}(q_2sq_2^{-1})+d_{q_2}(s).
\]
The cocycle and its action factor through $Q/S_A=A_0$.

By Lemma~\ref{lem:colors-local-abelianization}, its coefficient group is
the finite group
\[
 \Hom_{\mathrm{cont}}(S_A,I)
   =\prod_{v\in A}\Hom(T_v,I).
\]
On the $v$-component the action of $a\in A_0$ is the power of $\sigma_v$
specified by $\ord_v(a_v)$.  The corresponding component of the
discrepancy is zero whenever $\ord_v(a_v)=0$.  Indeed, choose a
representative of $a$ in $Q$ whose $v$-component is an unramified unit
of reduced norm $a_v$.  Such a unit exists by surjectivity of unramified
norms on units.  It commutes with every Teichm\"uller lift of $T_v$,
viewed as an element of $S_A$ supported at $v$.  Lemma~\ref{lem:colors-commuting-lifts}
then makes Equation~\eqref{eq:colors-discrepancy} zero on these lifts,
which determine the character.

It follows that both this component of the cocycle and its action factor
through the single valuation map $A_0\to\bbZ$.  This map is onto:
weak approximation in $U(k)$ allows its $v$-component to be close to
a local element of reduced-norm valuation one.  Let $d_v$ be the
discrepancy for such a generator.  It vanishes on $T_v^{\sigma_v}$.
Indeed, the Teichm\"uller lifts of these fixed residue elements are
central in $\mathcal D_v^\times$, so the same commuting-lifts test
applies for any representative of the generator.

For a finite abelian group $T$ with automorphism $\sigma$, the image of
\[
 \sigma^*-1:\Hom(T,I)\longrightarrow\Hom(T,I)
\]
is exactly the characters annihilating $T^\sigma$.  This follows by
duality from the exact sequence for $\sigma-1$ on $T$, or directly by
extending a character from its image to $I$.  Consequently $d_v$ is a
coboundary.  Choose a correcting character in each component and
replace $j(s)$ by $j(s)\lambda(s)$ for the resulting continuous character
$\lambda:S_A\to I$.  Equation~\eqref{eq:colors-discrepancy} then has
zero discrepancy for every $q$.  Thus we may, and do, assume that
\begin{equation}\label{eq:colors-equivariant-section}
 \widetilde q\,j(s)\,\widetilde q^{-1}=j(qsq^{-1})
 \quad(q\in Q,\ s\in S_A),
\end{equation}
so that $j(S_A)$ is normal in $\widetilde Q$.

\subsection{Removing the determinant commutator pairing}

The quotient by this normal subgroup is a central extension
\begin{equation}\label{eq:colors-determinant-extension}
 1\longrightarrow I\longrightarrow\widetilde Q/j(S_A)
 \longrightarrow A_0\longrightarrow1.
\end{equation}
Since $A_0$ is abelian, commutators in this extension give an alternating
biadditive pairing, written multiplicatively in its arguments,
\[
 \omega:A_0\times A_0\longrightarrow I.
\]
By Lemma~\ref{lem:colors-commuting-lifts}, $\omega(a,b)=0$ whenever,
at every $v\in A$, the two determinants $a_v,b_v$ are norms of elements
of one common commutative subfield of $\mathcal D_v$.

In particular, if every $a_v$ is an $n$th power, represent it by a
central scalar at each place; it commutes with any representative of
$b_v$.  Thus $\omega$ factors through the finite group
\begin{equation}\label{eq:colors-power-quotient}
 K_A=\prod_{v\in A}k_v^\times/(k_v^\times)^n.
\end{equation}
The map $A_0\to K_A$ is onto.  Local reduced norms are surjective, and
weak approximation in $U(k)$ realizes any prescribed classes because
the local power subgroups are open.  Mixed terms between two distinct
factors of Equation~\eqref{eq:colors-power-quotient} vanish: use central
representatives for each locally trivial power class.  We may therefore
treat the pairing one place at a time.

Fix $v$, suppress it from the notation, and let $\pi$ be a uniformizer.
Unit-unit terms vanish because units are norms from the unramified
maximal subfield.  Alternation kills uniformizer-uniformizer terms, so
the local pairing is specified by the character
\[
 u\longmapsto\omega_v(\pi,u),\qquad u\in\mathcal O_v^\times.
\]
This character vanishes on $1+\mathfrak p_v$.  To prove it first take
$c\in k_v$ of valuation one.  The Eisenstein extension defined by
$\theta^n=c$ has degree $n$, is totally ramified, and embeds in
$\mathcal D_v$ by the local invariant criterion.  Since $n$ is odd,
\[
 N(\theta)=c,\qquad N(1+\theta)=1+c.
\]
The common-subfield test gives $\omega_v(c,1+c)=0$; writing $c$ as a
uniformizer times a unit and using unit-unit vanishing gives
$\omega_v(\pi,1+c)=0$.  If $\ord_v(d)\geq2$, choose any such $c$ and
write
\[
 1+d=\frac{(1+c)(1+d)}{1+c}.
\]
Both numerator and denominator have the form $1+c'$ with
$\ord_v(c')=1$.  This proves vanishing on all principal units.
The remaining parameter is therefore a character of
$\bbF_q^\times$ killed by $n$.

We still have freedom to change $j$ by characters of $S_A$ invariant
under $Q$.  At the fixed place these are the $\sigma$-invariant
characters of $T_v$.  Such a change preserves
Equation~\eqref{eq:colors-equivariant-section}.  Its effect on the
pairing is evaluation, up to the fixed sign convention for commutators,
on the residue of the ordinary local commutator of determinant
representatives.  For a prime element $\Pi$ and an unramified unit with
residue $r$, that residue is
\[
 \frac{\sigma(r)}{r}.
\]
The residue norm of $r$ is the residue of the unit determinant $u$.
Replacing $r$ by another norm preimage changes this expression by an
element of $(\sigma-1)T_v$.  Hence we have a well-defined map
\begin{equation}\label{eq:colors-residue-commutator}
 \bbF_q^\times\longrightarrow T_v/(\sigma-1)T_v,
 \qquad N(r)\longmapsto\sigma(r)/r.
\end{equation}

This map is onto.  To see this explicitly, put
\[
 M=\frac{q^n-1}{q-1},\qquad d=\gcd(q-1,n),
\]
and choose a generator $g$ of $\bbF_{q^n}^\times$.  The group $T_v$ is
generated by $g^{q-1}$.  Since $(h,n)=1$,
\[
 |T_v/(\sigma-1)T_v|
 =\gcd(M,q^h-1)=\gcd(M,q-1)=d.
\]
The image of the generator $g^M$ of $\bbF_q^\times$ in
Equation~\eqref{eq:colors-residue-commutator} is represented, in the
generator $g^{q-1}$, by the exponent
\[
 \frac{q^h-1}{q-1}\equiv h\pmod d.
\]
It is a generator because $(h,d)=1$.  Dualizing this surjection shows
that invariant characters of $T_v$ produce exactly all residue unit
characters killed by $n$.  They can therefore cancel the remaining
parameter $u\mapsto\omega_v(\pi,u)$.  A unit adjustment between
$\Nrd(\Pi)$ and the chosen $\pi$ makes no difference, since unit-unit
terms already vanish.

Perform this adjustment at every place of $A$.  The new section still
has normal image, and the commuting-lifts test still holds; in
particular, all the preceding reductions of the pairing remain valid.
Its only possible parameters have now been cancelled, so $\omega=0$.
The extension in Equation~\eqref{eq:colors-determinant-extension} is
therefore abelian.  The circle group is divisible and hence injective
as an abelian group, so this extension splits.  A homomorphism from the
discrete group $A_0$ is automatically continuous.

Let $C_0$ be the inverse image in $\widetilde Q$ of the image of such
a section of Equation~\eqref{eq:colors-determinant-extension}.  Then
$C_0\cap I=1$ and $p(C_0)=Q$: for any lift of an element of $Q$, its
central coordinate in the quotient can be removed using $I$.
Consequently $p:C_0\to Q$ is a group isomorphism.  Locally on each
determinant fiber its inverse is a translate of $j$, so the inverse is
continuous.  We have proved that Equation~\eqref{eq:colors-circle-extension}
has a continuous homomorphic section
\begin{equation}\label{eq:colors-global-section}
 s:Q\longrightarrow\widetilde Q.
\end{equation}

\subsection{Extracting a nonzero finite character}

Compare the continuous section in Equation~\eqref{eq:colors-global-section}
with the specified abstract lift of rational points.  Their difference
is a character
\begin{equation}\label{eq:colors-difference-character}
 \chi_0:U(k)\longrightarrow I,
 \qquad \ell(u)=s(\rho(u))\,\chi_0(u).
\end{equation}
It has finite image on $S(k)$.  Indeed, for $u\in R'$ the canonical
formula in Equation~\eqref{eq:colors-canonical-lift} reads
$\ell(u)=c(u_A)$.  The difference of $c$ and $s|_{P_0}$ is a continuous
character of the profinite group $P_0$.  Its image in the circle is
finite: a quotient of a profinite group is profinite, whereas a closed
subgroup of the circle is either finite or the whole circle.  Thus
$\chi_0(R')$ is finite.  Since $R'$ has finite index in $S(k)$, the image
$\chi_0(S(k))$ is a finite union of cosets of that finite subgroup and
is finite.

It is also nonzero.  Choose $\beta\ne1$ in $B$ and $u\in V_0$
representing it.  Because $R$ is dense in $P_0$ and $u_A\in P_0$,
there is a sequence $r_j\in R$ such that
\[
 (ur_j)_A\longrightarrow1.
\]
All $ur_j$ still represent $\beta$ modulo $R'$.  If $\chi_0$ vanished
on $S(k)$, Equations~\eqref{eq:colors-canonical-lift} and
\eqref{eq:colors-difference-character} would give
\[
 c((ur_j)_A)\,\iota(\beta)=s(\rho(ur_j)).
\]
Both continuous sections tend to the identity, which would imply
$\iota(\beta)=0$, contrary to injectivity of $\iota$.

Finally we make the character finite on all of $U(k)$ without changing
its restriction to $S(k)$.  The multiplicative group of a number field
is free abelian modulo its finite roots of unity: use the ideal map and
the finite generation of the unit group.  Therefore its subgroup $A_0$
has the form
\[
 A_0=T\oplus\bigoplus_{j\in J}\bbZ a_j,
\]
where $T$ is finite and $J$ may be infinite.  Choose $u_j\in U(k)$
with $\det u_j=a_j$, and choose a character $\psi:A_0\to I$ satisfying
$\psi(a_j)=\chi_0(u_j)$ and $\psi|_T=0$.  Then
\[
 \chi=\chi_0-\psi\circ\det
\]
vanishes on all the chosen $u_j$ and equals $\chi_0$ on $S(k)$.
If $m$ kills the finite group $\chi_0(S(k))$ and $t$ kills $T$, then
$mt$ kills $\chi(U(k))$: remove the free determinant part using the
$u_j$, and the $t$th power of the remaining element lies in $S(k)$.
The subgroup of the circle killed by $mt$ is finite.  Thus $\chi$ is a
finite character and remains nonzero on $S(k)$.

When $A=\varnothing$, the same construction has $Q=A_0$ and trivial
$S_A$.  Commuting approximation makes the circle extension abelian
immediately, and all subsequent arguments apply unchanged.  We have
proved alternative~\textup{(i)} in the remaining case, completing the
proof of Proposition~\ref{prop:color-dichotomy}.

The finite defect can therefore be nontrivial only if the full unitary
group admits one of the two homomorphisms in that proposition.  The
character argument in Section~\ref{sec:characters} and the color argument
in Sections~\ref{sec:palettes}--\ref{sec:finite-groups} exclude these
alternatives, retaining the division and odd-degree hypotheses fixed here.
==== END The-Margulis-Platonov-conjecture-over-number-fields//sections/colors.tex ====

==== BEGIN The-Margulis-Platonov-conjecture-over-number-fields//sections/exceptional-types.tex ====
\section{Groups of type \texorpdfstring{$D_4$}{D4}}\label{sec:d4}

For type $D_4$ there are no anisotropic nonarchimedean factors.  Our
objective is therefore to show that every finite quotient of the rational
group is trivial.  The argument first kills rational noncentral
involutions and then represents every power-subgroup coset by a product of
two such involutions.  The second step requires a maximal torus on which
inversion lifts rationally, not just geometrically.

\begin{theorem}\label{thm:d4}
Let $k$ be a number field and let $G$ be an absolutely almost simple simply
connected $k$-group of type $D_4$, including the trialitarian forms.  Every
noncentral abstract normal subgroup of $G(k)$ is $G(k)$.
\end{theorem}

The isotropic case is among the established cases recalled in
Section~\ref{sec:foundations}.  We may thus assume throughout this section
that $G$ is $k$-anisotropic.  Fix a positive integer $e$, put
$R=G(k)^e$, and write $q:G(k)\to\mathcal Q=G(k)/R$.  By
Proposition~\ref{prop:finite-defect}, this is a finite quotient and its
cosets have the approximation properties used below.  It will suffice to
prove that $\mathcal Q=1$.

\subsection{Root involutions and a quadratic splitting field}

We record explicitly the lattice calculations used both here and in
Section~\ref{sec:e6}.  With roots of squared length two, the coroot lattice
of the simply connected group is
\[
 Q_4=\{(x_1,x_2,x_3,x_4)\in\bbZ^4:x_1+x_2+x_3+x_4\equiv0\pmod2\},
 \qquad \Phi_4=\{\pm e_i\pm e_j:i\ne j\}.
\]
The two-torsion of a split maximal torus is $Q_4/2Q_4$.  Its four central
elements are the classes having all coordinates of the same parity.  The
other twelve classes are represented by the twelve pairs of opposite
roots.  These representatives are distinct: if two roots differ by an
element of $2Q_4$, their coordinate parities agree, and the defining
even-sum condition then leaves only equality or opposition.  The Weyl
group is transitive on the roots, so all noncentral involutions form one
geometric conjugacy class.

For $n=\alpha^\vee(-1)$, the roots of $C_G(n)$ are precisely the four
orthogonal axes represented, after a Weyl conjugacy, by
\[
 \alpha=e_1-e_2,\qquad \beta_1=e_1+e_2,
 \qquad\beta_2=e_3-e_4,\qquad\beta_3=e_3+e_4.
\]
The semisimple centralizer is connected, by the connected-centralizer
theorem for simply connected groups
\cite[Section~2.10, p.~53]{SteinbergRegular1965}. The lattice
spanned by these four roots has index two in $Q_4$, and
\[
 \alpha+\beta_1+\beta_2+\beta_3=2(e_1+e_3)\in2Q_4.
\]
Consequently the centralizer and its simply connected covering have the
geometric description
\begin{equation}\label{eq:d4-centralizer}
 C_G(n)_{\bar k}\simeq
 (\SL_2^4)/\langle(-I,-I,-I,-I)\rangle.
\end{equation}
The product of the four standard reflection representatives acts by $-1$
on the torus and has square one in this quotient.  Every other
representative of inversion has the same square: for $a$ in the torus
and an inverter $x$, one has $(ax)^2=a\,xax^{-1}x^2=x^2$.

\begin{lemma}\label{lem:d4-kill-involutions}
Every noncentral involution in $G(k)$ is killed by $q$.
\end{lemma}

\begin{proof}
Let $n\in G(k)$ be such an involution.  The root pair representing $n$ is
unique, so its axis $\alpha$ is Galois-stable.  The simply connected cover
of $C_G(n)$ is therefore a product of an inner form of $\SL_2$ over $k$
and restrictions of inner forms of $\SL_2$ over a cubic etale $k$-algebra
$E$.  Equivalently these factors are the norm-one groups of quaternion
algebras over $k$ and $E$.

There is a quadratic field $L'/k$ whose scalar extension embeds as a
maximal etale quadratic algebra in every one of these quaternion
algebras.  To see this, impose local conditions only at their ramified
places.  At a finite ramified place we require $L'_v$ to be a field and
to remain a field over each completion of $E$ where the corresponding
quaternion algebra is ramified.  An odd-degree completion cannot contain
$L'_v$.  Since $\dim_k E=3$, at most one completion of $E$ has degree two;
thus at most one quadratic local field is excluded.  There are other
quadratic extensions of a nonarchimedean local field.  At real ramified
places choose $L'_v=\bbC$.  Weak approximation on a defining square class,
with a further nonsquare condition if necessary, realizes all these
requirements.  The local invariant theorem for quaternion algebras then
shows that $L'$ splits each algebra after scalar extension, which is
equivalent to the asserted quadratic embedding
\cite{PRbook,Reiner,KMRT}.

The resulting maximal norm tori give a maximal torus of $G$.  Orient the
four root axes using their common quadratic field $L'$.  Galois then acts
by a common sign, together with permutations of $\beta_1,\beta_2,\beta_3$
fixing $\alpha$.  In particular the two roots
\[
 \alpha,\qquad
 \gamma=\frac{-\alpha+\beta_1+\beta_2+\beta_3}{2}
\]
span a Galois-stable subsystem of type $A_2$: both have squared length
two and $(\alpha,\gamma)=-1$.  The corresponding connected root subgroup
$J$ is defined over $k$ and is split over $L'$, because both roots and
coroots of this subsystem are fixed over $L'$.  The element $n$ is the
image of the rational element $\alpha^\vee(-1)$ in the simply connected
cover $\widetilde J$.

The group $\widetilde J$ has no anisotropic finite places.  In the inner
case its degree-three central simple algebra is split by a quadratic
extension, so its index divides both three and two.  In the outer case
the algebra over the quadratic center is split; a ternary Hermitian form
over a quadratic extension of a nonarchimedean local field is isotropic.
The established inner-type-$A$ result and Theorem~\ref{thm:outer-a} now
give (MP) for $\widetilde J$.  Lemma~\ref{lem:restriction-mp}, with an empty
anisotropic-place product, makes its image under $q$ trivial.  In
particular $q(n)=1$.
\end{proof}

\subsection{Local representatives of inversion}

The next two lemmas give the local conditions imposed on a representative
of an arbitrary coset of $R$.  At a prescribed place we can choose a
suitable open set freely.  At the other places we will use a uniform
condition on integral reduction.

\begin{lemma}\label{lem:d4-local-openings}
For every completion $F$ of $k$ there is a nonempty open set of regular
semisimple elements $u\in G(F)$ such that inversion on $C_G(u)$ has a
representative in $G(F)$.
\end{lemma}

\begin{proof}
We first produce a noncentral involution in $G(k)$, which can then be
localized at any $F$.  The unique geometric class of noncentral
involutions is a closed semisimple orbit preserved by Galois, so it
descends to a $k$-homogeneous space $X$ under $G$.  Its full geometric
stabilizer is the connected semisimple centralizer in
Equation~\eqref{eq:d4-centralizer}.  Borovoi's theorem for a homogeneous
space under a simply connected group with connected geometric
stabilizer having no torus quotient requires only archimedean local
points to conclude $X(k)\ne\varnothing$
\cite[Proposition~3.4(i)]{Borovoi}.

These points exist over $\bbC$.  Over $\bbR$, the compact rank of every
real form of $D_4$ is at least three, by the real orthogonal
classification, including the quaternionic orthogonal form
\cite{KMRT,PRbook}.  A compact torus of dimension at least three has at
least eight rational two-torsion points, whereas the center has order
four.  It therefore contains a noncentral involution.  Borovoi's theorem
now supplies $n\in G(k)$.  This argument applies to all forms, including
trialitarian ones; it imposes no finite-place restriction.

View $n$ in $G(F)$.  Geometrically it is conjugate
to an inversion representative by Equation~\eqref{eq:d4-centralizer} and
the preceding lattice calculation.  Hence the map
$g\mapsto gng^{-1}n$ takes some geometric values that are regular
semisimple: on a torus inverted by $n$, its values include all squares.
The regular locus is a nonempty Zariski open condition on $g$, and
$G(F)$ is Zariski dense.  We may therefore choose such a $g$ rationally.
Set $u=gng^{-1}n$ and $T=C_G(u)$.  The involution $n$ normalizes $T$ and
sends $u$ to $u^{-1}$.  Its Weyl action is inversion.  Indeed inversion
belongs to the $D_4$ Weyl group; composing the two Weyl actions would fix
$u$, whose Weyl stabilizer is trivial by connectedness of its semisimple
centralizer and regularity.  Finally the conjugation map
$G(F)\times T(F)\to G(F)$ is a submersion at $(1,u)$, since no root takes
the value one on $u$.  Thus nearby regular elements have conjugate tori,
and the required property holds on an open neighborhood of $u$.
\end{proof}

\begin{lemma}\label{lem:d4-one-wall}
Outside a fixed finite set of finite places, the following assertion
holds.  Let $u\in G(k_v)$ be regular semisimple and integral.  If the
semisimple part of its reduction lies on at most one pair of opposite
root walls, then inversion on $C_G(u)$ has a rational representative.
\end{lemma}

\begin{proof}
Write $p$ for the residue characteristic and exclude $p=2$ and the fixed
bad-model places.  The compact closure of the cyclic group generated by
$u$ has a prime-to-$p$ finite-order factor.  Its corresponding element
$s$ reduces to the semisimple part of the reduction of $u$.  This follows
also by decomposing the finite-order part and the pro-$p$ part of this
procyclic group, whose reduction kernel is pro-$p$.  In particular
$T=C_G(u)$ is contained in $C_G(s)$.

The centralizer $C_G(s)$ has an unramified reductive model with the roots
specified by the reduction of $s$.  One way to check this assertion is to
pass to a finite unramified extension splitting the reduced data.  A
prime-to-$p$ torsion lift is integrally conjugate to the corresponding
element of a split torus: at each successive congruence level the
conjugacy obstruction is cohomology of a finite cyclic group of order
prime to $p$ on a residue Lie algebra, and averaging makes this
cohomology vanish.  The resulting centralizer has precisely the root
groups on which $s$ has value one.  Descent gives the asserted unramified
model.  Equivalently one may use smoothness of fixed points of an
automorphism of invertible finite order together with the
connected-centralizer theorem.

If there are no root walls, this centralizer is the unramified torus
$T$.  Inversion is a rational point of its Weyl-group quotient.  It has a
representative over the residue field by Lang's theorem for the torus,
and that representative lifts through the smooth unramified normalizer.

If there is one root pair $\{\pm\alpha\}$, its element
$n_0=\alpha^\vee(-1)$ is Galois-invariant, even if Galois exchanges
$\alpha$ and $-\alpha$.  It is an integral noncentral involution central
in $C_G(s)$.  Thus $T\subset C_G(n_0)$.  The latter centralizer is again
unramified, and its simply connected covering is a product of
restrictions of split $\SL_2$'s over unramified extensions: a semisimple
unramified group is quasi-split, and a quasi-split group of type $A_1$ is
split.  The covering torus above $T$ is a product of maximal tori of
these $\SL_2$'s.

For each such torus, represent it as the norm-one group of a quadratic
etale algebra.  Conjugation of that algebra, acting on its underlying
two-dimensional vector space, is an inverter in $\GL_2$ of determinant
$-1$.  Multiply all these inverters, over all factors and embeddings,
by the same scalar $i$ with $i^2=-1$.  The resulting tuple lies in
$\SL_2^4$ and inverts the covering torus.  A Galois automorphism either
preserves $i$ or changes its sign simultaneously in all four factors.
The latter discrepancy is exactly the kernel in
Equation~\eqref{eq:d4-centralizer}.  The image of the tuple therefore
descends to a $k_v$-rational element of $C_G(n_0)$ inverting $T$.  Notice
that this argument permits the individual quadratic tori to be ramified.
\end{proof}

\subsection{A torus with no local-to-global obstruction}

Write $P_4=\Hom(Q_4,\bbZ)$ for the weight lattice and
$W=W(D_4)$ for its Weyl group.  The following calculation explains why
forcing the full Weyl group into a torus's Galois action is sufficient.

\begin{lemma}\label{lem:d4-cyclic-cohomology}
Let $\Gamma$ be a subgroup of $\Aut(\Phi_4)$ containing $W$, acting on
$P_4$.  Every class in $H^2(\Gamma,P_4)$ whose restriction to every cyclic
subgroup vanishes is zero.  Consequently, if a maximal $k$-torus $T$ of
a simply connected group of type $D_4$ has character action containing
$W$, then $\Sha^1(k,T)=0$.
\end{lemma}

\begin{proof}
Let $M=P_4$ and let $\xi\in H^2(\Gamma,M)$ restrict to zero on every
cyclic subgroup.  The element $c=-I\in W$ is central in $\Gamma$ and
acts by $-1$ on $M$.  Apply Lemma~\ref{lem:foundations-sign-extension}
to an extension representing $\xi$.  It gives a class
\[
 \beta\in H^1(\Gamma/\langle c\rangle,M/2M)
\]
whose vanishing is equivalent to the splitting of the extension.  We
view a representative of this class as a cocycle on $\Gamma$ and show
that it is a coboundary.

For a simple reflection $s_\alpha$, cyclic triviality and the last
assertion of Lemma~\ref{lem:foundations-sign-extension} give a lift
$d_{s_\alpha}\in M$ of $\beta(s_\alpha)$ with
$(1+s_\alpha)d_{s_\alpha}=0$.  Therefore
\[
 d_{s_\alpha}\in P_4\cap\mathbb Q\alpha=\mathbb Z\alpha.
\]
The last equality follows by pairing with a neighboring simple coroot.
The fundamental weights allow us to prescribe all the coefficients
$\langle b,\alpha^\vee\rangle$ modulo two independently.  Adding the
coboundary $(1-\gamma)b$ for a single $b\in P_4$ therefore makes
$\beta$ vanish on every simple reflection, and hence on $W$.

Moreover,
\begin{equation}\label{eq:d4-no-mod-two-invariants}
 (P_4/2P_4)^W=0.
\end{equation}
Indeed a root is primitive in $P_4$; for example its pairing with an
adjacent simple coroot is $-1$.  If $p$ is fixed modulo $2P_4$ by every
simple reflection, the formula
$p-s_\alpha p=\langle p,\alpha^\vee\rangle\alpha$ therefore makes
every simple-coroot pairing even.  Expansion in fundamental weights
gives $p\in2P_4$.  Since $W$ is normal in $\Gamma$, the cocycle identity
and $\beta|_W=0$ imply that $\beta(\gamma)$ is $W$-invariant for every
$\gamma$.  Equation~\eqref{eq:d4-no-mod-two-invariants} gives
$\beta=0$ on all of $\Gamma$.  The extension consequently splits by
Lemma~\ref{lem:foundations-sign-extension}, proving the first assertion.

For the arithmetic conclusion, take the minimal Galois splitting field
of $T$ and identify $X^*(T)$ with $P_4$.  The first assertion and
Lemma~\ref{lem:foundations-cyclic-detection} give
$\Sha^2_\omega(k,X^*(T))=0$.  The torus-duality conclusion of that
lemma then gives $\Sha^1(k,T)=0$, as required.
\end{proof}

We now construct the torus to which this lemma will apply.  Fix a
splitting field and a pinning of $G$ there.  At finitely many sufficiently
large finite places that split completely in this field, prescribe
unramified maximal tori with Frobenius in each conjugacy class of $W$.
Such tori exist by Weyl twisting over the residue field and smooth
lifting.  Each has rational inversion by the no-wall argument in
Lemma~\ref{lem:d4-one-wall}.  Choose regular elements in them and impose
small regular neighborhoods with the same local tori.

The labeling here matters.  At each chosen place fix a completion
embedding of the splitting field and use the fixed pinning.  After a
global identification of a new maximal torus with the pinned torus by
inner conjugation, these local identifications differ by Weyl elements,
not arbitrary diagram automorphisms.  The image of the Galois group
fixing the chosen splitting field thus meets every $W$-conjugacy class
inside $W$.  This image is all of $W$.  Indeed a proper subgroup of a
finite group cannot meet every conjugacy class: its transitive action on
cosets would have no element without a fixed point, contradicting the
average fixed-point count of one.  The full character action $\Gamma$
therefore contains $W$.

At each place, define the allowed set as the union of the elements that
are not regular semisimple and the regular semisimple elements
whose centralizer torus admits rational inversion. The auxiliary
regular neighborhoods just chosen force the resulting rational element
to be regular semisimple, hence regular semisimple at every completion.
For this element, membership in the allowed sets therefore gives
local rational inversion everywhere.

Apply Proposition~\ref{prop:power-approximation} to any prescribed coset
of $R$, with these auxiliary conditions and the openings from
Lemma~\ref{lem:d4-local-openings} at the other fixed exceptional places.
We spell out the two uniform conditions needed for that proposition.

At integral good places, exclude the closed set where the semisimple
part lies on two independent root walls.  In a maximal torus these
intersections have codimension at least two, as does their image under
the finite quotient by $W$.  The conjugacy-quotient map has fibers of
dimension $\dim G-\rank G$, so its inverse image still has codimension
at least two.  To see the fiber dimension directly, a fiber is a union
of conjugates of $su$, where $s$ is fixed semisimple and $u$ ranges over
the unipotent variety of $C_G(s)$; the latter has dimension
$\dim C_G(s)-\rank G$.  These descriptions and codimension bounds spread
to the integral models after excluding finitely many places
\cite[Theorem~6.11, p.~64]{SteinbergRegular1965}.
At every remaining integral place, Lemma~\ref{lem:d4-one-wall} applies
if the element is regular semisimple; otherwise it is in the allowed
set by definition.

At a single pole, the place splits in the fixed splitting field used in
Proposition~\ref{prop:power-approximation}.  Use a split eight-dimensional
vector representation.  Choose the cocharacter to take distinct values
on its eight weights, also after every relevant diagram translate.  This
requires avoidance of only finitely many hyperplanes in the cocharacter
lattice.  Order the exponents as $d_1<\cdots<d_8$.  In the single-pole
normal form $g_L\lambda(t)^e g_R$, require all initial principal minors
of $g_Rg_L$ in this order to be nonzero upon reduction.  These conditions
are nonempty open conditions: they hold at the identity.  For each pole
factor, at least one of the two disjoint complete basic parameter lists
lies entirely before or after that factor.  After cyclically combining
the side factors as $g_Rg_L$, this list occurs between fixed invertible
translates.  Its product map is dominant, so it meets the simultaneous
principal-minor open set.  This argument includes poles in the first or
last factor; no boundary factor needs parameters on both sides.  The
$j$th characteristic coefficient then has a unique
least-valuation term, of valuation
$e(d_1+\cdots+d_j)\ord_v(t)$.  The Newton polygon has eight segments of
horizontal length one with distinct integral slopes.  Hensel lifting
therefore splits the characteristic polynomial into distinct linear
factors over $k_v$.  The centralizer torus is split: the vector
representation has finite kernel, so its weights span the character
space over $\bbQ$, on which Galois now acts trivially.  Split tori have
rational inversion representatives by the lattice calculation above.
The finitely many extra model or denominator exceptions are treated by
the local openings, as provided by
Proposition~\ref{prop:power-approximation}.

We have proved that every coset of $R$ has a regular semisimple
representative $u$ for which $T=C_G(u)$ has everywhere locally rational
inversion and character action containing $W$.

\begin{proof}[Proof of Theorem~\ref{thm:d4}]
For the torus just constructed, the fiber over inversion in
$N_G(T)\to N_G(T)/T$ is a $T$-torsor.  It has a point over every
completion, and Lemma~\ref{lem:d4-cyclic-cohomology} makes its class zero.
Choose a rational representative $x$ of inversion.  The common-square
calculation following Equation~\eqref{eq:d4-centralizer} gives
$x^2=(ux)^2=1$.  Neither involution is central, because each induces
nontrivial inversion on the positive-dimensional torus $T$.  Thus
$u=(ux)x$ is killed by Lemma~\ref{lem:d4-kill-involutions}.  Every coset
of $R$ is trivial, so $G(k)/G(k)^e=1$.  By
Proposition~\ref{prop:finite-defect}, an arbitrary noncentral abstract
normal subgroup has finite index and contains such a power subgroup.
It is consequently all of $G(k)$.
\end{proof}

\section{Groups of type \texorpdfstring{$E_6$}{E6}}\label{sec:e6}

The last case reduces a generic rational element to a product from a
subgroup of type $D_4$ and the centralizer of a root involution.  That
centralizer has types $A_1$ and $A_5$.  The geometric product
decomposition alone is insufficient: we will identify its rational
descent obstruction as a torsor under a simply connected semisimple
group, and arrange the required local points.

\begin{theorem}\label{thm:e6}
Let $k$ be a number field and let $G$ be an absolutely almost simple simply
connected $k$-group of type $E_6$.  Every noncentral abstract normal
subgroup of $G(k)$ is $G(k)$.
\end{theorem}

Again $A=\varnothing$, and only the $k$-anisotropic case needs proof.
Fix $e\ge1$, set $R=G(k)^e$, and write
$q:G(k)\to\mathcal Q=G(k)/R$.  Our goal is to kill each coset of this
finite quotient.

\subsection{A rational root involution}

\begin{lemma}\label{lem:e6-root-involution}
The group $G(k)$ contains an involution geometrically conjugate to
$\alpha^\vee(-1)$ for a root $\alpha$.  Its centralizer $H$ is connected
semisimple of type $A_1A_5$, and its simply connected covering has
geometric form
\begin{equation}\label{eq:e6-centralizer-cover}
 \pi:\SL_2\times\SL_6\longrightarrow H_{\bar k},
 \qquad \ker\pi=\langle(-I_2,-I_6)\rangle.
\end{equation}
\end{lemma}

\begin{proof}
First work geometrically.  The roots centralizing $\alpha^\vee(-1)$ are
$\pm\alpha$ and the roots perpendicular to $\alpha$.  The latter form
$A_5$.  For example, the identity
$\sum_{\beta\in\Phi(E_6)}(\alpha,\beta)^2=4h^\vee=48$, with
$h^\vee=12$, shows that forty of the seventy roots other than
$\pm\alpha$ have pairing $\pm1$, leaving thirty perpendicular roots.
The resulting rank-five subsystem is $A_5$.  The determinant comparison
$\det(A_1)\det(A_5)/\det(E_6)=2\cdot6/3=4$ gives index two for its
coroot lattice together with $\alpha$ in the $E_6$ coroot lattice.
Each individual summand is saturated: the square of any nontrivial
saturation index would divide two or six, respectively.  The kernel is
therefore the diagonal subgroup of order two, proving
Equation~\eqref{eq:e6-centralizer-cover}.  Connectedness again follows
from the semisimple-centralizer theorem.

The geometric root-involution class is a closed semisimple orbit
preserved by Galois, so it descends to a $k$-homogeneous space $X$ under
$G$.  Its full geometric stabilizer is the connected semisimple group
just described.  The ambient group is simply connected, and the
stabilizer has no torus quotient.  Borovoi's theorem therefore gives a
$k$-point of $X$ as soon as $X$ has points at every archimedean place
\cite[Proposition~3.4(i)]{Borovoi}.  No finite-place existence assertion
is needed.  We verify the real places explicitly.

Over $\bbR$, take a real maximal torus and identify its cocharacter
lattice with $Q(E_6)$.  Complex conjugation is an involution on the
six-dimensional vector space $Q(E_6)/2Q(E_6)$, so its fixed space has
dimension at least three.  The quadratic form
\[
 \overline x\longmapsto (x,x)/2\pmod2
\]
on this space is nondegenerate, since $\det(E_6)=3$ is odd.  The
thirty-six root pairs give thirty-six distinct vectors of value one.
Indeed if $\alpha-\beta=2x$, then
$(x,x)=1-(\alpha,\beta)/2$; evenness of the root lattice permits this
only for $\beta=\pm\alpha$.  A nondegenerate quadratic form in dimension
six over $\bbF_2$ has either twenty-eight or thirty-six vectors of value
one.  Ours consequently has minus type and Witt index two, with the root
pairs exhausting its value-one vectors.  A three-dimensional fixed
subspace cannot be totally singular.  Its value-one vector is a real
two-torsion point in the required root-involution class.

Existence over $\bbC$ is immediate.  All the archimedean hypotheses of
Borovoi's theorem are now satisfied, giving a rational root involution
$n\in G(k)$ and completing the proof.
\end{proof}

Fix $n$ as in the lemma, put $H=C_G(n)$, and denote its simply connected
cover over $k$ by $\pi:\widetilde H\to H$.  Its almost simple factors
have types $A_1$ and $A_5$, for which (MP) is now known.

\subsection{The generic \texorpdfstring{$D_4$}{D4} subgroup}

\begin{lemma}\label{lem:e6-generic-d4}
There is a nonempty $k$-open subset $\mathcal U\subset G$ with the
following property.  For $u\in\mathcal U(k)$, set
$n'=unu^{-1}$ and $t=n'n$.  Then $t$ is regular semisimple, with maximal
torus $T=C_G(t)$, and the roots of $T$ that conjugation by $n$ sends to
their negatives generate a simply connected $k$-subgroup $D_*$ of type
$D_4$.  The group $D_*$ contains $n$ and $n'$ as geometrically conjugate
elements, and contains $t$ in its maximal torus $T_*=D_*\cap T$, on
which $n$ acts by inversion.
\end{lemma}

\begin{proof}
Work first over $\bar k$.  Number the $E_6$ diagram by the chain
$1,3,4,5,6$, with node $2$ attached to node $4$, and let $\delta$ be its
diagram involution, as in Figure~\ref{fig:e6-diagram}.  The four roots
\begin{figure}[tbp]
\centering
\begin{tikzpicture}[x=10mm,y=9mm,every label/.style={fill=white,inner sep=1pt}]
 \draw[gray,densely dashed] (2,-0.65)--(2,1.65);
 \draw (0,0)--(4,0) (2,0)--(2,1);
 \node[circle,draw,fill=white,inner sep=2pt,label=below:$1$] at (0,0) {};
 \node[circle,draw,fill=white,inner sep=2pt,label=below:$3$] at (1,0) {};
 \node[circle,draw,fill=white,inner sep=2pt,label=below:$4$] at (2,0) {};
 \node[circle,draw,fill=white,inner sep=2pt,label=below:$5$] at (3,0) {};
 \node[circle,draw,fill=white,inner sep=2pt,label=below:$6$] at (4,0) {};
 \node[circle,draw,fill=white,inner sep=2pt,label=above:$2$] at (2,1) {};
\end{tikzpicture}
\caption{The $E_6$ numbering used in
Equation~\eqref{eq:e6-fixed-roots}.  Reflection in the dashed axis is
$\delta$: it exchanges $1$ with $6$ and $3$ with $5$, and fixes $2,4$.}
\label{fig:e6-diagram}
\end{figure}
\begin{equation}\label{eq:e6-fixed-roots}
 \alpha_2,\quad \alpha_4,\quad
 \alpha_3+\alpha_4+\alpha_5,\quad
 \alpha_1+\alpha_3+\alpha_4+\alpha_5+\alpha_6
\end{equation}
have the $D_4$ Gram matrix: the first has pairing $-1$ with each of the
other three, which are mutually perpendicular.  They span the full
$\delta$-invariant sublattice of $Q(E_6)$.  Indeed successive differences
give $\alpha_3+\alpha_5$ and $\alpha_1+\alpha_6$, along with
$\alpha_2,\alpha_4$.  Their lattice is thus saturated.  Its
squared-length-two vectors are exactly its $D_4$ roots, so the
$\delta$-fixed roots form this subsystem.  The associated subgroup $D$
is simply connected, since its coroot lattice is saturated in that of
$G$.

Let $T_0$ be the ambient maximal torus and $S$ the rank-four maximal
torus of $D$.  There is no root perpendicular to $S$.  Such a root
would belong to the anti-invariant space of $\delta$ and satisfy
$\delta\alpha=-\alpha$, impossible because a diagram automorphism
preserves positive roots.  Thus a nonempty open subset $S_{\mathrm{reg}}$
consists of elements regular in $G$, with centralizer $T_0$.

Inversion in $D$ acts on the full root space as $-1$ on the
$\delta$-invariant space and as $1$ on its perpendicular complement.
It therefore represents the longest Weyl element $w_0=-\delta$ of
$E_6$.  By the $D_4$ calculation it has a representative of order two,
which is conjugate inside $D$ to a root involution.  We may conjugate our
chosen $n$ geometrically so that it is this representative.  For every
$s\in S$, the element $sn$ is conjugate to $n$ inside $D$: choose
$r\in S$ with $r^2=s$ and use $rnr^{-1}=sn$.

Consider the morphism into the $G$-conjugacy class of $n$ given by
\[
 H\times S_{\mathrm{reg}}\longrightarrow \operatorname{Cl}_G(n),
 \qquad (h,s)\longmapsto h(sn)h^{-1}.
\]
Its fibers have dimension two.  In fact equality with the value at
$(h_0,s_0)$ implies, on multiplying by $n$, that
$a=h_0^{-1}h\in H$ satisfies $asa^{-1}=s_0$.  Regularity forces
$a\in H\cap N_G(T_0)$, and $a$ determines $s$.  Conversely each such
$a$ preserves $S=\im(1-n:T_0\to T_0)$, since it commutes with $n$;
this holds for every component of $H\cap N_G(T_0)$.  It also preserves
$S_{\mathrm{reg}}$, so it gives the corresponding input in the same
fiber.  The identity
component of $H\cap N_G(T_0)$ is the fixed subtorus of $w_0=-\delta$,
of dimension two.  As $\dim H=3+35=38$, the image has dimension
\[
 38+4-2=40=78-38=\dim\operatorname{Cl}_G(n).
\]
It therefore contains an open subset of the class.

For every point in this image, the asserted subgroup is $hDh^{-1}$ and
its torus is $hSh^{-1}$: multiplication by $n$ gives the regular element
$hsh^{-1}$.  Intrinsically, these are recovered from the roots on which
$n$ acts as minus one.  This intrinsic description is Galois-invariant.
The resulting property holds on a Galois-stable dense constructible
subset, which contains a nonempty $k$-open subset.  Pulling it back under
$u\mapsto unu^{-1}$ gives $\mathcal U$.  The same intrinsic description
then defines $D_*$ and $T_*$ over $k$, with all the claimed properties.
\end{proof}

\subsection{The full pullback and the factorization torsor}

We next justify the rational descent of the preceding geometric
factorization.  It is important here to identify the entire pullback
subgroup, not just its identity component.

\begin{lemma}\label{lem:e6-pullback}
For $u\in\mathcal U(k)$, let $D_*$ be as in
Lemma~\ref{lem:e6-generic-d4}, and put
\[
 J=C_{D_*}(n)=D_*\cap H,
 \qquad \widetilde J=\pi^{-1}(J)\subset\widetilde H.
\]
Then $\widetilde J$ is connected semisimple simply connected.  The
variety
\begin{equation}\label{eq:e6-factorization-torsor}
 Y_u=\{(w,\widetilde q)\in D_*\times\widetilde H:
                    w\pi(\widetilde q)=u\}
\end{equation}
is a torsor under $\widetilde J$ and has points over every
nonarchimedean completion of $k$.
\end{lemma}

\begin{proof}
Over $\bar k$, conjugate $n$ inside $D_*$ to the root representative.
The four rank-one factors covering $J$ then map to
$\widetilde H_{\bar k}=\SL_2\times\SL_6$ by
\begin{equation}\label{eq:e6-four-block-map}
 (a,b,c,d)\longmapsto\bigl(a,\diag(b,c,d)\bigr).
\end{equation}
The distinguished root factor maps to the $A_1$ factor, and the three
perpendicular factors map to three disjoint two-dimensional blocks of
$A_5$.  This is an injective homomorphism.  The simultaneous minus one
maps to $(-I_2,-I_6)$, the full kernel of $\pi$, and the resulting
quotient is exactly $J$, by Equation~\eqref{eq:d4-centralizer}.  Hence
the image of Equation~\eqref{eq:e6-four-block-map} contains the kernel
of $\pi$ and surjects onto $J$.  It is therefore the full inverse image
$\pi^{-1}(J)$, not merely its connected component.  This identifies that
inverse image geometrically with $\SL_2^4$; descent proves that
$\widetilde J$ is connected semisimple simply connected over $k$.

The variety $Y_u$ is geometrically nonempty, since $n'$ is conjugate to
$n$ inside $D_*$.  Indeed if $wnw^{-1}=n'$, then $w^{-1}u\in H$, and
this element lifts through $\pi$ over $\bar k$.  The right action
\[
 (w,\widetilde q)\cdot y
       =\bigl(w\pi(y),y^{-1}\widetilde q\bigr),
       \qquad y\in\widetilde J,
\]
is simply transitive on geometric points.  For two factorizations, the
ratio of their $D_*$ factors lies in $D_*\cap H=J$, and the ratio of
their covering factors is its unique specified lift in
$\widetilde J$.  Thus Equation~\eqref{eq:e6-factorization-torsor} is a
$\widetilde J$-torsor.  Local $H^1$-vanishing for simply connected
semisimple groups gives its points at every nonarchimedean place
\cite{PRbook}.
\end{proof}

To use this torsor, we must choose local factorizations that are small
in the $\widetilde H$ factor.  The next elementary observation ensures
that the square root used for this purpose remains in the varying torus
$T_*$.

Fix a faithful representation of $G$.  At any completion, sufficiently
near the identity, the matrix analytic square root $t\mapsto t^{1/2}$
is defined and commutes with conjugation.  For all the tori $T_*$ in
Lemma~\ref{lem:e6-generic-d4}, one can choose this neighborhood uniformly
so that
\begin{equation}\label{eq:e6-uniform-root}
                  t\in T_*\quad\Longrightarrow\quad t^{1/2}\in T_*.
\end{equation}
Indeed these tori form one geometric conjugacy class of subtori, and
$t$ is regular semisimple, so its image in the fixed representation is
diagonalizable.  In such a diagonalization their character relations
are a fixed finitely generated lattice of monomial relations, up to a
permutation of the eigenvalues.  The square root is defined by a
convergent matrix power series and commutes with conjugation.  Its
eigenvalues are uniformly near one when the original matrix is near
one: a bound on the matrix norm bounds the absolute values of all its
eigenvalues, without any bound on a diagonalizing matrix.  Every
defining monomial evaluated on the root eigenvalues has square one,
because $t\in T_*$.  After shrinking a single neighborhood using the
finite list of relations, each such monomial must be one, proving
Equation~\eqref{eq:e6-uniform-root}.  This argument applies also at the
places above two, by choosing a smaller analytic neighborhood.

\begin{proof}[Proof of Theorem~\ref{thm:e6}]
The restriction of $q\circ\pi$ to $\widetilde H(k)$ is controlled by
(MP) for its $A_1$ and $A_5$ factors, using Theorem~\ref{thm:outer-a} in
the outer case.  By Lemma~\ref{lem:restriction-mp}, it extends to a
continuous homomorphism on their finite products of anisotropic finite
local groups.  Choose a finite set $B$ containing these places and all
archimedean places.  At each finite place in $B$, a sufficiently small
identity neighborhood in $\widetilde H(k_v)$ is killed by this local
homomorphism.

Take any coset of $R$.  The approximation conclusion of
Proposition~\ref{prop:finite-defect} gives a representative
$u\in\mathcal U(k)$ arbitrarily close to one at all places of $B$.
Use the notation of Lemma~\ref{lem:e6-generic-d4}.  For $v\in B$, the
element $t=unu^{-1}n$ is close to one.  Set
$w_v=t^{1/2}\in T_*(k_v)$, using
Equation~\eqref{eq:e6-uniform-root}.  Since $n$ inverts $T_*$, we have
\[
 w_v n w_v^{-1}=w_v^2n=tn=n'.
\]
Thus $w_v^{-1}u\in H(k_v)$ is close to one.  The central isogeny
$\pi$ is an analytic local isomorphism at one, so this element lifts to
$\widetilde q_v\in\widetilde H(k_v)$ close to one.  At the finite
places in $B$ the lift can be placed in the chosen kernel neighborhoods.
At the archimedean places this construction supplies the local point of
$Y_u$ that is needed for the Hasse principle.

Lemma~\ref{lem:e6-pullback} supplies points of $Y_u$ at all other finite
places.  The Hasse principle for torsors under simply connected
semisimple groups gives a rational point.  After trivializing the torsor
by this point, weak approximation for the same groups gives a rational
point $(w,\widetilde q)$ approximating the specified local points at $B$
\cite{PRbook}.  Consequently
\[
                         u=w\pi(\widetilde q).
\]
Theorem~\ref{thm:d4} kills $w\in D_*(k)$ under $q$, since a group of type
$D_4$ has no anisotropic finite places.  The local kernel conditions and
Lemma~\ref{lem:restriction-mp} kill $\pi(\widetilde q)$.  Thus $q(u)=1$.
Every coset of $R$ has been killed, so $G(k)/G(k)^e=1$.  The finite-index
conclusion of Proposition~\ref{prop:finite-defect} now gives the stated
assertion for every noncentral abstract normal subgroup.
\end{proof}

\subsection{Completion of the normal-subgroup theorem}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
The established cases recalled in Section~\ref{sec:foundations},
Theorem~\ref{thm:outer-a}, Theorem~\ref{thm:d4}, and
Theorem~\ref{thm:e6} exhaust the absolutely almost simple simply
connected groups over number fields.  Their proofs give, in every case,
the vanishing of the finite defect for the abstract power subgroup
$R=G(k)^e$:
\[
 R=\delta^{-1}(P_0),\qquad
 P_0=\overline{\delta(R)}\triangleleft\prod_{v\in A}G(k_v),
 \qquad P_0\ \text{open}.
\]
For the established cases this equality follows directly from (MP);
for the remaining cases it is the conclusion of the preceding
arguments.

Let now $N\triangleleft G(k)$ be any abstract normal subgroup not
contained in the center.  Proposition~\ref{prop:finite-defect} gives
$[G(k):N]<\infty$.  Choose $e$ to be the exponent of the finite group
$G(k)/N$, so that $R\subset N$.  Density of the diagonal, and openness
of $P_0$, identify
\[
 G(k)/R\simeq
 \left(\prod_{v\in A}G(k_v)\right)/P_0.
\]
The normal subgroup $N/R$ therefore corresponds to a normal subgroup of
this finite local quotient.  Its full inverse image $W$ in
$\prod_{v\in A}G(k_v)$ is open and normal, and
$N=\delta^{-1}(W)$.

If $A$ is empty, the local product, its quotient, and $W$ are all the
one-element group.  The same argument gives $R=G(k)$ and hence
$N=G(k)$.  No finite-generation or closedness hypothesis on $N$ has
entered the argument.
\end{proof}
==== END The-Margulis-Platonov-conjecture-over-number-fields//sections/exceptional-types.tex ====

==== BEGIN The-Margulis-Platonov-conjecture-over-number-fields//sections/finite-groups.tex ====
\section{The finite simple-group obstruction}\label{sec:finite-groups}

We prove Lemma~\ref{lem:finite-obstruction}. The argument uses the
classification of finite simple groups
\cite[Classification Theorem, p.~737]{Aschbacher2004}.
Two elementary tests handle the
alternating, linear, and sporadic families. A counting argument in a full
unitary group handles the other classical families, and small maximal tori
handle the exceptional families.

Throughout this section, $\mathcal F$ is a finite nonabelian simple group,
$C(B)=C_{\mathcal F}(B)$, and
\[
 I(\mathcal F)=\{g\in\mathcal F:g^2=1\},\qquad
 k(\mathcal F)=\text{the number of conjugacy classes of }\mathcal F.
\]
For commuting $W,Z\in\mathcal F$, the condition to be excluded is
\begin{equation}\label{eq:finite-relation}
 ZW\in C(B)(ZW^{-1})C(BZ^2)
 \quad\text{for every }B\in\mathcal F.
\end{equation}
The choice of the nonempty proper conjugacy-invariant set $\mathcal S$
will be specified in each family. In every case we show that
Equation~\eqref{eq:finite-relation} is incompatible with
$W\in\mathcal S$ and $Z\notin\mathcal S$.

\subsection{Two elementary tests}

First take
\begin{equation}\label{eq:finite-square-set}
 \mathcal S=\{W\in\mathcal F:W^2\ne1\}.
\end{equation}
This set is nonempty, since a group of exponent two is abelian, and it
does not contain the identity. For commuting $W\in\mathcal S$ and
$Z\notin\mathcal S$, put $v=ZW$. Then $Z^2=1$, $v^2=W^2\ne1$, and
Equation~\eqref{eq:finite-relation} becomes
\begin{equation}\label{eq:finite-inverse-coset}
 v\in C(B)v^{-1}C(B)\quad\text{for every }B\in\mathcal F.
\end{equation}
Suppose that $\mathcal F$ acts on a set and that some point $p$ satisfies
$v^2p\ne p$. If $C(B)$ fixes $p$ and $vp$ pointwise, writing
$v=a v^{-1}b$ with $a,b\in C(B)$ gives
$vp=a v^{-1}p$. Applying $a^{-1}$ gives $vp=v^{-1}p$, a contradiction.
We call this the point-pair test. We will also use its incidence version:
if $C(B)$ fixes a line $p$ and preserves a subspace $L$, while $vp\subset L$
and $v^{-1}p\not\subset L$, the same equation is impossible.

The second test uses a self-centralizing abelian subgroup.
\begin{lemma}\label{lem:finite-abelian-test}
Suppose that $C\subset\mathcal F$ is an abelian subgroup of odd order
$c>1$ such that $C_{\mathcal F}(t)=C$ for every $1\ne t\in C$. If
\begin{equation}\label{eq:finite-class-bound}
 |\mathcal F|>k(\mathcal F)c^4,
\end{equation}
then the union $\mathcal S$ of the conjugates of $C\setminus\{1\}$
has the property required in Lemma~\ref{lem:finite-obstruction}.
\end{lemma}
\begin{proof}
The set is nonempty and excludes the identity. If $W\in\mathcal S$ and
$Z$ commutes with $W$, then $Z$ lies in the same conjugate of $C$ as $W$.
Consequently $Z\notin\mathcal S$ forces $Z=1$.

Conjugate $W$ into $C$. Testing Equation~\eqref{eq:finite-inverse-coset}
with every conjugate of a fixed nonidentity element of $C$ shows that
every conjugate $v$ of $W$ belongs to $Cv^{-1}C$. If
$v=a v^{-1}b$, $a,b\in C$, then
\[
 (v b^{-1})^2=a b^{-1}\in C.
\]
If this square is nonidentity, $v b^{-1}$ centralizes a nonidentity
element of $C$, so $v\in C$. Otherwise $v\in I(\mathcal F)C$.
Since $I(\mathcal F)$ is conjugacy-invariant, $I(\mathcal F)C=CI(\mathcal F)$.
Thus the whole conjugacy class of $W$ lies in $CI(\mathcal F)$, and
\[
 \frac{|\mathcal F|}{c}\le c|I(\mathcal F)|.
\]
The Frobenius--Schur formula and Cauchy--Schwarz give
\[
 |I(\mathcal F)|
 =\sum_{\chi\in\operatorname{Irr}(\mathcal F)}\nu_2(\chi)\chi(1)
 \le\sum_\chi\chi(1)
 \le\sqrt{k(\mathcal F)|\mathcal F|}.
\]
These inequalities contradict Equation~\eqref{eq:finite-class-bound}.
\end{proof}

\subsection{Alternating, linear, and sporadic groups}

For alternating and projective special linear groups we use
Equation~\eqref{eq:finite-square-set}. In $A_n$, choose $p$ with
$v^2p\ne p$. There is an even permutation $B$ fixing exactly $p,vp$
and having distinct odd cycle lengths greater than one on the remaining
points, except when $n=6$ or $8$. Indeed, for odd $n$ use the single
cycle of length $n-2$, and for even $n\ge10$ use lengths $3,n-5$.
Every permutation centralizing these moving cycles is even on their
support. Therefore a centralizer element in $A_n$ cannot interchange the
two fixed points. The point-pair test applies. The two exceptions are
$A_6\simeq\PSL_2(9)$ and $A_8\simeq\PSL_4(2)$.

Now let $\mathcal F=\PSL_l(q)$, $l\ge3$. Choose a line $p$ such that
$v^2p\ne p$. The three lines $p,vp,v^{-1}p$ are distinct. A hyperplane
$L$ can be chosen to contain $vp$ and contain neither $p$ nor $v^{-1}p$:
in the quotient by $vp$, choose a linear functional nonzero on both
remaining nonzero vectors. The union of two proper hyperplanes in the
dual is not the entire dual, also over $\bbF_2$.

On the direct sum $p\oplus L$, take $B$ to be the identity on $p$ and
an irreducible norm-one field multiplication on $L$. Such an element
exists in the cyclic group of order $(q^{l-1}-1)/(q-1)$: a generator
cannot lie in a proper subfield, since that group's order exceeds
$q^{(l-1)/2}-1$ when $l-1\ge2$. A projective centralizer of $B$ is an
actual centralizer. Indeed, scalar multiplication must preserve its
unique base-field eigenvalue, which is $1$, and hence the multiplier is
$1$. The centralizer fixes $p$ and preserves $L$, contradicting the
incidence version of Equation~\eqref{eq:finite-inverse-coset}.

For simple $\PSL_2(q)$, the pointwise stabilizer of two points of the
projective line is a split torus. It contains an element whose
centralizer fixes both points, unless $q=5$: its order is $q-1$ for
even $q$ and $(q-1)/2$ for odd $q$, and a noninvolution in it is regular
with that torus as centralizer. The case $q=5$ is $A_5$. Choosing the
torus for $p,vp$ proves the assertion in all these cases.

For the sporadic groups other than $M_{12}$, and also for the Tits group,
Table~\ref{tab:finite-sporadic} gives a prime $c$ for which a cyclic
subgroup of order $c$ is the centralizer of each of its nonidentity
elements. The centralizer orders, class numbers, and group orders are
those in the \emph{Atlas} and the online \emph{ATLAS of Finite Group
Representations}~\cite{Atlas,AtlasOnline}. In each row, an element $x$
in the class labeled $c\mathrm A$ in that group's conjugacy-class table
has $C(x)=\langle x\rangle$ of order $c$. Since $c$ is prime, every
nonidentity power of $x$ has the same centralizer.
The displayed $k$ is the exact class number. In every row the group
order exceeds $kc^4$, so Lemma~\ref{lem:finite-abelian-test} applies.
For example, the smallest margin is $10\cdot5^4=6250<7920=|M_{11}|$.
One may also check the other inequalities using $k\le200$, except that
$k\le25$ suffices for $M_{22},M_{23},J_1$.

\begin{table}[htbp]
\centering
\small
\begin{tabular}{lrr@{\qquad}lrr}
\toprule
Group & $c$ & $k$ & Group & $c$ & $k$\\
\midrule
$M_{11}$ & 5 & 10 & $\mathrm{O'N}$ & 31 & 30\\
$M_{22}$ & 11 & 12 & $\mathrm{Co}_1$ & 23 & 101\\
$M_{23}$ & 23 & 17 & $\mathrm{Co}_2$ & 23 & 60\\
$M_{24}$ & 23 & 26 & $\mathrm{Co}_3$ & 23 & 42\\
$J_1$ & 7 & 15 & $\mathrm{Fi}_{22}$ & 13 & 65\\
$J_2$ & 7 & 21 & $\mathrm{Fi}_{23}$ & 23 & 98\\
$J_3$ & 19 & 21 & $\mathrm{Fi}_{24}'$ & 29 & 108\\
$J_4$ & 43 & 62 & $\mathrm{HN}$ & 19 & 54\\
$\mathrm{HS}$ & 11 & 24 & $\mathrm{Ly}$ & 67 & 53\\
$\mathrm{McL}$ & 11 & 24 & $\mathrm{Th}$ & 31 & 48\\
$\mathrm{He}$ & 17 & 33 & $\mathbb B$ & 47 & 184\\
$\mathrm{Ru}$ & 29 & 36 & $\mathbb M$ & 71 & 194\\
$\mathrm{Suz}$ & 13 & 43 & ${}^2F_4(2)'$ & 13 & 22\\
\bottomrule
\end{tabular}
\caption{Self-centralizing prime-order subgroups for the sporadic and
Tits groups.}\label{tab:finite-sporadic}
\end{table}

For $M_{12}$ use its sharply $5$-transitive action of degree $12$ and
the \emph{Atlas} class $8B$, of cycle shape $1^2\,2\,8$
\cite{Atlas,AtlasOnline}; in the online database these are the
$M_{12}$ degree-$12$ representation and its class $8B$.
An element centralizing $B\in8B$ restricts to a cyclic shift on its
eight-point orbit. This restriction is injective: its kernel fixes
eight points, and an element fixing five points is the identity.
The centralizer is therefore exactly $\langle B\rangle$, and fixes
the two fixed points individually. Two-transitivity moves this pair to
$p,vp$, so the point-pair test applies with
Equation~\eqref{eq:finite-square-set}.

\subsection{An auxiliary unitary count}

The remaining classical groups need a torus argument on a large
irreducible block. The relevant obstruction takes place in the full
unitary isometry group, which need not itself be simple.

\Needspace{9\baselineskip}
\begin{lemma}\label{lem:finite-unitary-count}
Let $p\ge3$ be an odd prime, let $H=\U_p(q)$ be the full isometry
group of a nondegenerate Hermitian form over $\bbF_{q^2}$, and let
$T\subset H$ be an irreducible torus of order $q^p+1$. If $w\in T$
has field degree $p$ over $\bbF_{q^2}$, then
\[
 w\in T^h w^{-1}T^h\quad\text{for every }h\in H
\]
is impossible.
\end{lemma}
\begin{proof}
Put
\[
 \overline H=H/Z(H),\qquad \overline T=T/Z(H),\qquad
 c=\frac{q^p+1}{q+1},\qquad d=(p,q+1).
\]
The actual centralizer of $w$ is $T$. If $g$ centralizes its projective
image, then $gwg^{-1}=\lambda w$ for a scalar $\lambda$ of norm one.
Taking determinants gives $\lambda^p=1$, so
\begin{equation}\label{eq:finite-unitary-centralizer}
 |C_{\overline H}(\overline w)|\le dc.
\end{equation}

Suppose the displayed condition in the lemma holds. Every conjugate
$v$ of $w$ then lies in $Tv^{-1}T$. The calculation used in
Lemma~\ref{lem:finite-abelian-test} gives an element $u=v b^{-1}$ whose
square lies in $T$. If $u^2$ is nonscalar, it has field degree $p$,
because $p$ is prime; its centralizer is $T$, and hence $u\in T$.
If $u^2$ is scalar, $\overline u$ has square one. Consequently the
entire projective conjugacy class of $\overline w$ lies in
$\overline T I(\overline H)$.

The determinant quotient of $\overline H$ has order $d$, and
$\overline T$ maps onto it. To see this, write $T$ as the norm-one
subgroup of $\bbF_{q^{2p}}^\times$ over $\bbF_{q^p}$. The determinant
of a field multiplication is its norm to $\bbF_{q^2}$. On this torus,
that norm takes a generator to a generator of the scalar norm-one
group of order $q+1$; its exponent is congruent to $c$ modulo
$q^p+1$. Modding out by scalar matrices mods the determinant out by
$p$th powers, giving the quotient of order $d$.

All members of the conjugacy class have the same determinant in this
quotient. For each fixed $i\in I(\overline H)$, at most $c/d$ elements
$t\in\overline T$ have $ti$ with that determinant. Combining this
with Equation~\eqref{eq:finite-unitary-centralizer} gives
\[
 \frac{|\overline H|}{dc}
 \le \frac c d |I(\overline H)|,
 \qquad\text{and therefore}\qquad
 |\overline H|\le c^2|I(\overline H)|.
\]
We show that the reverse strict inequality always holds.

Every projective involution lifts to an involution in $H$. Indeed,
if $g^2=\alpha I$, then $\alpha^p=\det(g)^2$ in the scalar cyclic
group of order $q+1$. As $p$ is odd, $\alpha$ is a square in that
group, and scalar rescaling gives an involution.

For odd $q$, an involution is determined up to conjugacy by the
dimensions $a,p-a$ of its two nondegenerate eigenspaces. Projectively
the unordered partitions are indexed by $1\le a<p/2$, and their
centralizers have order at least
\begin{equation}\label{eq:finite-unitary-odd-centralizer}
 \frac{|\U_a(q)|\,|\U_{p-a}(q)|}{q+1}.
\end{equation}
For even $q$, the Jordan type is $2^a1^b$, where $b=p-2a$. Its full
unitary centralizer has order
\begin{equation}\label{eq:finite-unitary-even-centralizer}
 q^{a^2+2ab}|\U_a(q)|\,|\U_b(q)|.
\end{equation}
Here is the structure behind this formula. In the ambient general
linear group the connected matrix centralizer has reductive quotient
$\GL_a\times\GL_b$ and unipotent radical of dimension $a^2+2ab$.
The Hermitian Frobenius gives the unitary forms on both multiplicity
spaces. Equivalently, for $g=1+N$ of order two, $N=N^*$; the induced
forms on $\ker N/\im N$ and, using $N$, on the quotient by $\ker N$
are the forms on these two spaces. Nondegenerate Hermitian forms of
a given dimension over a finite field are isometric. Connectedness
and Lang's theorem give one full-unitary class for each occurring
type and the stated order. Dividing by $q+1$ gives a lower bound
for its projective centralizer.

The unitary order formula implies
\begin{equation}\label{eq:finite-unitary-order-lower}
 |\U_j(q)|=q^{j^2}\prod_{r=1}^j(1-(-q)^{-r})\ge q^{j^2}.
\end{equation}
For completeness, pair consecutive factors: their product is
\[
 (1+q^{-(2r-1)})(1-q^{-2r})
 =1+q^{-2r}(q-1-q^{-(2r-1)})>1.
\]
An unpaired final factor also exceeds one. Thus every nonidentity
projective involution class has centralizer order at least
\[
 M=\frac{q^{(p^2+1)/2}}{q+1}.
\]
There are at most $(p-1)/2$ such classes, and including the identity
gives
\[
 \frac{|I(\overline H)|}{|\overline H|}\le\frac{p+1}{2M}.
\]
When $p\ge5$, we have $c<q^{p-1}$ and
\[
 \frac{M}{q^{2p-2}}
 =\frac{q^{(p^2-4p+5)/2}}{q+1}
 \ge\frac{2^p}{3}>\frac{p+1}{2}.
\]
The first inequality uses $(p^2-4p+5)/2\ge p$ and monotonicity in
$q\ge2$. This proves $c^2|I(\overline H)|<|\overline H|$.

It remains to check $p=3$. Now
\[
 c=q^2-q+1,\qquad h=|\overline H|=q^3(q^3+1)(q^2-1).
\]
There is one nonidentity involution class. In odd characteristic its
centralizer lower bound is $M=q(q-1)(q+1)^2$, and $h/M=q^2c$.
In even characteristic it is $M=q^3(q+1)$, and $h/M=(q^2-1)c$.
In both cases $|I(\overline H)|\le1+q^2c$, whereas
\[
 h-c^2(1+q^2c)
 =c\bigl((3q-4)q^4+(q-2)q^2+q-1\bigr)>0
 \quad(q\ge2).
\]
This includes $\U_3(2)$ and finishes the proof.
\end{proof}

\subsection{Symplectic and orthogonal groups}

Let $\mathcal F$ be a projective symplectic or orthogonal simple group
on its natural space of dimension $2l$ or $2l+1$. We first treat
$l\ge3$, using the standard low-rank identifications and treating
$\PSp_4$ separately below. Choose an odd prime
\begin{equation}\label{eq:finite-block-prime}
 l/2<p\le l,
\end{equation}
requiring $p<l$ in plus orthogonal type. For $l\ge4$, set
$m=\lfloor l/2\rfloor\ge2$. Bertrand's postulate gives
$m<p<2m$, hence $l/2<p<l$, and this prime is odd.
For $l=3$ take $p=3$, apart from plus type, which is $\PSL_4$ and
has already been treated. In odd-dimensional orthogonal type we may
assume odd characteristic, since in even characteristic the group
has the corresponding symplectic realization.

We use a nondegenerate block $E$ of dimension $2p$ carrying a torus
of order $q^p+1$. In orthogonal type this block has minus type; the
complement can be chosen with the sign needed for the ambient form.
For minus type $p=l$ the block is the whole space, whereas the
requirement $p<l$ in plus type leaves a complement of the necessary
sign. The torus is the norm-one subgroup of
$\bbF_{q^{2p}}^\times$ over $\bbF_{q^p}$, acting on $E$ by field
multiplication.

Define $\mathcal S$ to consist of those projective elements whose
square, on the natural space, has an irreducible self-dual block of
dimension $2p$. This condition is independent of the choice of a
scalar lift and is conjugacy-invariant. The block is unique, because
$4p>2l+1$, and is nondegenerate: its dual would otherwise require a
second block of the same degree. The set excludes identity and is
nonempty. Indeed, if $t$ generates the torus of order $q^p+1$, use
$t^2$ on the block and identity on the complement. Squaring meets
the orthogonal component and spinor conditions, whose quotients
have exponent two. Moreover $t^4$ still has degree $2p$. To see
this, the quotient of the norm-one torus by its intersection with
$\bbF_{q^2}^{\times}$ has odd order
\[
 \frac{q^p+1}{q+1}>1.
\]
Thus $t^4\notin\bbF_{q^2}$. The only other maximal proper subfield
is $\bbF_{q^p}$, where a norm-one element is $\pm1$ and hence already
lies in $\bbF_{q^2}$.

Suppose now that $W\in\mathcal S$, that $Z\notin\mathcal S$
commutes with $W$, and that Equation~\eqref{eq:finite-relation} holds.
Lift $W,Z$ to natural isometries, using $\Omega$ in orthogonal type.
Their possible projective commutation multiplier has order at most
two. They therefore commute actually with $W^2$ and preserve its
unique marked block $E$. On $E$ they act as field multiplications
$w,z\in\bbF_{q^{2p}}^\times$ of norm one. Since $Z\notin\mathcal S$,
$z^2$ lies in a proper subfield. The norm-one condition puts $z^2$
in $\bbF_{q^2}$, and then $z$ itself is in $\bbF_{q^2}$: its degree
over that field divides both $p$ and $2$.

There is a full unitary group $H=\U_p(q)$ acting on $E$ and
containing its field torus $T$. One way to see this directly is to
write the invariant form in field coordinates. Its bilinear form
has the shape
\[
 \Tr_{\bbF_{q^{2p}}/\bbF_q}(\kappa x y^*),\qquad x^*=x^{q^p}.
\]
Take the trace first to $\bbF_{q^2}$. If the resulting form is
anti-Hermitian and the characteristic is odd, multiply it by
$\delta\in\bbF_{q^2}^{\times}$ with $\delta^q=-\delta$; this makes
it Hermitian without changing its isometry group. In characteristic
two an anti-Hermitian form is already Hermitian. This gives the
required unitary structure. In the
quadratic case the form is
\[
 Q(x)=\Tr_{\bbF_{q^p}/\bbF_q}(\kappa xx^*),
 \qquad\kappa\in\bbF_{q^p}^\times.
\]
In characteristic two this expression follows from the polarization
and $W^2$-invariance: two invariant quadratic forms with the same
polarization differ by the square of an invariant linear form,
and an irreducible block of degree greater than one has no such
nonzero form. These formulas show that $H$ preserves the original
form on $E$. The element $z$ is a scalar of this unitary group.

For each $h\in H$, choose $B$ acting on $E$ by a square of a
generator of $T^h$ and as identity on the complement. This is an
allowed ambient element. Even if $h$ is not in $\Omega$, conjugation
by an isometry preserves $\Omega$, so the orthogonal component
condition is unchanged. Both $B$ and $BZ^2$ have a unique marked
square block $E$: multiplication of a full-degree element by a
base-field scalar preserves its degree over $\bbF_{q^2}$, and the
norm-one condition then gives full degree $2p$ over $\bbF_q$.

Each projective centralizer of $B$ or $BZ^2$ commutes actually with
its square, preserves $E$, and restricts there into $T^h$. This
does not require the action on the complement to be semisimple;
the complement is too small to have an irreducible factor of
degree $2p$. Restricting Equation~\eqref{eq:finite-relation} to $E$
and absorbing its scalar multipliers and the scalar $z$ into $T^h$
therefore gives
\[
 w\in T^h w^{-1}T^h\quad\text{for every }h\in H.
\]
The element $w$ has degree $p$ over $\bbF_{q^2}$. This contradicts
Lemma~\ref{lem:finite-unitary-count}.

\subsection{Projective unitary groups and rank two symplectic groups}

Let $\mathcal F=\PSU_l(q)$, $l\ge3$, excluding the nonsimple pair
$(l,q)=(3,2)$. Choose an odd prime $l/2<p\le l$. Define
$\mathcal S$ by requiring the square of a natural lift to have an
irreducible block of degree $p$ over $\bbF_{q^2}$. As above this
block is unique and nondegenerate. If $l>p$, the determinant of a
chosen torus element on the block can be corrected on its nonzero
dimensional orthogonal complement. If $l=p$, use instead the
determinant-one torus of order
\[
 c=\frac{q^p+1}{q+1}.
\]
Its scalar intersection has order $d=(p,q+1)$. Except for
$(p,q)=(3,2)$, $c>d$: for $p\ge5$,
$c\ge q^{p-2}(q-1)+1\ge2^{p-2}+1>p$, and for $p=3,q\ge3$,
$c\ge7>3$. As $c$ is odd, a generator and its square are both
nonscalar and hence have degree $p$. This proves that
$\mathcal S$ is nonempty; it again excludes identity.

We check the projective-centralizer issue before applying the block
argument. Suppose $gAg^{-1}=\lambda A$, where $A\in\SU_l(q)$ has
the marked square block. Multiplication by a base-field scalar
preserves irreducible degrees, so $g$ preserves the unique block.
The determinant on the block gives $\lambda^p=1$, and the total
determinant gives $\lambda^l=1$. If $l>p$, then $p<l<2p$, so these
conditions force $\lambda=1$. If $l=p$ and $\lambda\ne1$, multiplication
by $\lambda$ cycles all $p$ distinct eigenvalues. Their product is
$w^p$, since $p$ is odd, and determinant one gives $w^p=1$.
But $\lambda$ has order $p$ in the scalar group, so $p\mid q+1$;
all $p$th roots of unity then lie in $\bbF_{q^2}$, contradicting
the full degree of $w$. Thus these projective centralizers are
actual centralizers on the block.

For commuting $W\in\mathcal S$ and $Z\notin\mathcal S$, the
restrictions are consequently field scalars $w,z$, with
$z\in\bbF_{q^2}$ by the same odd-degree argument as before.
For every conjugate $T^h$ in the full block unitary group choose
$B$ with irreducible square on that torus, correcting its determinant
on the complement if necessary. The square of $BZ^2$ retains full
degree. Applying the preceding projective-centralizer argument
to $B$ and $BZ^2$, Equation~\eqref{eq:finite-relation} restricts to
the relation prohibited by Lemma~\ref{lem:finite-unitary-count}.

For $\PSp_4(q)$ take $\mathcal S$ to be the set whose square is
irreducible on the natural four-dimensional space. It is nonempty
by the torus of order $q^2+1$ and excludes identity. Lifting a
commuting pair $W\in\mathcal S$, $Z\notin\mathcal S$ gives norm-one
field scalars $w,z\in\bbF_{q^4}^\times$. Since $z^2\in\bbF_{q^2}$,
the norm-one condition gives $z^2=\pm1$. Thus $Z^2=1$ projectively,
and $v=ZW$ still has an irreducible square.

Choose a nonzero vector $x$ such that $x,vx$ span an isotropic
plane $L$. Such an $x$ exists in the field description of the
alternating form: the condition
$\Tr_{\bbF_{q^4}/\bbF_q}(\kappa x(vx)^*)=0$ is one homogeneous
$\bbF_q$-linear condition on $xx^*\in\bbF_{q^2}$. It has a nonzero
solution, and the field norm $x\mapsto xx^*$ is surjective.
Irreducibility ensures that $x,vx$ are independent and that
$v^{-1}x\notin L$; otherwise $v$ would satisfy a quadratic polynomial.

Choose a regular unipotent $B$ whose unique line-plane flag is
$\langle x\rangle\subset L$. Its existence in every characteristic
can be seen in a symplectic flag basis from the matrix
\[
 \begin{pmatrix}
 1&1&0&0\\0&1&1&-1\\0&0&1&-1\\0&0&0&1
 \end{pmatrix},
\]
for the alternating form with nonzero upper anti-diagonal entries
$1,1$. This matrix preserves the form and has one Jordan block,
also in characteristic two. Its centralizer preserves
$\ker(B-1)=\langle x\rangle$ and $\ker((B-1)^2)=L$.
A projective centralizer is actual, since a scalar multiple of
a unipotent matrix can have the same eigenvalues only when the
scalar is one. The incidence test now contradicts
Equation~\eqref{eq:finite-inverse-coset}. The nonsimple group
$\Sp_4(2)$ is not needed; its simple derived group is $A_6$.

\subsection{Small tori in exceptional groups}

We have proved the obstruction for all classical simple groups.
For most exceptional families we apply
Lemma~\ref{lem:finite-abelian-test} to a rational maximal torus.
We use the usual simply connected algebraic realization followed
by its central quotient. Rational tori are obtained by Weyl-group
twisting, and their orders are determinants of the Frobenius action
minus one on their character lattices; see \cite{Steinberg} and
\cite[Propositions~25.1--25.3]{MalleTesterman}. Semisimple centralizers
in a simply connected semisimple group are connected
\cite[Section~2.10]{SteinbergRegular1965}. The tori below and their
self-centralizing property in the simple quotient occur in
\cite[Corollary~8.2 and Lemma~8.7]{SegevSeitz2002}.
We verify that property here by a root-lattice estimate before applying
Lemma~\ref{lem:finite-abelian-test}. Their orders outside type $G_2$
are also recorded in \cite[Section~3]{GarionLarsenLubotzky}; the two
$G_2$ orders follow directly from its order-three and order-six Weyl
actions.

Write $\Phi_j$ for the $j$th cyclotomic polynomial. Use the tori
in Table~\ref{tab:finite-exceptional-tori}, before passing to the
central quotient. For $G_2(q)$ choose the sign avoiding a factor
$3$, and for $q=3$ choose the smaller order $7$.
\begin{table}[htbp]
\centering
\begin{tabular}{lll}
\toprule
Group & Torus order & Weyl action\\
\midrule
$G_2(q)$, $q\ge3$ & $q^2\pm q+1$ & Order $3$ or $6$\\
${}^3D_4(q),F_4(q)$ & $\Phi_{12}(q)$ & Characteristic polynomial $\Phi_{12}$\\
${}^{\epsilon}E_6(q)$ & $\Phi_9(\epsilon q)$ & $\Phi_9$, with diagram sign\\
$E_8(q)$ & $\Phi_{30}(q)$ & Coxeter action\\
\bottomrule
\end{tabular}
\caption{Tori used for the exceptional simple groups;
$\epsilon=1$ denotes split $E_6$ and $\epsilon=-1$ twisted $E_6$.}
\label{tab:finite-exceptional-tori}
\end{table}

These actions are irreducible over $\bbQ$. For $G_2,F_4,E_8$ use
a Coxeter element or an appropriate power. In trialitarian $D_4$,
let $b$ be the central node, $a$ an outer node, and $\gamma$ the
diagram cycle of the outer nodes. The action $s_a s_b\gamma$ has
characteristic polynomial $X^4-X^2+1$. For $E_6$, the regular
order-nine Weyl element in Springer's
tables~\cite[Section 5.4, Table 1]{Springer} has
a primitive ninth-root eigenvalue. Its rational characteristic
polynomial, of degree six, is therefore $\Phi_9$. Its negative
belongs to the nontrivial diagram coset and gives the twisted
case.

After passing to the central quotient, dividing by $(3,q-\epsilon)$
in type $E_6$, every prime divisor of these torus orders is a
primitive cyclotomic prime. Their respective lower bounds are
\[
 7,\qquad13,\qquad19,\qquad31.
\]
Indeed a prime dividing $\Phi_m(q)$ either has multiplicative
order $m$ for $q$ modulo that prime, or is an exceptional prime
dividing $m$. The displayed polynomials have no such exceptional
factor except for $3$ in $\Phi_9(\epsilon q)$; when present this
factor has valuation one and is exactly removed by the center.
All resulting torus orders are odd and greater than one.

We next prove that \emph{every} nonidentity element of each resulting
torus has centralizer exactly that torus. It suffices to do so for
an element $t$ of prime order $r$ upstairs, with $r$ one of these
primitive primes. If a root $\alpha$ vanished on $t$, Frobenius
invariance would make every root in its twisted Weyl orbit vanish
on $t$. By irreducibility that orbit spans the rational character
space. Choose a linearly independent subset. The lattice it
generates has full rank in the weight lattice, and its index must
be divisible by $r$, since evaluation at $t$ is a nontrivial
character of order $r$ trivial on this sublattice.

Normalize long roots to have squared length two. Hadamard's
inequality bounds these lattice indices by
\begin{equation}\label{eq:finite-root-index}
 2\sqrt3,\qquad 8,\qquad 8\sqrt3,\qquad16
\end{equation}
in the respective rows of Table~\ref{tab:finite-exceptional-tori}.
The respective weight-lattice covolumes are
\[
 1/\sqrt3,\qquad1/2,\qquad1/\sqrt3,\qquad1.
\]
The second value holds for both $D_4$ and $F_4$.
Each bound is smaller than the corresponding $r$.
Thus $t$ is regular. Connectedness of its algebraic centralizer
makes that centralizer exactly the torus.

Any nonidentity element of a torus in the simple quotient has a
nontrivial prime-order power of this kind. The argument survives
projective centralization: one can lift that power with order $r$
coprime to the center, and a conjugate differing from it by a
central element must have that central factor equal to one,
by preservation of its order. Therefore its projective centralizer
is precisely the torus in the quotient. The hypotheses on $C$
in Lemma~\ref{lem:finite-abelian-test} are now verified.

For the numerical inequality, we use
\begin{equation}\label{eq:finite-exceptional-class-bound}
 k(\mathcal F)\le100q^l,
\end{equation}
where $l$ is the absolute rank. This follows, with a smaller
constant, from \cite[Theorem~1.1, p.~3024]{FulmanGuralnick} for the simply
connected fixed-point groups; taking a quotient cannot increase
the number of conjugacy classes. The standard group order
formulas \cite[Tables~24.1--24.2]{MalleTesterman} give
\[
 \begin{array}{c|ccccc}
 \mathcal F&G_2(q)&{}^3D_4(q)&F_4(q)&{}^{\epsilon}E_6(q)&E_8(q)\\
 \hline
 |\mathcal F|>&0.7q^{14}&0.7q^{28}&q^{52}/2&q^{78}/6&q^{248}/2.
 \end{array}
\]
In the two small $G_2$ cases the direct comparisons are
\[
 \begin{aligned}
 |G_2(3)|&=4245696>100\cdot3^2\cdot7^4,\\
 |G_2(4)|&=251596800>100\cdot4^2\cdot13^4.
 \end{aligned}
\]
For $q\ge5$ the chosen $G_2$ torus has order $c\le1.24q^2$, and
$0.7q^{14}>100q^2(1.24q^2)^4$ already at $q=5$, with increasing
ratio thereafter. In trialitarian type $c<q^4$, so it is enough
that $0.7q^8>100$, true at $q=2$. For $F_4,E_6,E_8$ use respectively
$c<q^4,c<2q^6,c<2q^8$; the displayed lower bounds exceed
$100q^l c^4$ already at $q=2$, and thereafter with increasing
ratio. This proves the desired inequality in every row, so
Lemma~\ref{lem:finite-abelian-test} completes these families.

\subsection{Type \texorpdfstring{$E_7$}{E7} and the Suzuki--Ree groups}

For $E_7$ we transfer the torus obstruction from a suitable $E_6$
subgroup. In the simply connected algebraic group choose a
maximal-rank subgroup $M$ geometrically a Levi subgroup with
derived group $D_M=[M,M]$ of type $E_6$ and one-dimensional center. Both
rational forms $E_6$ and ${}^2E_6$ can be obtained: the ambient
Weyl group contains $-1$, which normalizes this subsystem and
induces its nontrivial diagram option, so rational Weyl twisting
gives the second form. Choose the sign $\epsilon$ such that
$(3,q-\epsilon)=1$.

Conjugation on the derived group gives a map $\pi$ from $M(q)$ to
adjoint ${}^{\epsilon}E_6(q)$. Its image is the simple group of
that type. Indeed the simply connected derived subgroup already
maps onto it: both the finite center and the Frobenius cohomology
of its order-three algebraic center vanish under the chosen
coprimality condition. Denote this finite simple image by
$\mathcal F_0$, and let $C_0\subset\mathcal F_0$ be the torus used
above. The kernel $K_M=\ker\pi$ is central, with order
dividing a product of $q-\epsilon$ and powers of $3$. It has no
prime factor in common with $|C_0|$.
Since $\pi(D_M(q))=\mathcal F_0$, every element of $M(q)$ differs
from an element of $D_M(q)$ by an element of $K_M$. Thus
\[
 M(q)=D_M(q)K_M,\qquad
 M(q)/D_M(q)\simeq K_M/(K_M\cap D_M(q)).
\]
In particular, for every primitive prime $r$ dividing $|C_0|$,
this quotient has order prime to $r$, and every element of
$r$-power order in $M(q)$ belongs to $D_M(q)$.

Define $\mathcal S$ in projective $E_7(q)$ to be the union of the
conjugates of the images of those elements of $M(q)$ whose
projection belongs to $C_0\setminus\{1\}$. The preimage in $M$
of the algebraic torus containing $C_0$ is a maximal torus of
$M$, and also of $E_7$. Thus all these elements are semisimple.
The set is nonempty by surjectivity, and it excludes identity:
the ambient central elements project trivially to adjoint $E_6$.

We need their primitive-prime powers to be regular not only in
$E_6$ but in $E_7$. Such a power lies in $D_M(q)$ by the quotient
calculation just given. Every $E_7$
root restricts to a nonzero weight on this $E_6$. To check
nonvanishing, the fundamental weight perpendicular to the $E_6$
subsystem has squared norm $3/2$; a root proportional to it would
have rational proportionality factor of square $4/3$, impossible.
The restriction has length at most $\sqrt2$. Its orbit under
the irreducible twisted Weyl action spans the six-dimensional
weight space. The same index argument as in
Equation~\eqref{eq:finite-root-index} bounds the resulting
weight-lattice index by $8\sqrt3<19$, smaller than the primitive
prime. Hence these powers are regular in $E_7$ as well.

Now conjugate $W\in\mathcal S$ into the specified torus in $M(q)$.
Every element commuting with it projectively belongs to this
torus: the center of simply connected $E_7$ has order at most
two, so the primitive-prime power is actually centralized, and
its connected centralizer is the torus. In particular a commuting
$Z\notin\mathcal S$ lies in $M(q)$ and projects to $1$ in
$\mathcal F_0$; a nonidentity projection would lie in $C_0$ and
put $Z$ in $\mathcal S$.

For each conjugate of a nonidentity element of $C_0$ in
$\mathcal F_0$, choose a lift $B$ in $M(q)$. Both $B$ and $BZ^2$
have that nonidentity projection, so the preceding regularity
argument applies to each. Their projective centralizers are
toral and project into the corresponding conjugate of $C_0$.
Projecting Equation~\eqref{eq:finite-relation} to $\mathcal F_0$
therefore puts the nonidentity projection $w$ of $W$ in
\[
 C_0^h w^{-1}C_0^h\quad\text{for every }h\in\mathcal F_0.
\]
The counting proof of Lemma~\ref{lem:finite-abelian-test}, already
verified for this $E_6$ group, excludes this. This proves the
obstruction in $E_7$.

For the simple Suzuki and rank-one Ree groups, use
Equation~\eqref{eq:finite-square-set} and their doubly transitive
rank-one BN-pair action. A pair stabilizer has a torus of order
$q-1$, containing a prime-order element of order greater than
three. In Suzuki type $q=2^{2a+1}\ge8$, and $q-1$ is odd and not
divisible by three. In Ree type $q=3^{2a+1}\ge27$, the number
$q-1$ has 2-adic valuation one and an odd factor greater than one,
not divisible by three.

Such a prime-order element $t$ is regular. On the standard
Frobenius-stable pair torus, the exceptional Frobenius $F$ acts
on roots by $F^*\alpha=\ell^j\alpha'$, where $\ell$ is the
characteristic and $\alpha'$ is a root of the opposite length;
the factor $\ell^j$ includes the field-Frobenius part.
If $\alpha(t)=1$, then $F(t)=t$ gives
$\alpha'(t)^{\ell^j}=1$. Since the order of $t$ is prime to
$\ell$, also $\alpha'(t)=1$. The two roots are nonparallel,
because a reduced root system has no parallel roots of different
lengths. The index of the
lattice they span in the weight lattice is at most $2\sqrt2$
in $B_2$ and at most $2$ in $G_2$, by the same length and
covolume calculation as above. Neither index can be divisible
by the chosen prime. Its centralizer is therefore the pair torus,
which fixes both points individually. Choose the pair to be
$p,vp$ and apply the point-pair test.

Finally consider ${}^2F_4(q)$, $q=2^{2a+1}\ge8$. Let $\gamma$ be
the normalized diagram reversal in $F_4$. The twisted Weyl
action $s_1s_2\gamma$ has square $s_1s_2s_4s_3$, a Coxeter
element. The resulting finite torus $C$ is contained in the
Frobenius-square torus of order $\Phi_{12}(q)$. Its order is one
of the two factors
\[
 q^2\pm\sqrt{2q^3}+q\pm\sqrt{2q}+1,
\]
with the signs chosen together; these are Malle's two cyclic tori,
as recorded in \cite[Theorem~8]{GarionLarsenLubotzky}, and their
orders multiply to $\Phi_{12}(q)$. Equivalently, the
determinant formula bounds its order between
$(\sqrt q-1)^4$ and $4q^2$. It is odd and greater than one.
Every prime divisor is at least thirteen, and the $F_4$
root-lattice argument proves that every nonidentity element
has centralizer precisely $C$. The standard group order gives
$|{}^2F_4(q)|>q^{26}/2$. Also
$k({}^2F_4(q))\le100q^4$; the sharper bound
$q^2+4q+17$ is given in \cite[Table~1, p.~3049]{FulmanGuralnick}.
Thus
\[
 |{}^2F_4(q)|>100q^4(4q^2)^4\ge k({}^2F_4(q))|C|^4
 \qquad(q\ge8),
\]
and Lemma~\ref{lem:finite-abelian-test} applies.

\begin{proof}[Completion of the proof of Lemma~\ref{lem:finite-obstruction}]
The classification consists of the alternating groups, the classical
and exceptional groups of Lie type, and the sporadic groups. All
families have been covered above. The low-rank orthogonal groups
have their usual linear, unitary, or symplectic realizations;
in particular plus type $l=3$ is $\PSL_4$, and rank two symplectic
type was treated separately. The exceptional small derived simple
groups are $G_2(2)'\simeq\PSU_3(3)$,
${}^2G_2(3)'\simeq\PSL_2(8)$, and the Tits group, already included
in Table~\ref{tab:finite-sporadic}. Every chosen $\mathcal S$ is
nonempty, proper, and conjugacy-invariant, and for it
Equation~\eqref{eq:finite-relation} has been excluded for every
commuting pair with $W\in\mathcal S$, $Z\notin\mathcal S$.
\end{proof}
==== END The-Margulis-Platonov-conjecture-over-number-fields//sections/finite-groups.tex ====

==== BEGIN The-Margulis-Platonov-conjecture-over-number-fields//sections/foundations.tex ====
\section{Finite quotients and approximation}\label{sec:foundations}

The first objective is to isolate what local approximation does not already
prove.  Every noncentral normal subgroup has finite index, but its closure
at the anisotropic finite places need not a priori recover the subgroup.
We express this possible discrepancy as a finite quotient.  We then
develop two tools for eliminating it: an approximation procedure that
retains the abstract rational coset, and a Hasse principle with weak
approximation for a particular norm torus.

\subsection{Conventions and established inputs}

Throughout, \(k\) is a number field, \(V_k\) is its set of places, and
\(G\) is an absolutely almost simple simply connected \(k\)-group.
Write
\[
 A=\{v\in V_k:\ v\text{ is nonarchimedean and }
                    \rank_{k_v}G=0\},\qquad
 G_A=\prod_{v\in A}G(k_v).
\]
The diagonal homomorphism to this product is denoted by \(\delta\).
Thus \(G_A=H_A\) and \(\delta=\delta_A\) in the notation of the introduction.
An empty product is the trivial group.  The set \(A\) is finite: outside
finitely many places the group has a reductive model and is unramified
and quasi-split, hence has positive relative rank.
At a nonarchimedean place an absolutely simple anisotropic group has
type \(A\).

For a finite set of places \(B\), a subscript \(B\) on a rational element
denotes its tuple of local images.  Products over finitely many places
have their ordinary product topology.  In an adelic product we use the
restricted product with respect to fixed integral compact open subgroups
outside a finite set.  Weak approximation means density at finitely many
completions; strong approximation means density in the specified
restricted adelic product.

For an integer \(e\geq 1\), the notation
\[
                 G(k)^e=\langle g^e:g\in G(k)\rangle
\]
always denotes the abstract subgroup generated by the powers.  No closure
is included in this definition.  Degrees of central simple algebras are
reduced degrees; for example, \(\SL_1(M_r(\Delta))\) has type
\(A_{r\deg\Delta-1}\).  Restrictions of scalars will be treated over
their defining extension fields when applying an induction or an
approximation theorem.

We use the strong approximation theorem for simply connected absolutely
almost simple groups: if \(G(k_w)\) is noncompact, then \(G(k)\) is dense
in the adelic product away from \(w\)
\cite[Theorem 7.12]{PRbook}.  We also use the lattice theorem
for semisimple \(S\)-arithmetic groups
\cite[Theorem~5.7(1)]{PRbook}, the normal subgroup theorem for
irreducible \(S\)-arithmetic lattices of total rank at least two, and the
local Kneser--Tits and simplicity theorems.  In the last assertion, for
a nonarchimedean \(v\) with positive rank, \(G(k_v)\) is perfect and
\(G(k_v)/Z(G(k_v))\) is abstractly simple.
These results are used in their number-field forms
\cite[Chapters VII and IX]{PRbook}\cite[Chapter VIII, Theorem A]{Margulis}.

We refer to the assertion of Theorem~\ref{thm:main} for a particular
group over a particular number field as \emph{(MP)}.  We require the
following established cases.

\begin{theorem}[Known cases of (MP)]\label{thm:known-mp}
The assertion (MP) holds except possibly for anisotropic groups of
outer type \(A\), trialitarian type \(D_4\), and type \(E_6\).
In particular it holds for every inner form of \(\SL_n\), over every
number field.
\end{theorem}

The formulation, including the results for multiplicative groups of
division algebras that enter the inner type \(A\) case, is recalled in
\cite[Sections 3.4--3.5]{PRsurvey}. For the isotropic completion see
\cite[Theorem~8.1]{Gille2009}; for inner type \(A\) see
\cite{RapinchukPotapchik1996,Segev1999,SegevSeitz2002}, with the subsequent
strengthening in \cite{RapinchukSegevSeitz2002}.
Thus, when we address one of the remaining types, we may assume that
the group is anisotropic over its ground field.

We will also use the following form of the Hasse principle.  A torsor
under a \(k\)-group \(H\) means a principal homogeneous space for \(H\)
over \(k\).

\begin{theorem}[Simply connected Hasse principle]
\label{thm:foundations-hasse}
Let \(H\) be a simply connected semisimple group over a number field
\(k\).  Then \(H^1(k_v,H)=1\) at every nonarchimedean place.  A torsor
\(X\) under \(H\) that has a point over every completion of \(k\) has
a \(k\)-point.  Moreover, \(X(k)\) is dense in
\(\prod_{v\in B}X(k_v)\) for every finite set \(B\) of places.
\end{theorem}

The local and global vanishing assertions are the Kneser--Harder--Chernousov
Hasse principle; we use the standard number-field statements in
\cite[Theorems~6.4 and~6.6]{PRbook}.  For the last assertion, a rational point identifies
\(X\) with \(H\).  Decompose \(H\) as a product of restrictions of scalars
of simply connected absolutely almost simple groups.  For each factor,
choose an auxiliary split finite place outside the given approximation
set and apply strong approximation away from it.  This gives the stated
weak approximation, also for products and restrictions of scalars.

We use the usual local and global theory of central simple algebras
and involutions, including local Brauer invariants, the embedding
criterion for fields, the Hasse--Schilling norm theorem, and the
classification of hermitian forms
\cite{PRbook,Reiner,KMRT}.  When an etale algebra is a product of fields,
norm and local arguments are understood factor by factor.  All fields
have characteristic zero unless they are explicitly residue fields.
In root computations the roots of a simply laced system have squared
length two.  A superscript \(1\) on a multiplicative group denotes
the kernel of the indicated norm.

\subsection{The finite quotient not detected by closure}

\begin{proposition}[Finite-defect reduction]\label{prop:finite-defect}
Every abstract noncentral normal subgroup of \(G(k)\) has finite index.
Fix \(e\geq1\), and put
\[
 R=G(k)^e,\qquad
 P_0=\overline{\delta(R)}\subset G_A,\qquad
 V_0=\delta^{-1}(P_0),\qquad V=V_0/R.
\]
Then \(R\) has finite index, \(P_0\) is open and normal in \(G_A\),
and \(V\) is finite.  For every positive-rank nonarchimedean place
\(w\), the closure of \(R\) in the adelic product away from \(w\)
is exactly the inverse image of \(P_0\).

To prove (MP) for \(G\), it suffices to show \(V=1\) for every \(e\).
If \(\gamma\in V_0\), representatives of \(\gamma R\) can be chosen
arbitrarily close to \(1\) at any prescribed finite set of places.
Away from \(A\), these near-identity conditions may be replaced by
arbitrary nonempty local open conditions.  Finitely many nonempty
Zariski open conditions may be imposed simultaneously.
\end{proposition}

\begin{proof}
Let \(N\lhd G(k)\) be noncentral, and choose a noncentral \(n\in N\).
Take a finite set \(T\) containing the archimedean places, \(A\),
the denominator places of \(n\), and auxiliary split finite places
whose total local rank is at least two.  For fixed integral compact
opens outside \(T\), set
\[
 K^T=\prod_{v\notin T}G(\mathcal O_v),\qquad
 \Gamma=G(k)\cap K^T,
\]
where changing finitely many integral models changes neither argument.
The group \(\Gamma\) is the corresponding \(T\)-arithmetic group and
contains \(n\).

The lattice theorem \cite[Theorem~5.7(1)]{PRbook} makes \(\Gamma\) a lattice in the product of the
local groups at \(T\).  Remove its compact factors.  This preserves
discreteness: the inverse image of a compact subset of the remaining
product is compact after the compact factors are restored, and hence
meets \(\Gamma\) in finitely many points.  It also preserves finite
covolume.  The rational embedding remains faithful.  Strong
approximation away from each noncompact factor supplies the projection
density required for irreducibility.  The \(S\)-arithmetic normal
subgroup theorem now applies to \(\Gamma\cap N\), which is normal and
noncentral, and yields
\[
                         [\Gamma:\Gamma\cap N]<\infty .
\]
If one uses the formulation with a finite-normal-subgroup alternative,
that alternative is central: the centralizer of a finite normal
subgroup contains a finite-index Zariski-dense subgroup of \(\Gamma\),
and hence is all of the connected group \(G\).

Strong approximation, omitting a positive-rank place in \(T\), shows
that \(\Gamma\) is dense in \(K^T\).  Thus the closure of
\(\Gamma\cap N\) has finite index in \(K^T\).  It is closed, so it
is open.  It follows that the closure \(C\) of \(N\) in the restricted
product away from \(T\) is open.  It is normal because \(N\) is
normal and \(G(k)\) is dense there.

Every place outside \(T\) has positive rank.  For such a place \(v\),
the intersection \(C\cap G(k_v)\), with the local group supported at
that single place, is an open normal subgroup.  It cannot be central,
since the center is finite.  Local simplicity modulo center and
perfectness give \(C\cap G(k_v)=G(k_v)\).
Finite-support products are dense in the restricted product, so
\(C\) is the whole restricted product.  Given \(g\in G(k)\), we can
therefore choose \(a\in N\) with \(ag\in K^T\), and hence \(ag\in\Gamma\).
The map from \(\Gamma/(\Gamma\cap N)\) to \(G(k)/N\) is surjective.
This proves the finite-index assertion.

The power map has differential \(e\) times the identity at \(1\);
it is therefore dominant in characteristic zero.  Rational points
of a connected linear algebraic group over \(k\) are Zariski dense.
Consequently \(R\) is Zariski dense and noncentral, so the preceding
assertion applies to \(R\).
Density of \(G(k)\) in \(G_A\) follows from strong approximation away
from an auxiliary positive-rank finite place.  Since \(R\) has finite
index, its closure \(P_0\) has finite index in \(G_A\), is normal, and
is open.

Now omit any positive-rank finite place \(w\).  The closure \(C_w\)
of \(R\) in the remaining adelic product is again closed, normal,
and of finite index: finitely many translates of it already contain
the dense subgroup \(G(k)\).  Thus it is open.  The local argument
just given shows that \(C_w\) contains every positive-rank
nonarchimedean factor.  It contains every archimedean factor as well:
the group of real points of a simply connected semisimple algebraic
group is connected, and so is the group of complex points.  Their
maps into the finite discrete quotient by \(C_w\) are therefore
trivial.  Taking closures of finite-support products proves that
\(C_w\) contains the kernel of the projection to \(G_A\).
Its image there is closed, contains \(\delta R\), and is contained
in \(P_0\); hence it is exactly \(P_0\).  This proves the closure
description.

For an original noncentral normal subgroup \(N\), the finite quotient
\(G(k)/N\) has some exponent \(e\), so \(R\subset N\).  If \(V=1\),
then \(R=\delta^{-1}(P_0)\), and density gives an isomorphism
\[
                         G(k)/R \simeq G_A/P_0 .
\]
The normal subgroup \(N/R\) corresponds to a normal subgroup of this
finite quotient.  Its inverse image in \(G_A\) is an open normal
subgroup \(W\), and \(N=\delta^{-1}(W)\).

Finally, for \(\gamma\in V_0\), the closure of \(\gamma R\) away
from \(w\) is the same inverse image of \(P_0\), since
\(\gamma_A P_0=P_0\).  It contains the identity at the places in \(A\)
and all tuples at other places.  Choose \(w\) outside any prescribed
finite approximation set to obtain the asserted local choices.
Nonempty local open subsets are Zariski dense in the smooth connected
group; intersecting one such opening with any additional nonempty
Zariski open subset preserves nonemptiness.  This supplies the last
assertion.
\end{proof}

The following consequence records exactly how an already established
case of (MP) will be used for subgroups.  In particular it does not
require a finite homomorphism to be continuous in advance.

\begin{lemma}[Restriction to a group satisfying (MP)]
\label{lem:restriction-mp}
Let \(\psi:G(k)\to F_0\) be a homomorphism to a finite group of
exponent dividing \(e\).  Let \(f:H\to G\) be a \(k\)-homomorphism,
where \(H\) is a product of restrictions of scalars of simply connected
absolutely almost simple groups for which (MP) is known over their
respective number fields.  The homomorphism \(\psi\circ f\) on \(H(k)\)
extends continuously to the product of the anisotropic finite local
groups of those factors.  Its image still has exponent dividing \(e\).
Consequently \(\psi(f(h))=1\) if \(h\) is an \(e\)-th power at every
one of these local factors.
\end{lemma}

\begin{proof}
For one absolutely almost simple factor, the kernel of the restricted
finite homomorphism has finite index and is Zariski dense, so it is
not central.  By (MP) it is the inverse image of an open normal
subgroup \(W_H\) of the anisotropic local product.  Density identifies
the rational quotient with the finite quotient by \(W_H\).
The homomorphism therefore extends through that quotient.
For a product, extend the maps factor by factor.  Their images
commute, as they do on rational points, so the extensions combine.
Apply this argument over the extension field for each restriction
of scalars.  The extended image is unchanged, and every local
\(e\)-th power maps to \(1\).
\end{proof}

\subsection{An affine avoidance lemma}

The closure calculation permits finitely many local conditions.
The next argument supplies an additional condition at almost all
finite places.  It will be applied to an affine space of parameters,
not to integral points of a torus.
Codimension-two avoidance in affine space is an instance of arithmetic
purity of strong approximation; see \cite[Proposition~3.6]{CaoXu2018}
and \cite[arXiv version, Lemma~1.1]{Wei2019}. We give the fixed-direction version with
an unbounded parameter at the omitted place needed below.

\begin{lemma}[Avoidance along a line]\label{lem:foundations-affine-sieve}
Let \(E\) be an \(N\)-dimensional \(k\)-vector space, and let
\(Z\subset E\) be closed of geometric codimension at least two.
Choose \(d\in E(k)\setminus\{0\}\) whose point at infinity is outside
the projective closure of \(Z\), and let
\(\pi:E\to E/kd\) be the quotient map.  Fix a nonarchimedean place
\(v_0\), linear coordinates, and a rational linear section of \(\pi\).
Outside a finite set of model exceptions, require that these data
give an integral direct sum \(E=kd\oplus E'\), with \(d\) an integral
basis for the first summand, that \(\pi|_Z\) remains finite on
reduction, and that every geometric fiber has cardinality at most
\(D\).

Let \(\Sigma\) be a finite set containing \(v_0\), the archimedean
places, those model exceptions, and all places with residue
cardinality at most \(D\).  For every
\(v\in\Sigma\setminus\{v_0\}\), prescribe a nonempty open subset
\(\mathcal O_v\subset E(k_v)\).
There are a vector \(y\in E(k)\) and elements \(l\in k\) with
\(|l|_{v_0}\) arbitrarily large such that \(a=y+ld\) belongs to
\(\mathcal O_v\) at these prescribed places, is integral outside
\(\Sigma\), and its reduction avoids \(Z\) at every finite place
outside \(\Sigma\).
\end{lemma}

\begin{proof}
The condition on the point at infinity makes \(\pi|_Z\) finite.
For example, extend the projection to projective space: the only
indeterminacy is the chosen point at infinity, which is disjoint
from the projective closure of \(Z\).  Over an affine point of the
target, the only possible point at infinity on its fiber line is
that same point.  Thus the restricted morphism is proper and has
finite fibers, and is finite.  Its image \(Y\subset E/kd\) is closed
and proper, since
\(\dim Z\leq N-2<\dim(E/kd)\).
Use the fixed linear coordinates and integral models; their
exceptional places already belong to \(\Sigma\).

Linear projection is open at each completion.  Additive strong
approximation away from \(v_0\) gives
\[
 q\in (E/kd)(k)\setminus Y(k)
\]
that is integral outside \(\Sigma\) and belongs to
\(\pi(\mathcal O_v)\) at every prescribed place.
To ensure \(q\notin Y\), first shrink one of the prescribed
nonempty openings in the quotient to miss \(Y\); a proper algebraic
subset has empty interior.  Choose a rational linear lift \(y\) of
\(q\), integral outside \(\Sigma\).

Because \(q\notin Y\), its reduction belongs to the reduction of
\(Y\) at only finitely many finite places outside \(\Sigma\).
Indeed a polynomial vanishing on \(Y\), but not at \(q\), is a
nonzero integral value away from finitely many places.
At each of the remaining exceptional places, the forbidden points
in the fiber line number at most \(D\).  The residue field has
more than \(D\) elements, so an integral value of \(l\) avoiding
them exists.  This is a nonempty local congruence condition on \(l\).
At a prescribed place \(v\), the condition \(q\in\pi(\mathcal O_v)\)
likewise gives a nonempty local open set of \(l\) with
\(y+ld\in\mathcal O_v\).

Apply additive strong approximation once more, now to \(l\), imposing
these finitely many conditions and integrality at all other finite
places outside \(\Sigma\).  The local openings may be chosen
relatively compact away from \(v_0\).  There are infinitely many
such rational \(l\), by density in the adeles away from \(v_0\).
They cannot all remain bounded at \(v_0\), since the diagonal
copy of \(k\) is discrete in its full adele ring.  We may therefore
choose \(|l|_{v_0}\) arbitrarily large.  This gives all the assertions.
\end{proof}

\subsection{Approximation by abstract power factors}

We now use anisotropy to make the unrestricted direction in
Lemma~\ref{lem:foundations-affine-sieve} harmless at \(v_0\).
Assume \(G\) is \(k\)-anisotropic.  Fix a maximal \(k\)-torus
\(T_0\subset G\), a finite Galois extension \(P/k\) splitting \(T_0\)
and any fixed auxiliary data, and a regular cocharacter
\(\lambda:\bbG_{m,P}\to T_{0,P}\).
The map
\begin{equation}\label{eq:foundations-eta}
 \eta:\Res_{P/k}\bbG_m\longrightarrow T_0,\qquad
 \eta(a)=N_{P/k}\bigl(\lambda(a)\bigr)
\end{equation}
is defined over \(k\).  Its restriction to the diagonal \(\bbG_m\)
has cocharacter \(\sum_{\sigma\in\Gal(P/k)}\lambda^\sigma\).
This is a \(k\)-cocharacter and hence is zero by anisotropy.
In particular
\begin{equation}\label{eq:foundations-scalar-invariance}
                         \eta(la)=\eta(a)\quad(l\in k^\times).
\end{equation}

For \(x\in G(k)\), or for \(x\) in a fixed rational algebraic
coset \(X=x_*G\) inside a larger linear algebraic group, consider
\begin{equation}\label{eq:foundations-power-map}
 \Phi(a_1,\ldots,a_r)
       =x\prod_{j=1}^r k_j\eta(a_j)^e k_j^{-1},
       \qquad k_j\in G(k),\quad a_j\in P^\times .
\end{equation}
Its values lie in the same right coset \(xR\), with
\(R=G(k)^e\).  Its parameter domain is the open torus
\((\Res_{P/k}\bbG_m)^r\) in the affine \(k\)-space
\(E=(\Res_{P/k}\bbA^1)^r\).
Over \(P\), its boundary consists of the hyperplanes
\(a_{j,\sigma}=0\), one for each absolute coordinate.

We call a reduction a \emph{single-pole reduction} if exactly one
of these absolute coordinates vanishes.  Away from the other
hyperplanes, Equation~\ref{eq:foundations-power-map} then takes the
form
\begin{equation}\label{eq:foundations-single-pole}
             \Phi=g_L\,\lambda^\sigma(t)^e\,g_R,
                    \qquad t=a_{j,\sigma},
\end{equation}
where the side factors are regular group- or coset-valued functions.
At a good finite place with this reduction, Frobenius fixes the unique
vanishing coordinate.  Its action on the embeddings of the Galois
extension \(P/k\) is regular.  Thus the place splits completely in
\(P\), \(t\) has positive valuation, and the side factors are integral.
The local analysis of Equation~\ref{eq:foundations-single-pole} may
therefore be carried out in split coordinates.

Here is the precise criterion used in later sections.  Its hypothesis
about the single-pole configurations is substantive and will be
verified in each application.  A consecutive list of factors in
Equation~\ref{eq:foundations-power-map} is called a \emph{basic list}
if its product map is dominant and has surjective differential
somewhere.

\begin{proposition}[Power-factor approximation]
\label{prop:power-approximation}
Let \(G\) be \(k\)-anisotropic, with \(P\), \(\eta\), \(R\), and
\(X=x_*G\) as above.  Fix \(\gamma\in G(k)\), the rational coset
\(\mathcal C=x_*\gamma R\subset X(k)\), a
geometrically codimension-at-least-two closed subset \(D\subset X\),
and a finite set \(B\) containing the archimedean places, \(A\), and
the exceptions for fixed integral models and sufficiently small
residue fields.  Include in \(B\) a positive-rank finite place \(v_0\).
At \(B\) prescribe nonempty local open conditions compatible with
the closure of \(\mathcal C\) in
Proposition~\ref{prop:finite-defect}.

At each finite place outside \(B\), prescribe an allowed set
\(\mathcal U_v\subset X(k_v)\).  Assume:
\begin{enumerate}
 \item every integral point whose reduction avoids \(D\) belongs
       to \(\mathcal U_v\);
 \item \(\mathcal U_v\) contains a nonempty local open set;
 \item single-pole configurations are tested by finitely many
       algebraic nonvanishing conditions over \(P\), independent
       of \(v\).  Their nonemptiness is compatible with the following
       order of choices.  First choose at least two disjoint basic
       lists, with their conjugator tuple in a nonempty Zariski
       open subset of its product of copies of \(G\).  For each
       such tuple there is a nonempty Zariski open subset of \(X\)
       in which to choose \(x\).  After \(x\) is fixed, for every
       finite number of additional factors there is a nonempty
       Zariski open subset of the corresponding product of copies
       of \(G\) in which to choose their conjugators.  For every
       tuple allowed by these successive choices, the tests
       restrict to a nonempty open subset of each absolute
       boundary hyperplane away from the others.  They guarantee
       membership in \(\mathcal U_v\) at a single-pole reduction,
       outside finitely many model exceptions for the fixed data.
\end{enumerate}
In the third condition the tests are imposed with all their Galois
conjugates.  Appending additional factors is required to preserve
the tests on an existing hyperplane when the new parameters are
set to \(1\).  Thus their insertion does not remove the previously
available open subsets.  The additional factors permit the local
moves required by the second condition.

Then there is \(x'\in\mathcal C\) satisfying the prescribed conditions
at \(B\) and belonging to \(\mathcal U_v\) for every finite
\(v\notin B\).  Finitely many nonempty Zariski open conditions on
the initial representative and conjugators can be included in their
choice, provided they are compatible with the third condition.
\end{proposition}

\begin{proof}
We give the construction in an order that keeps the finite model
exceptions separate from the infinitely many avoidance conditions.
When the preliminary choices enlarge \(B\), use at each newly added
place a nonempty opening inside \(\mathcal U_v\), as supplied by the
second hypothesis.  Such places lie outside \(A\), so these added
conditions are compatible with the coset closure.  Meeting them
preserves the conclusion requested for the original \(B\).

\emph{Dominant parameter lists.}
The adjoint translates of the differential of a nonzero cocharacter
span \(\Lie(G)\): their span is a nonzero adjoint-invariant subspace
of the simple Lie algebra.  Since \(G(k)\) is Zariski dense, finitely
many rational conjugators make the product of their \(\eta^e\)-maps
have surjective differential at some point.  Such a product map is
dominant.  Choose at least two disjoint consecutive lists with this
property in the nonempty open set supplied by the third hypothesis.
We may intersect that set with any further prescribed nonempty
Zariski open conditions.  The product of copies of \(G\) is
geometrically irreducible, so such intersections remain nonempty.

On the open parameter torus the total product map is smooth whenever
one of its basic list maps is smooth.  Each list has a proper closed
nonsmooth locus.  Since the two lists have disjoint parameter sets,
their simultaneous nonsmooth locus has codimension at least two.
The inverse image of \(D\) has codimension at least two on the smooth
locus, and any remaining part lies in that same codimension-two
nonsmooth locus.  Therefore its closure in \(E\) has codimension at
least two.  Adding further independent lists does not change this
conclusion.

We also need a residue-field observation about one fixed basic list.
After discarding finitely many model exceptions, its dominance
persists on reduction.  For any integral translating point \(x\),
the inverse image of the dense open \(X\setminus D\) is a nonempty
open in its affine parameter space.  The equations excluded here,
including the norm equations defining the torus boundary, have
bounded degree independently of the translating point.  The
elementary bound for zeros of a nonzero polynomial over a finite
field therefore supplies a rational parameter in this open whenever
the residue field is sufficiently large.  Thus unit parameters in
one basic list, all other parameters set to \(1\), can give a
reduction outside \(D\).  Put these fixed exceptions and small
residue fields into \(B\).

\emph{The initial point and its denominator places.}
Choose \(x\in\mathcal C\) satisfying the conditions at \(B\), using
Proposition~\ref{prop:finite-defect}; to approximate also at \(v_0\),
omit a different positive-rank place.  Any prescribed compatible
Zariski open conditions can be included; in particular, take \(x\)
in the open subset supplied by the third hypothesis after the
basic conjugators have been fixed.  Let \(B_1\) be the finite
set of places outside \(B\) where \(x\) is not integral.
At each such place choose a nonempty open subset of \(\mathcal U_v\).

The local normal subgroup generated by
\(\eta((P\otimes_k k_v)^\times)^e\) is all of \(G(k_v)\).
Indeed this image is Zariski noncentral, as can be seen in the
independent absolute coordinates, and local simplicity modulo center
and perfectness apply.  Every element of that generated subgroup
is a finite product of conjugates of such powers; inverse factors
are again of the same form.  Consequently finitely many extra
lists can move \(x_v\) into the chosen opening at each \(v\in B_1\).
Fix the finite number of factors needed for these moves, padding
the local lists by factors with parameter \(1\) where necessary.
The third hypothesis supplies a nonempty Zariski open subset for
this whole tuple of extra conjugators, with the initial \(x\)
and basic conjugators unchanged.  Each product of the required
local openings is Zariski dense and hence meets this subset.
Choose their conjugators globally by strong approximation away from
\(v_0\), approximating the required local conjugators and remaining
integral outside \(B\cup B_1\).  Additional nonempty Zariski open
conditions are compatible with this choice.
The basic parameters can be taken near \(1\) on \(B\cup B_1\);
the extra parameters can be taken near \(1\) on \(B\).  We now have
nonempty open conditions on the full affine parameter space at
every place of \(B\cup B_1\).

\emph{The closed subset to be avoided.}
Fix the constants just chosen.  In \(E\), let \(Z\) be the union of
the closure of \(\Phi^{-1}(D)\), intersections of distinct boundary
hyperplanes, and the excluded proper closed subsets of each
individual hyperplane from the third hypothesis.
Take all Galois conjugates, so \(Z\) is defined over \(k\).
Every component has codimension at least two: this was proved for
the first part, is immediate for the intersections, and follows
from nonemptiness of the specified open tests for the last part.
For parameters integral at a good place and reducing outside \(Z\),
there are exactly two possibilities.  Either every torus coordinate
is a unit and \(\Phi\) is integral with reduction outside \(D\), or
there is one pole and its prescribed open test holds.  In both cases
\(\Phi\) belongs to \(\mathcal U_v\).

\emph{The direction and the residual exceptions.}
Choose a rational direction \(d=(d_1,\ldots,d_r)\in E(k)\)
whose point at infinity avoids the projective closure of \(Z\).
Choose each \(d_j\) sufficiently close to \(1\) at \(v_0\) so that
\(\Phi(d)\) still satisfies the prescribed condition there.
These requirements are compatible: avoiding a proper algebraic
subset is dense in any local open set.
The projection along \(d\), its linear splitting, the tests defining
\(Z\), and the fiber bound introduce only finitely many further
exceptions.  They are fixed before the quotient point in the sieve
is chosen.

At a new exceptional place outside \(B\cup B_1\), the point \(x\)
and all conjugators are integral.  Use the residue-field observation
for a basic list to choose unit parameters giving reduction outside
\(D\), and take the extra parameters near \(1\).  This provides a
nonempty local opening where \(\Phi\in\mathcal U_v\), even though
the projection or single-pole model might be unusable there.
Any additional small residue fields needed for the fiber bound
are treated in exactly this way; the smaller fields for which the
basic-list observation itself fails already lie in \(B\).
Let \(\Sigma\) contain \(B\cup B_1\) and all these residual
exceptions.

We can now apply Lemma~\ref{lem:foundations-affine-sieve} with the
parameter openings just specified, omitting the condition at
\(v_0\).  It gives parameters \(a=y+ld\), integral outside
\(\Sigma\), reducing outside \(Z\) there, and meeting all the
openings at \(\Sigma\setminus\{v_0\}\).  As
\(|l|_{v_0}\to\infty\), Equation~\ref{eq:foundations-scalar-invariance}
gives
\[
 \eta(a_j)=\eta(d_j+l^{-1}y_j)\longrightarrow\eta(d_j).
\]
For sufficiently large \(|l|_{v_0}\), every \(a_j\) is nonzero and
\(\Phi(a)\) satisfies the required condition at \(v_0\).
It satisfies the conditions at all other exceptional places by
construction, and lies in \(\mathcal U_v\) elsewhere by avoidance
of \(Z\).  Finally \(\Phi(a)\in xR=\mathcal C\), since every factor
in Equation~\ref{eq:foundations-power-map} is an abstract \(e\)-th
power.  This completes the construction.
\end{proof}

The proposition allows poles, but only in a controlled form at places
that split completely in \(P\).  In particular, it makes no assertion
that the torus parameters can be integral units everywhere.
The later applications will specify \(D\), prove the nonemptiness
of every single-pole test, and check its local consequence.

\subsection{Cohomology detected on cyclic subgroups}

Our final preliminary tool solves simultaneous norm equations with
prescribed local approximations.  We first separate the arithmetic
duality input from the finite-group calculation.

For a discrete Galois module \(M\), write
\[
 \Sha^i(k,M)=
 \ker\left(H^i(k,M)\longrightarrow\prod_{v\in V_k}H^i(k_v,M)\right).
\]
The notation \(\Sha_\omega^i(k,M)\) means classes whose localizations
vanish at all but finitely many places.  For a group \(H\), the
degree-one notation has its pointed-set meaning.  Galois cohomology
and its cochains are continuous throughout.
A \emph{lattice} is a finitely generated free abelian group with a
continuous Galois action, which necessarily has finite image.

\begin{lemma}[Cyclic detection]\label{lem:foundations-cyclic-detection}
Let \(M\) be a \(k\)-lattice split by a finite Galois extension
\(P/k\), and put \(\Gamma=\Gal(P/k)\).  Inflation identifies
\(\Sha_\omega^2(k,M)\) with
\[
 \ker\left(
 H^2(\Gamma,M)\longrightarrow
             \prod_{C\subset\Gamma\ {\rm cyclic}}H^2(C,M)
 \right).
\]
If this group is zero and \(T\) is a \(k\)-torus with character
lattice \(M\), every everywhere locally soluble torsor under \(T\)
has a rational point and satisfies weak approximation.
\end{lemma}

\begin{proof}
Over \(P\) the action on \(M\) is trivial.  The exact sequence with
rational coefficients identifies
\[
 H^2(P,M)\simeq H^1(P,M\otimes(\bbQ/\bbZ)).
\]
The right side consists of continuous finite-image characters.
If a class is locally zero almost everywhere, its restriction to
\(P\) is zero by Chebotarev.  Moreover
\(H^1(P,M)=0\), since a continuous homomorphism from a profinite
group to a torsion-free discrete abelian group has finite, hence
trivial, image.  Inflation--restriction therefore identifies such
a class with a unique class in \(H^2(\Gamma,M)\).

At a place unramified in \(P\), the decomposition group is cyclic.
Local inflation is injective by the same vanishing of degree-one
cohomology over the splitting field.  Chebotarev realizes, up to
conjugacy, every element of \(\Gamma\) as a Frobenius element.
Thus local vanishing almost everywhere is equivalent to vanishing
on every cyclic subgroup.  Conversely, a class with those cyclic
restrictions zero is locally zero at every unramified place, which
proves the identification.

Torus duality identifies the Hasse-principle obstruction for torsors
under \(T\) with the dual of \(\Sha^2(k,M)\), and controls the weak
approximation defect of \(T\) by \(\Sha_\omega^2(k,M)\)
\cite[Proposition 9.8 and Corollary 8.14]{Sansuc}.
The stated vanishing therefore gives
a rational point on every locally soluble torsor and weak
approximation on \(T\).  Translation by that rational point gives
weak approximation on the torsor.
\end{proof}

To calculate this kernel, it is convenient to use extensions by
lattices.  The following lemma will also apply when the finite
group is not a direct product.

\begin{lemma}[A central sign element]\label{lem:foundations-sign-extension}
Let \(\Gamma\) be a finite group and \(M\) a torsion-free
\(\Gamma\)-lattice.  Suppose that \(c\in Z(\Gamma)\) has order two
and acts on \(M\) as multiplication by \(-1\).
An extension
\[
       1\longrightarrow M\longrightarrow E
        \longrightarrow\Gamma\longrightarrow1
\]
determines a class
\(\beta(E)\in H^1(\Gamma/\langle c\rangle,M/2M)\), and this class
is zero if and only if the extension splits.
If the restriction to the subgroup generated by an involution
\(r\in\Gamma\) splits, the value at the image of \(r\) of every
cocycle representing \(\beta(E)\) lifts to an element
\(d_r\in M\) with \((1+r)d_r=0\).
\end{lemma}

\begin{proof}
Write the subgroup \(M\) additively, with translations on the left.
Let \(z\) lift \(c\).  Since \(z^2\in M\) commutes with \(z\),
it lies in \(M^c=0\); hence \(z\) is an involution.
Choose lifts \(s_g\) of \(g\in\Gamma\), with \(s_1=1\), and write
\[
 s_gs_h=f(g,h)s_{gh},\qquad
                 z s_gz^{-1}=d_gs_g.
\]
Comparing the two expressions for the conjugate of \(s_gs_h\) gives
\[
                    d_{gh}=d_g+g d_h+2f(g,h).
\]
Thus \(g\mapsto d_g\bmod 2M\) is a cocycle.
Replacing \(s_g\) by \(a_gs_g\) replaces \(d_g\) by
\(d_g-2a_g\).  Replacing \(z\) by \(az\) replaces \(d_g\) by
\(d_g+(1-g)a\), a coboundary change modulo two.
Choose \(s_c=z\); then \(d_c=0\), and the cocycle descends to
\(\Gamma/\langle c\rangle\), whose action on \(M/2M\) is well
defined.

If its class is zero, change \(z\) so that all \(d_g\) are even,
and then replace \(s_g\) by \((d_g/2)s_g\).  All these lifts commute
with \(z\).  The centralizer \(C_E(z)\) consequently maps onto
\(\Gamma\), and its kernel is \(M^c=0\).  The inverse of this
isomorphism is a splitting.  A splitting plainly gives the zero
class.  Finally, if \(s_r\) is an involutive lift of \(r\), conjugating
its square by \(z\) gives \((1+r)d_r=0\).
Changing the cocycle by a coboundary replaces this lift of its value
by \(d_r+(1-r)a\) for some \(a\in M\), and preserves the identity
because \((1+r)(1-r)a=0\).
\end{proof}

\subsection{The simultaneous norm torus}

\begin{lemma}[Hasse principle and approximation for a norm torus]
\label{lem:norm-torus}
Let \(L/k\) be quadratic, let \(F/k\) be separable of degree
\(m\geq3\), and suppose that the normal closure of \(F\), jointly
with \(L\), has Galois group \(S_m\times C_2\), with its natural
action on the embeddings of \(F\).  Set
\[
 T=\ker\left(
 \Res_{F/k}\bigl(\Res^1_{LF/F}\bbG_m\bigr)
       \xrightarrow{\,N_{LF/L}\,}\Res^1_{L/k}\bbG_m
 \right).
\]
This kernel is a torus.  Every torsor under \(T\) with points
over all completions has a rational point, and its rational points
are dense at any prescribed finite set of completions.
\end{lemma}

\begin{proof}
Let \(c\) denote the generator of the second factor \(C_2\).
On character lattices the norm map is the diagonal inclusion
\[
 \bbZ_{\mathrm{sign}}\longrightarrow
               \bbZ^m\otimes\mathrm{sign}_{C_2}.
\]
It is primitive.  Thus its cokernel is torsion-free, the kernel
of the norm map is connected, and its character lattice is
\begin{equation}\label{eq:foundations-norm-lattice}
       M=(\bbZ^m/\bbZ\mathbf1)\otimes\mathrm{sign}_{C_2}.
\end{equation}
Here \(S_m\) permutes the coordinates and \(c\) acts by \(-1\).
By Lemma~\ref{lem:foundations-cyclic-detection}, it suffices to split
every extension of \(S_m\times C_2\) by \(M\) that splits over
all cyclic subgroups.

Fix such an extension.  Lemma~\ref{lem:foundations-sign-extension}
gives a class
\[
                 \beta\in H^1(S_m,M/2M).
\]
We show that \(\beta=0\).  Put
\[
       V_m=\bbF_2^m,\qquad Q_m=V_m/\bbF_2\mathbf1=M/2M .
\]
The permutation-module sequence
\[
 0\longrightarrow\bbF_2\mathbf1
   \longrightarrow V_m\longrightarrow Q_m\longrightarrow0
\]
has a connecting map
\[
 H^1(S_m,Q_m)\longrightarrow H^2(S_m,\bbF_2).
\]
By Shapiro's lemma, the preceding map on degree-one cohomology
is the restriction
\[
 H^1(S_m,\bbF_2)\longrightarrow H^1(S_{m-1},\bbF_2).
\]
It is an isomorphism for \(m\geq3\): both sides are generated by
the permutation sign character.  Therefore the connecting map
is injective.  Its image consists of classes whose restriction to
the point stabilizer \(S_{m-1}\) is zero.

Let \(\widehat S_m\) be the central extension by \(\bbF_2\)
corresponding to the image of \(\beta\), and denote its nonidentity
central kernel element by \(\zeta\).  It splits on each point
stabilizer, by conjugacy.  Consequently every transposition has
involutive lifts.  For \(m\geq5\), two disjoint transpositions fix
a common point, so their lifts commute.  For \(m=3\) there is no
disjoint pair of Coxeter generators to consider.

In either of these cases, choose involutive lifts \(t_i\) of
the adjacent transpositions \((i,i+1)\).
Write
\[
                 (t_it_{i+1})^3=\zeta^{\epsilon_i},
                       \qquad \epsilon_i\in\bbF_2 .
\]
Multiplying \(t_i\) by \(\zeta^{a_i}\), with
\(a_1=0\) and \(a_{i+1}=a_i+\epsilon_i\), makes all these cubes
equal to \(1\).  It does not change the squares or the commutation
of distant generators.  The Coxeter presentation of \(S_m\)
therefore gives a splitting of \(\widehat S_m\).
By injectivity of the connecting map, \(\beta=0\).

It remains to treat \(m=4\), where disjoint transpositions have
no common fixed point.  The only remaining Coxeter relation is
commutation of lifts of
\(\tau=(12)\) and \(\nu=(34)\).
Fix the sign lift \(z\) in the proof of
Lemma~\ref{lem:foundations-sign-extension}, and use the cocycle
\(g\mapsto d_g\bmod2M\) produced there.  Choose representatives
\(b(g)\in\bbF_2^4\) of its values, with \(b(1)=0\).
Its connecting factor set is given by
\[
 \epsilon(g,h)\mathbf1=b(g)+g b(h)-b(gh).
\]
Since \(\tau\nu=\nu\tau\), failure of their lifts to commute would
give
\begin{equation}\label{eq:foundations-four-obstruction}
       (1+\nu)b(\tau)+(1+\tau)b(\nu)=\mathbf1 .
\end{equation}

We now return to the original extension by the lattice \(M\),
not the central extension \(\widehat S_4\).
Its restriction to \(\langle\tau\rangle\) splits by hypothesis.
Replace the chosen lift of \(\tau\) by an involutive lift.
This changes its translation parameter by twice an element of \(M\),
so the fixed cocycle modulo two is unchanged.  By
Lemma~\ref{lem:foundations-sign-extension}, the associated
translation parameter \(d\in M\) satisfies
\((1+\tau)d=0\), and its reduction represents \(\beta(\tau)\).
Choose an integral representative
\(\widetilde d\in\bbZ^4\).  For some \(a\in\bbZ\) we have
\[
           \widetilde d+\tau\widetilde d=a\mathbf1,
                \qquad 2\widetilde d_3=a=2\widetilde d_4.
\]
Hence \(\widetilde d_3=\widetilde d_4\).
Every representative \(b(\tau)\) has equal third and fourth
coordinates: adding a diagonal vector modulo two preserves that
equality.  The third and fourth coordinates of
\((1+\nu)b(\tau)\) are therefore zero, while those of
\((1+\tau)b(\nu)\) are zero because \(\tau\) fixes them.
This contradicts Equation~\ref{eq:foundations-four-obstruction}.
The disjoint lifts commute after all, and the same Coxeter-generator
adjustment splits \(\widehat S_4\).  Thus \(\beta=0\) also for \(m=4\).

In every degree \(m\geq3\),
Lemma~\ref{lem:foundations-sign-extension} now splits the original
lattice extension.  Lemma~\ref{lem:foundations-cyclic-detection}
gives \(\Sha_\omega^2(k,M)=0\) and the asserted Hasse principle
and weak approximation.
\end{proof}

We have obtained two distinct kinds of control.  The approximation
procedure chooses a representative inside a fixed abstract coset
while controlling its local centralizers.  The norm-torus lemma
then solves the simultaneous norm equations in those centralizers.
The next section uses this combination to construct commuting
rational elements with specified determinants.
==== END The-Margulis-Platonov-conjecture-over-number-fields//sections/foundations.tex ====

==== BEGIN The-Margulis-Platonov-conjecture-over-number-fields//sections/introduction.tex ====
\section{Introduction}\label{sec:introduction}

Let $k$ be a number field and let $G$ be an absolutely almost simple
simply connected algebraic $k$-group. For a place $v$ of $k$, write $k_v$
for the completion. The local group $G(k_v)$ has its usual topology.
Let
\[
 A=A(G,k)=\{v\text{ nonarchimedean}:\rank_{k_v}G=0\},
 \qquad H_A=\prod_{v\in A}G(k_v),
\]
and let $\delta_A:G(k)\to H_A$ be the diagonal homomorphism.
The set $A$ is finite. The product $H_A$ is given its product topology;
if $A$ is empty, it is the trivial group.

The Margulis--Platonov conjecture asserts that the noncentral normal
subgroups of $G(k)$ are accounted for by this finite collection of compact
local groups. Normal subgroups in this assertion are abstract: no topology
or finite-generation condition is imposed on them. We prove the conjecture
in this form.

\begin{theorem}\label{thm:main}
Let $k$ be a number field and $G$ an absolutely almost simple simply
connected algebraic $k$-group. If $N\triangleleft G(k)$ is not contained
in $Z(G(k))$, then there is an open normal subgroup $W\triangleleft H_A$
such that
\[
                         N=\delta_A^{-1}(W).
\]
\end{theorem}

Thus the result resolves the Margulis--Platonov conjecture positively
over number fields, including globally anisotropic groups. In the case
$A=\varnothing$, the conclusion is $N=G(k)$, not merely that $N$ has
finite index or is dense in a local topology.

\subsection{Background and significance}

The problem joins two kinds of rigidity. At isotropic completions,
generation by unipotent subgroups and simplicity modulo the center leave
little room for normal quotients. At anisotropic nonarchimedean
completions, compactness allows finite quotients and their open kernels.
The assertion is that there is no additional abstract normal-subgroup
obstruction invisible to these local factors.

The conjecture grew from Kneser's quaternionic simplicity question and
Platonov's local-to-global simplicity proposal. Margulis formulated the
description by open subgroups of the anisotropic finite local product
\cite[Russian original, Section~2.4.8]{Margulis1979}.
Strong approximation and Margulis's normal subgroup theorem supply the
arithmetic framework and finite-index reduction \cite{PRbook,Margulis}.
The isotropic case follows from the Kneser--Tits theorem over global
fields; Gille's account includes the final outer-$E_6$ cases, using work
of Garibaldi and Chernousov--Timoshenko
\cite[Theorem~8.1]{Gille2009}. For anisotropic groups, Chernousov proved
projective simplicity of the rational points when the absolute rank is
at least two and the group splits over a quadratic extension
\cite[Theorem~1]{Chernousov1990}. Tomanov and Sury established substantial
classical and low-index unitary cases \cite{Tomanov1990,Sury1991}.

For inner type $A$, the arithmetic work of Platonov--Rapinchuk and
Raghunathan \cite{PR1984,Raghunathan1988} preceded the reduction of
Rapinchuk--Potapchik to finite simple quotients of the multiplicative
group of a division algebra \cite{RapinchukPotapchik1996}.
Segev obtained a commuting-graph obstruction, and Segev--Seitz verified
the required finite-simple-group properties, completing this case
\cite{Segev1999,SegevSeitz2002}. Rapinchuk--Segev--Seitz later proved
that every finite quotient of the multiplicative group of a
finite-dimensional division algebra is solvable
\cite{RapinchukSegevSeitz2002}; Rapinchuk gave a shorter proof of the
inner-$A$ case using two-generation of finite simple groups
\cite{Rapinchuk2006}. The established cases are surveyed in
\cite[Sections~3.4--3.5]{PRsurvey}. They include isotropic groups and
all inner forms of type $A$, as well as the other types outside the
remaining anisotropic outer $A$, trialitarian $D_4$, and $E_6$ cases.
We use that reduction in Theorem~\ref{thm:known-mp} and supply the
remaining arguments here. In particular, the previously established
inner-type-$A$ theorem is available over every number field that occurs
in a restriction-of-scalars subgroup.

The conjecture also enters the congruence subgroup problem: it is a
normal-subgroup input to several descriptions of congruence kernels.
That relationship does not justify using a theorem which assumes the
conjecture to prove it. Our central-extension argument instead uses the
unconditional finite-nonarchimedean injectivity in the metaplectic-kernel
computation of Prasad and Rapinchuk \cite[Proposition~2.6]{PRmetaplectic}.
The distinction between metaplectic vanishing and congruence-kernel
conclusions under the normal-subgroup hypothesis is also explicit in
\cite[Section~2]{Rapinchuk2026}.

\subsection{A positive-density congruence consequence}

Theorem~\ref{thm:main} supplies the Margulis--Platonov hypothesis in
Rapinchuk's positive-density theorem, giving the following consequence.

\begin{corollary}[Positive-density congruence subgroup property]
\label{cor:positive-density-csp}
Let $k$ be a number field and $G$ an absolutely almost simple simply
connected algebraic $k$-group. Let $M/k$ be the minimal Galois extension
over which $G$ becomes an inner form of the split group, and let
$\operatorname{Spl}(M/k)$ be the set of nonarchimedean places of $k$ that
split completely in $M$. Suppose that $S$ is a set of places of $k$ which
contains all archimedean places, contains no nonarchimedean place $v$
with $\rank_{k_v}G=0$, and satisfies
\[
 d_k\bigl(S\cap\operatorname{Spl}(M/k)\bigr)>0,
\]
where $d_k$ denotes Dirichlet density and the indicated density is assumed
to exist. Then the $S$-congruence kernel is trivial:
\[
 C^S(G)=\{1\}.
\]
Equivalently, for any fixed faithful $k$-defined representation of $G$,
the associated group $G(\mathcal O(S))$ over the ring of $S$-integers
has the classical congruence subgroup property: every finite-index normal
subgroup contains a principal congruence subgroup of some nonzero ideal
level.
\end{corollary}

\begin{proof}
Theorem~\ref{thm:main} is the Margulis--Platonov statement assumed in
\cite[Theorem~1.1]{Rapinchuk2026}, for the same group and number field.
The other hypotheses are exactly those imposed above, so that theorem
gives $C^S(G)=\{1\}$. The equivalence with the classical congruence
subgroup property is recalled in \cite[Section~2]{Rapinchuk2026}.
\end{proof}

For an inner form of the split group, $M=k$, and the density condition is
positive Dirichlet density of the nonarchimedean places in $S$. The
corollary is restricted to the stated positive-density range: it does not
address Serre's finite-$S$ conjecture, arbitrary infinite $S$, or
centrality or finiteness of congruence kernels outside that range.

\subsection{The finite obstruction}

We first prove that a noncentral normal subgroup $N$ has finite index.
Choose a positive integer $e$ that annihilates the finite group $G(k)/N$,
and let
\[
 R=G(k)^e=\langle g^e:g\in G(k)\rangle,\qquad
 P_0=\overline{\delta_A(R)},\qquad
 V_0=\delta_A^{-1}(P_0).
\]
Here $G(k)^e$ is the subgroup generated by powers, not just the set of
powers. The group $P_0$ is open and normal in $H_A$. It remains to show
that the finite group $V_0/R$ is trivial. This is the distinction between
the desired abstract equality and the closure statement furnished by
approximation.

Section~\ref{sec:foundations} develops two tools for this finite
obstruction. First, it constructs representatives of a fixed coset
modulo $R$ satisfying specified local openings and avoiding a prescribed
codimension-two reduction locus at almost all other places. The parameters
are products of conjugates of torus powers. Scalar invariance permits an
affine sieve while retaining control at the omitted approximation place;
possible single poles are treated explicitly. Second, a signed permutation
lattice calculation gives the Hasse principle and weak approximation for
the norm torus needed to prescribe commuting elements with exact
determinants.

\subsection{Odd-degree division algebras}

The first unresolved case is a special unitary group $S=\SU(D,*)$, where
$D$ is an odd-degree central division algebra over a quadratic extension
$L/k$ and $*$ is a unitary involution. Let $U=\U(D,*)$ be its full
unitary group. Section~\ref{sec:colors} proves that a nontrivial finite
defect for $S(k)$ yields one of two objects on $U(k)$: a finite abelian
character nonzero on $S(k)$, or a homomorphism to a finite nonabelian
simple group which is surjective on the defect preimage $V_0$.
The commuting approximation and central-extension arguments are needed
to obtain this dichotomy on the full unitary group. The class-preserving
extension step adapts the inner-type argument of Rapinchuk--Potapchik
\cite[Section~2]{RapinchukPotapchik1996}; here the commuting approximation
must also prescribe determinants. A quotient of a unitary subgroup need
not extend to the multiplicative group of its algebra.

Section~\ref{sec:characters} rules out the first object. An algebraic
difference function on $D$ has enough exact elementary invariances to be
constant on the required orbits. The exact additive span of $U(k)$ in
$D$, together with the known inner-type-$A$ theorem and a real-signature
interpolation argument, forces any finite abelian character to vanish
on $S(k)$.

Sections~\ref{sec:palettes} and~\ref{sec:finite-groups} rule out the
second object. A finite simple quotient assigns a color to each rational
unitary element. The distribution of these colors is not assumed
continuous. We construct a translation-invariant probability law on
their finite patterns whose individual color distributions are uniform.
Fractional unitary transformations turn additive recurrence into a
centralizer double-coset condition. A finite-simple-group lemma, stated
in Section~\ref{sec:palettes} and proved in
Section~\ref{sec:finite-groups}, excludes that condition. Elementary
transformations then force an impossible invariance under all left
color translations.

The construction of the uniform-marginal law is an important part of the
proof: additive averaging by itself need not preserve all colors.
Conformal weights on rational points of a flag variety use Langlands's
Eisenstein-series convergence theorem \cite{Langlands}. Finite-volume
harmonicity and Howe--Moore matrix-coefficient decay
\cite{HoweMoore,Ciobotaru} then provide the required marginals before
additive averaging is performed. The later positive-density argument
uses the multidimensional theorem of Furstenberg--Katznelson
\cite{FurstenbergKatznelson}. The uniform-marginal construction, the
coset-preserving sieve, and the explicit norm-torus calculation are
developed before the final case reductions so that their hypotheses and
conclusions remain visible.

\subsection{Completion of the proof}

Section~\ref{sec:unitary-induction} proves the other outer-type-$A$
cases by induction on reduced degree, simultaneously over all number
fields. The ternary split-algebra case uses quadratic norm equations on a
Ch\^atelet surface. Even degree is reduced by a quaternion factorization
and a smaller unitary centralizer, with a separate degree-four determinant
correction. The approximation construction controls the finite places of
the subgroups as they vary.

\enlargethispage{\baselineskip}
Section~\ref{sec:d4} treats $D_4$ by finding, in each relevant rational
coset, a regular element whose torus admits a rational inverter. It is a
product of two involutions, each already controlled inside a type-$A_2$
subgroup. Section~\ref{sec:e6} reduces a generic $E_6$ element to a
simply connected $D_4$ subgroup and the cover of a root-involution
centralizer of type $A_1A_5$. The last step is a torsor under a connected
simply connected group, so its local factorizations can be globalized.
The proof ends by returning to the original abstract subgroup $N$.
==== END The-Margulis-Platonov-conjecture-over-number-fields//sections/introduction.tex ====

==== BEGIN The-Margulis-Platonov-conjecture-over-number-fields//sections/palettes.tex ====
\section{Excluding nonabelian simple colors}\label{sec:palettes}

We retain the odd-degree division-algebra hypotheses and notation of
Section~\ref{sec:colors}. Thus $L/k$ is quadratic, $D$ is a central division
$L$-algebra of odd degree $n\geq3$, the involution $*$ induces the nontrivial
automorphism of $L/k$, and $S=\SU(D,*)\subset U=\U(D,*)$.
Fix $i\in L^\times$ with $\bar i=-i$.
For $R=S(k)^e$ let $P_0$ and $V_0$ have the meaning of
Proposition~\ref{prop:finite-defect}. We now eliminate the nonabelian
alternative in Proposition~\ref{prop:color-dichotomy}.

\begin{theorem}\label{thm:no-simple-colors}
There is no homomorphism
\[
 \phi:U(k)\longrightarrow\mathcal F
 \quad\text{with}\quad
 R\subset\ker\phi,\qquad \phi(V_0)=\mathcal F,
\]
where $\mathcal F$ is a finite nonabelian simple group.
\end{theorem}

Suppose, for a contradiction, that such a homomorphism exists.
We call its values \emph{colors}. It is surjective, so central scalars in
$U(k)$ have color $1$: their images centralize $\mathcal F$.
The finite-group input is the following assertion. Its statement is
independent of the arithmetic constructions below.

\begin{lemma}\label{lem:finite-obstruction}
For every finite nonabelian simple group $\mathcal F$ there is a nonempty
proper conjugacy-invariant subset $\mathcal S\subset\mathcal F$ with the
following property. If $W\in\mathcal S$, $Z\notin\mathcal S$, and $WZ=ZW$,
then it is impossible that
\begin{equation}\label{eq:palette-obstruction}
 ZW\in C(B)(ZW^{-1})C(BZ^2)
 \qquad\text{for every }B\in\mathcal F,
\end{equation}
where $C(B)=C_{\mathcal F}(B)$ denotes the centralizer of $B$ in
$\mathcal F$.
\end{lemma}

The proof of Lemma~\ref{lem:finite-obstruction} occupies
Section~\ref{sec:finite-groups}. We use exactly its displayed
double-coset condition. Our task in this section is to force that condition
from sufficiently many additive returns of colors, and then to show that
avoiding it would make membership in $\mathcal S$ constant on
$\mathcal F$.

\subsection{The compact space of color configurations}

Let $\widetilde G$ be the simply connected special unitary group of the
hyperbolic binary hermitian form over $(D,*)$ with coefficients $1,-1$.
Let $X$ be its projective variety of isotropic right $D$-lines.
Since $D$ is division, the form has relative rank one, and the stabilizer
of such a line is a minimal $k$-parabolic of $\widetilde G$.
Every rational isotropic line has a unique representative $(u,1)$ with
$u\in U(k)$: the second entry cannot vanish, and division makes it
invertible. This identifies $X(k)$ with $U(k)$.

There is also an additive parametrization. Put
\[
 J=\{x\in D:x=x^*\},\qquad
 q(x)=(x+i)(x-i)^{-1}.
\]
Here $J$ is a $k$-vector space, and
\begin{equation}\label{eq:palette-cayley}
 X(k)=J(k)\sqcup\{\infty\},\qquad q(\infty)=1.
\end{equation}
Indeed $x-i$ is nonzero for $x=x^*$, and $u-1$ is invertible for every
$u\ne1$ in $U(k)$. The corresponding column on the additive chart is
$(x+i,x-i)$. At a completion $k_v$, these two parametrizations give open
charts isomorphic to $U(k_v)$ and $J(k_v)$, respectively. Their
complements are proper algebraic subvarieties and therefore have measure
zero for any smooth measure on $X(k_v)$. We shall use this assertion at
finitely many completions at a time.

Let $\mathcal H$ be the countable group of rational transformations of
$X$ induced by rational form similitudes. It contains $\widetilde G(k)$,
the left and right rotations
\[
 L_a(u)=au,\qquad R_a(u)=ua\quad(a\in U(k)),
\]
and the additive translations $T_b(x)=x+b$ for $b\in J(k)$.
Write $\tau_s=T_s$ when $s\in k\subset J(k)$.
For $p\in X(k)$ define a configuration
\[
 c(p)=(c_h(p))_{h\in\mathcal H},\qquad
 c_h(p)=\phi(u(hp))\in\mathcal F.
\]
We call this configuration a \emph{palette}, and let
$\mathcal C\subset\mathcal F^{\mathcal H}$ be the closure of the rational
palettes. The product topology makes $\mathcal C$ compact and metrizable.
The action is
\[
 (h c)_j=c_{jh};\qquad c(hp)=h c(p).
\]
Every pattern on finitely many coordinates of a palette in $\mathcal C$
occurs at a rational point, because the colors form a finite discrete
space. In particular, all finite-coordinate identities satisfied by
rational palettes hold throughout $\mathcal C$.

\subsection{A translation-invariant law with uniform marginals}

The next proposition supplies additive recurrence without losing any
color. Arbitrary additive averages of rational palettes need not have
uniform marginals, so we first construct a measure with controlled
conditional laws over the local flag varieties.

\begin{proposition}\label{prop:palette-measure}
There is a probability measure $\mu$ on $\mathcal C$ invariant under
$T_b$ for every $b\in J(k)$ such that
\[
 \mu\{c:c_h=a\}=\frac1{|\mathcal F|}
 \qquad(h\in\mathcal H,\ a\in\mathcal F).
\]
\end{proposition}

\begin{proof}
We construct a conformal joint law, identify its local projection, prove
that its conditional color marginals are uniform, and only then average
additively.

\paragraph{A joint law over all completions.}
For every place $v$, choose an Iwasawa maximal compact subgroup
$K_v\subset\widetilde G(k_v)$ and its invariant probability $m_v$ on
$X(k_v)$. The compact group is transitive on this flag variety.
Choose the good integral compacts at almost all finite places, and set
\[
 Y=\prod_v X(k_v),\qquad m=\prod_v m_v.
\]
Each $m_v$ is smooth and has full support. A fixed $h\in\mathcal H$
has smooth positive Jacobians on all local factors and preserves $m_v$
outside finitely many places. For the latter assertion, discard the
finitely many denominators and nonunit similitude multipliers of a
representative of $h$. At every remaining place it normalizes the good
integral compact of $\widetilde G$; its pushforward of $m_v$ is consequently
the same compact-invariant probability. Thus
\[
 \rho(h,y)=\frac{d(h_*m)}{dm}(y)
\]
is a continuous positive function on $Y$, given by a finite product, and
\begin{equation}\label{eq:palette-cocycle}
 \rho(ag,y)=\rho(a,y)\rho(g,a^{-1}y).
\end{equation}

Assign positive weights $w(p)$ to rational points by requiring the ratio
$w(hp)/w(p)$ to be the product of the forward local volume Jacobians of
$h$ at $p$. To construct them, normalize the weight at one rational
base point and transport by $\widetilde G(k)$. Rational conjugacy of
parabolics of the given type gives transitivity
\cite[Theorem~4.13(a,c)]{BorelTits1965}. If two transporters
give the same point, their quotient fixes the base point; the product
of its tangent determinants is $1$ by the product formula in $k$.
This proves independence of the transporter. The same reasoning at
rational fixed points proves the weight-ratio rule for every
$h\in\mathcal H$, not just for $\widetilde G(k)$.

We shall use
\begin{equation}\label{eq:palette-weight-sum}
 \sum_{p\in X(k)}w(p)^s<\infty\qquad(s>1).
\end{equation}
Here is the convergence theorem and normalization involved. Let $P$ be
the stabilizing minimal $k$-parabolic. For a local decomposition
$g=k_0p_0$ with $k_0\in K_v$ and $p_0\in P(k_v)$, the forward tangent
Jacobian at the base point is $\delta_P(p_0)^{-1}$, where $\delta_P$ is
the modular character of $P$. After replacing $g$ by $g^{-1}$, the
sum in Equation~\ref{eq:palette-weight-sum} is the spherical
minimal-parabolic Eisenstein sum with inducing character $\delta_P^s$.
The derived Levi is $k$-anisotropic, since $P$ is minimal and
$\widetilde G$ has relative rank one. More precisely, its Levi subgroup
$M$ is $k$-anisotropic modulo its maximal $k$-split central torus, so
$M(k)\backslash M(\bbA_k)^1$ is compact, where the superscript $1$
imposes adelic norm one for every $k$-rational character of $M$
\cite[Theorem~5.6]{PRbook}.
The constant inducing function is therefore square-integrable modulo
the split center and cuspidal, since $M$ has no proper $k$-parabolics.
The usual absolute-convergence
theorem applies in the Godement domain
$\operatorname{Re}\lambda>\rho_P$ for normalized parameter
$\rho_P+\lambda$, where $2\rho_P$ is the sum of the positive relative
roots with multiplicities; see
\cite[Chapter~4, Lemma~4.1, and Appendix~II]{Langlands}.
For $\delta_P^s$ this parameter is
$\lambda=(2s-1)\rho_P$, and the domain is exactly $s>1$.
We use the number-field adelic formulation, equivalently the arithmetic
formulation after restriction of scalars and finite adelic double-coset
decomposition.

Give $(p,c(p))\in Y\times\mathcal C$ mass proportional to $w(p)^s$,
and denote this probability by $\mu_s$. Its transformation rule is
\[
 \frac{d(h_*\mu_s)}{d\mu_s}(y,c)=\rho(h,y)^s.
\]
The reciprocal in this formula is worth noting: the mass at $hp$ after
pushforward is the old mass at $p$, so the forward Jacobian is inverted.
Take a weak limit $\mu^0$ as $s\downarrow1$. Compactness gives such a
limit, and uniform convergence of $\rho(h,\cdot)^s$ for each fixed $h$
gives
\begin{equation}\label{eq:palette-joint-conformal}
 h_*\mu^0=\rho(h,\cdot)\mu^0.
\end{equation}

\paragraph{The base projection is the product measure.}
Let $\nu$ be the projection of $\mu^0$ to $Y$. We prove $\nu=m$.
Fix a sufficiently large finite set $T$ of places containing the
archimedean and model-exceptional places. Use subscripts $T$ for products
over $T$, and superscripts $T$ for products over its complement.
An arithmetic lattice $\Gamma\subset\widetilde G_T$, integral outside
$T$, preserves the tail probability. Equation~\ref{eq:palette-joint-conformal}
therefore gives
\[
 \gamma_*\nu_T=\rho_T(\gamma,\cdot)\nu_T
 \qquad(\gamma\in\Gamma).
\]
Define the continuous positive function
\[
 f(g)=\int_{Y_T}\rho_T(g,y)\,d\nu_T(y)
 \qquad(g\in\widetilde G_T).
\]
The cocycle identity and the preceding transformation rule give
$f(\gamma g)=f(g)$.

Choose a full-support probability $\eta$ on $\widetilde G_T$ that is
left invariant by $K_T=\prod_{v\in T}K_v$. Compact transitivity and
$\int\rho_T(h,y)\,dm_T(y)=1$ imply
\[
 \int\rho_T(h,z)\,d\eta(h)=1\quad(z\in Y_T).
\]
Indeed average first over left multiplication of $h$ by $K_T$; the
result averages $\rho_T(h,\cdot)$ over $m_T$.
Equation~\ref{eq:palette-cocycle} now gives
\[
 f(g)=\int f(gh)\,d\eta(h).
\]
This is a positive harmonic function on the finite-volume space
$\Gamma\backslash\widetilde G_T$.
To see it is constant without an unproved integrability assertion, apply
Jensen's inequality to the bounded strictly concave function
$q(t)=t/(1+t)$ for $t>0$. Its nonnegative Jensen discrepancy has integral
zero over the quotient, by right invariance of the quotient volume.
Equality in strict Jensen makes $f(gh)$ constant for $\eta$-almost all
$h$, for almost every coset $g$. Full support of $\eta$ and continuity
then make $f$ constant everywhere. Its value is $f(1)=1$.

We still have to show that these integrals determine $\nu_T$.
Choose a sequence $a_j\in\widetilde G_T$ contracting the big cell of
$Y_T$ to a point $x_0$, using a split cocharacter for the parabolic at
each place. The complement of that cell has $m_T$-measure zero, so
$(a_j)_*m_T\to\delta_{x_0}$. For $b\in C(Y_T)$ put
\[
 B_j(y)=\int_{K_T}b(k_0x_0)\rho_T(k_0a_j,y)\,dk_0.
\]
Under the probability density $\rho_T(k_0a_j,y)\,dk_0$, the variable
$k_0^{-1}y$ has distribution $(a_j)_*m_T$: use compact invariance and
push Haar measure on $K_T$ onto $Y_T$. Compact equicontinuity shows that
$B_j(y)\to b(y)$ uniformly in $y$. On the other hand, $f(k_0a_j)=1$
gives
\[
 \int B_j\,d\nu_T
 =\int_{K_T}b(k_0x_0)\,dk_0
 =\int b\,dm_T.
\]
It follows that $\nu_T=m_T$. Exhausting the places proves $\nu=m$.

Disintegrate the joint law as
\begin{equation}\label{eq:palette-disintegration}
 d\mu^0(y,c)=dQ_y(c)\,dm(y).
\end{equation}
The spaces are compact metrizable, so regular conditional probabilities
$Q_y$ exist. Equation~\ref{eq:palette-joint-conformal} gives
$Q_{hy}=h_*Q_y$ almost everywhere for each $h$. Since $\mathcal H$ is
countable, all these equalities may be imposed on a common conull set.

\paragraph{Uniform conditional color marginals.}
Write $P(y)$ for the probability vector on $\mathcal F$ which is the
$c_1$-marginal of $Q_y$, and put $N=\ker\phi\cap S(k)$.
The exact palette identities
\[
 c_{L_a h}=\phi(a)c_h,\qquad
 c_{R_a h}=c_h\phi(a)
\]
imply that $P$ is invariant under the separate left and right rotations
by every $a\in N$. We first upgrade this invariance at one completion,
using a finite-volume unitary representation, and then use approximation
away from that completion. No topology is placed on the abstract map
$\phi$.

Enlarge $T$ so that it contains a finite place $v_0$ where $S$ has
positive rank. On the $u_T$-chart, $m_T$ is in the Haar measure class
of $U_T$. Disintegrate this class along the determinant map. On a fiber
of determinant $\lambda$ choose a measurable representative
$d_\lambda\in U_T$ and write
\[
 u_T=g d_\lambda,\qquad g\in S_T.
\]
The determinant map is a smooth group map onto its image, so the
conditional measure class along its fibers is Haar measure on $S_T$.
For almost every $\lambda$, the function $P(gd_\lambda,y^T)$ is invariant
under the diagonal left action of
\[
 \Gamma'=N\cap\{a\in S(k):a_v\text{ is integral for }v\notin T\}.
\]
This is a finite-index subgroup of an arithmetic lattice. Consequently
each component of $P$ defines a bounded function on
\begin{equation}\label{eq:palette-lattice-space}
 \Gamma'\backslash(S_T\times Y^T).
\end{equation}
The space has finite invariant measure, obtained from Haar measure on
the group factor and $m^T$ on the tail. For example, a measurable
fundamental domain for $\Gamma'$ in $S_T$, together with the tail,
constructs this measure. Left integrality ensures that the tail action
preserves $m^T$.

Right invariance by $a\in\Gamma'$ acts on
Equation~\ref{eq:palette-lattice-space} by
\[
 (g,y^T)\longmapsto
 (g d_\lambda a_Td_\lambda^{-1},\ R_a y^T).
\]
These actions commute with the diagonal left action. The right tail
rotations lie in a compact group of measure-preserving transformations,
since all their local coordinates are integral outside $T$.

There is a sequence $a_j\in\Gamma'$ escaping every compact set at
$v_0$ and bounded in the other factors of $S_T$. To obtain it, use the
finite-index closure of $N$ and strong approximation off $v_0$ to find
infinitely many elements in a fixed compact cylinder at all the other
places. The full arithmetic diagonal is discrete, so a subsequence
must escape at $v_0$. Pass to a subsequence on which the bounded factors
and the compact tail rotations converge. The fixed conjugations by
$d_\lambda$ preserve these escape and convergence properties.

Let $\mathcal L$ be the $L^2$ space of
Equation~\ref{eq:palette-lattice-space}, and decompose it into the
$S(k_{v_0})$-invariant vectors and their orthogonal complement.
Howe--Moore decay applies on the latter: $S(k_{v_0})$ is an isotropic
simple local group, and local Kneser--Tits identifies it with the group
generated by its root groups; see \cite{HoweMoore} or
\cite[Theorem~4.19 and Definition~4.23]{Ciobotaru}.
For completeness, let $v$ be the projection of a component of $P$ to
this complement. The simultaneous right actions fixing $P$ also fix
$v$. They are products $\pi(g_j)A_j$, where $g_j\to\infty$ in
$S(k_{v_0})$ and the commuting unitary operators $A_j$ converge strongly
to a unitary operator $A$. Therefore
\[
 \|v\|^2
 =\langle\pi(g_j)A_jv,v\rangle
 =\langle\pi(g_j)Av,v\rangle+o(1)\longrightarrow0.
\]
Thus $P$ is invariant under the full right $S(k_{v_0})$ action.
All descent and invariance statements above hold on almost every
determinant fiber by ordinary measure disintegration and countability
of the rational groups. Strong continuity of the local and compact
actions follows first on continuous compactly supported functions and
then on $L^2$.

Full right $S(k_{v_0})$-invariance makes $P(y)$ depend on $u_{v_0}$
only through its determinant, up to null sets. We now return to the
left invariance under $N$. Take $a\in V_0$. By
Proposition~\ref{prop:finite-defect}, there are $r_j\in R\subset N$
converging to $a$ in the restricted adelic product away from $v_0$.
On the space obtained by forgetting the determinant-one variable at
$v_0$, the left rotations $L_{r_j}$ converge to $L_a$. This convergence
passes to bounded measurable functions in the required measure class:
restricted adelic convergence places all sufficiently late $r_j$ and
$a$ in the same integral compacts outside one fixed finite set; those
tail actions preserve measure. At the finitely many remaining local
factors, the smooth actions converge and their Radon--Nikodym densities
are uniformly bounded. Approximate a bounded function in $L^1$ by
continuous cylinder functions to pass invariance to the limit.
The determinant at $v_0$ is unchanged by every element of $S$, so it
causes no additional action on this reduced space.
Since $P$ is invariant under each $L_{r_j}$, it follows that
$P(L_a y)=P(y)$ almost everywhere for every $a\in V_0$.

Conditional equivariance and the left-rotation palette identity also give
$P(L_a y)=\phi(a)P(y)$. The image of $V_0$ is all of
$\mathcal F$, so $P(y)$ is the uniform probability almost everywhere.
For arbitrary $h$, the equality $Q_{hy}=h_*Q_y$ identifies the
$c_h$-marginal at $y$ with the $c_1$-marginal at $hy$. Nonsingularity
therefore makes every conditional coordinate marginal uniform.

\paragraph{Additive averaging.}
We have obtained all colors with equal conditional probability, but
the joint law is still conformal rather than translation invariant.
On the additive chart over a finite set $T$, replace $m_T$ in
Equation~\ref{eq:palette-disintegration} by additive Haar measure on
$J_T$, keeping $Q_y$ and the tail base $m^T$. The resulting
$\sigma$-finite joint measure is invariant under every rational
translation $T_b$ integral outside $T$: Haar measure is translation
invariant on $J_T$, the good integral translations preserve $m^T$, and
the conditional laws are equivariant.
Restrict to expanding Folner windows in $J_T$, divide by their Haar
volumes, and project to $\mathcal C$. Any weak limit is invariant under
all these translations. Every averaging measure already has uniform
coordinate marginals, because the conditional marginals were uniform.
Finally exhaust the places by such finite sets $T$ and take another
weak limit. Every $b\in J(k)$ is integral off a sufficiently large
$T$, so this limit has all the asserted invariances and marginals.
\end{proof}

\subsection{Fractional transformations and simultaneous color returns}

Fix the conjugacy-invariant subset $\mathcal S$ supplied by
Lemma~\ref{lem:finite-obstruction}. We shall relate the colors of $w$ and
$\tau_s w$ to returns under a quadratic additive translation.
For $w\in U(k)$ and $s\in k$, put
\begin{equation}\label{eq:palette-boost-data}
 t=\frac{s}{s-2i},\qquad C=tw,\qquad
 R_C=(1-C^*)(1-C)^{-1},\qquad z=wR_C.
\end{equation}
These expressions are defined for every $s$: $i\notin k$, and
\[
 d:=1-t\bar t=\frac{-4i^2}{s^2-4i^2}\ne0.
\]
The elements $C$ and $C^*$ commute, and $CC^*=t\bar t$ is central.
Division and $d\ne0$ imply that $1-C$ and $1-C^*$ are invertible.
It follows that $R_C$ is unitary and commutes with $w$. Hence $z$ is
unitary and commutes with $w$. Direct Cayley substitution gives
\begin{equation}\label{eq:palette-z-color}
 z=(w-\bar t)(1-tw)^{-1}
   =\frac{2i-s}{2i+s}\,\tau_s(w).
\end{equation}
The scalar factor is central and has norm one, so
$\phi(z)=\phi(\tau_s(w))$.

For such a $C$, the fractional transformation
\[
 D_C(u)=(u+C)(C^*u+1)^{-1}
\]
is induced by a form similitude: its matrix
$\left(\begin{smallmatrix}1&C\\C^*&1\end{smallmatrix}\right)$ has
multiplier $1-t\bar t$. The denominator is invertible for unitary $u$;
otherwise it vanishes, which would force $t\bar t=1$.
Writing $u=wa$ shows that
\begin{equation}\label{eq:palette-boost-commutes}
 w^{-1}D_C(u)=(a+t)(\bar t a+1)^{-1}
 \quad\text{commutes with }a=w^{-1}u.
\end{equation}
This calculation does not commute $u$ with $w$.

Starting from a variable $u\in U(k)$, define
\[
 v=D_{C^*}(uR_C),\qquad u'=R_C D_{-C}(v).
\]
Apply Equation~\ref{eq:palette-boost-commutes} first with unitary
parameter $w^{-1}$ and scalar $\bar t$. It gives
$[wv,wuR_C]=1$; conjugating by $w^{-1}$ gives $[vw,uz]=1$.
Applying the same equation with scalar $-t$ gives
$[w^{-1}v,z^{-1}u']=1$. Thus, writing
\[
 W=\phi(w),\quad Z=\phi(z),\quad
 X_1=\phi(u),\quad Y_1=\phi(v),\quad X_1'=\phi(u'),
\]
we have
\begin{equation}\label{eq:palette-two-commutators}
 [Y_1W,X_1Z]=1,\qquad [W^{-1}Y_1,Z^{-1}X_1']=1.
\end{equation}

The transformation $u\mapsto u'$ is an additive translation. Indeed its
fractional matrix is
\begin{align*}
 &\diag(R_C,1)
 \begin{pmatrix}1&-C\\-C^*&1\end{pmatrix}
 \begin{pmatrix}1&C^*\\C&1\end{pmatrix}
 \diag(1,R_C^{-1})\\
 &\hspace{35mm}=
 \begin{pmatrix}d+C-C^*&-(C-C^*)\\ C-C^*&d-(C-C^*)\end{pmatrix}.
\end{align*}
After dividing by $d$ and applying it to the column $(x+i,x-i)$,
the result is translation by $2i(C-C^*)/d$. Expanding the latter gives
\begin{equation}\label{eq:palette-quadratic-translation}
 b_w(s)=-(w+w^*)s-\frac{w-w^*}{2i}s^2\in J(k).
\end{equation}
The matrix calculation includes the point $\infty$ of the additive
chart. Neither $w$ nor the variable point is required to have a finite
Cayley coordinate.

Suppose that, for this fixed translation $T_{b_w(s)}$, every color
occurs as $X_1=X_1'$ at some rational point; different colors may use
different points. Put $B=Z^{-1}X_1$. Equations~\ref{eq:palette-two-commutators}
then imply
\[
 A:=W^{-1}Y_1\in C(B),\qquad
 D_0:=Z^{-1}Y_1WZ\in C(BZ^2).
\]
Since $WZ=ZW$, direct multiplication gives
\[
 A^{-1}(ZW^{-1})D_0=ZW.
\]
As $X_1$ ranges over $\mathcal F$, so does $B$, and this is exactly
Equation~\ref{eq:palette-obstruction}. Consequently
\begin{equation}\label{eq:palette-return-obstruction}
 \begin{gathered}
 \phi(w)\in\mathcal S,\quad\phi(\tau_s w)\notin\mathcal S\quad\Longrightarrow\\
 \text{not every color can recur under }T_{b_w(s)}.
 \end{gathered}
\end{equation}
When $s=0$, the translation is the identity and $Z=W$, so this implication
has no degenerate exception.

\subsection{Uniform finite witnesses for recurrence}

The translation-invariant law of Proposition~\ref{prop:palette-measure} will now
force simultaneous returns. Uniformity is essential: it lets us pass
the resulting restrictions from rational palettes to their closure.
A subset $D_0$ of the additive group $k$ is called \emph{thick} if it
contains a translate of every finite subset of $k$.

\begin{lemma}\label{lem:palette-uniform-recurrence}
Let $0<\delta\leq1$.
\begin{enumerate}[label=\textup{(\roman*)}]
\item There is an integer $r=r(\delta)\geq2$ such that, for every
probability-preserving action $T$ of $\bbZ^2$ and every event $E$ of
probability $\delta$, some $1\leq j\leq r$ satisfies
\[
 \mathbb P(E\cap T_{(j,j^2)}^{-1}E)>0.
\]
\item For every thick set $D_0\subset k$, there is a finite subset
$F_0\subset D_0$, depending only on $D_0,k,\delta$, such that for every
probability-preserving action $T$ of the discrete group $k^2$ and every
event $E$ of probability $\delta$, some $s\in F_0$ satisfies
\[
 \mathbb P(E\cap T_{(s,s^2)}^{-1}E)>0.
\]
\end{enumerate}
\end{lemma}

\begin{proof}
For an action of either abelian group, the spectral measure $\sigma$ of
$1_E$ is a positive measure on the compact character group, of total
mass $\delta$, with
\begin{equation}\label{eq:palette-spectral-atom}
 \sigma(\{1\})=\|\operatorname{proj}_{\rm inv}1_E\|_2^2
 \geq\delta^2.
\end{equation}
Its Fourier coefficient at $a$ is the return correlation
$\mathbb P(E\cap T_a^{-1}E)$, in particular a nonnegative real number.
The atom lower bound is preserved under weak limits: for the closed
singleton $\{1\}$ the limit mass is at least the upper limit of the
approximating masses.

For \textup{(ii)}, suppose no finite $F_0$ works. Enumerate $D_0$ and
choose a counterexample to each initial finite set. Compactness of the
spectral measures gives a limit $\sigma$ satisfying
Equation~\ref{eq:palette-spectral-atom} and having Fourier coefficient
zero at $(s,s^2)$ for every $s\in D_0$.
Take finite Folner sets $F_j\subset(k,+)$ and translates
$E_j=t_j+F_j\subset D_0$. We show that the average over $E_j$ of
\[
 f(s)=\chi_1(s)\chi_2(s^2)
\]
tends to zero for every nontrivial character pair $(\chi_1,\chi_2)$.
If $\chi_2=1$, this is ordinary averaging of a nontrivial linear
character. Such averaging is uniform in $t_j$, since a translate only
multiplies the average by a scalar of absolute value one.
If $\chi_2\ne1$, choose $M$ distinct shifts $a_1,\ldots,a_M\in k$.
Folner invariance allows replacement of the average of $f$ by the
average of $M^{-1}\sum_\ell f(s+a_\ell)$, with error tending to zero
independently of the translate. Cauchy--Schwarz, followed by expansion,
therefore gives
\begin{align*}
 \limsup_j\left|\frac1{|E_j|}\sum_{s\in E_j}f(s)\right|^2
 &\leq\frac1M+
 \frac1{M^2}\sum_{\ell\ne q}
 \limsup_j\left|\frac1{|E_j|}\sum_{s\in E_j}
 \chi_2(2(a_\ell-a_q)s)\right|\\
 &=\frac1M.
\end{align*}
All off-diagonal characters are nontrivial because multiplication by
$2(a_\ell-a_q)\ne0$ is onto $k$. Letting $M\to\infty$ proves the
claim. Dominated convergence against $\sigma$ now says that the average
of its coefficients at $(s,s^2)$ over $E_j$ tends to
$\sigma(\{1\})>0$, contradicting their vanishing.

For \textup{(i)}, failure of every bound $r$ similarly gives a measure
$\sigma$ on $\bbR^2/\bbZ^2$ satisfying
Equation~\ref{eq:palette-spectral-atom} and having coefficient zero at
every $(j,j^2)$ with $j\geq1$. Write a character phase as
$\exp(2\pi\mathrm i(\alpha j+\beta j^2))$, where here $\mathrm i$
is the usual complex imaginary unit. Choose finitely many pairs with
both $\alpha,\beta$ rational, including the trivial pair, so that the
remaining rational-pair mass is less than
$\varepsilon<\delta^2$. Choose a positive integer $M$ divisible by
their denominators, for example a sufficiently large factorial.
Along $j=M\ell$, all selected phases are $1$. Every pair with at least
one irrational coefficient has average zero by the degree-two form of
Weyl's theorem \cite{Weyl}. For the unselected rational pairs the
absolute contribution is bounded by $\varepsilon$; their individual
averages exist by periodicity. Dominated convergence makes the limiting
average at least $\delta^2-\varepsilon>0$, again a contradiction.
Increasing the resulting bound to at least $2$ is harmless.
\end{proof}

Apply the lemma to independent copies of $(\mathcal C,\mu)$ indexed by
$a\in\mathcal F$, and to the event
\[
 E=\{(c^{(a)})_{a\in\mathcal F}:c^{(a)}_1=a
       \text{ for every }a\in\mathcal F\}.
\]
Its probability is the fixed number
$\delta=|\mathcal F|^{-|\mathcal F|}>0$.
Writing $b_w(s)=sA_w+s^2B_w$ as in
Equation~\ref{eq:palette-quadratic-translation}, use the $k^2$-action
\[
 (a,b)\longmapsto T_{aA_w+bB_w}
\]
on this product space. Positive return of $E$ yields every recurrent
color in some palette. Each required two-coordinate pattern then
occurs rationally, so Equation~\ref{eq:palette-return-obstruction}
applies. For part~\textup{(i)}, and any $u_0\in k$, use instead the
$\bbZ^2$-action
\[
 (a,b)\longmapsto T_{a u_0A_w+b u_0^2B_w}.
\]
We have obtained two statements uniform in $w$:
\begin{enumerate}[label=\textup{(\alph*)}]
\item for every $u_0\in k$, some $s\in\{u_0,2u_0,\ldots,ru_0\}$
gives simultaneous recurrence of all colors;
\item for every fixed thick $D_0\subset k$, a fixed finite subset of
$D_0$ contains such an $s$ for every $w$.
\end{enumerate}
The second finite subset is not asserted to be uniform over thick sets.

\subsection{Additive invariance of membership in the finite subset}

Fix $c\in\mathcal C$ and a prefix $h\in\mathcal H$, and set
\[
 C_0=\{x\in k:c_{\tau_xh}\notin\mathcal S\}.
\]
If $x\notin C_0$, then $C_0-x$ cannot contain
$\{u_0,2u_0,\ldots,ru_0\}$ for any $u_0\in k$.
Indeed, match the finitely many relevant coordinates by a rational
palette and apply statement~\textup{(a)} to its rational point
$w=u(\tau_xhp)$. This contradicts
Equation~\ref{eq:palette-return-obstruction}.
Nor can $C_0-x$ be thick: apply statement~\textup{(b)} to that particular
thick set and match the finitely many witness coordinates by a rational
palette. The same contradiction results.

We record explicitly the combinatorial consequence:
\begin{equation}\label{eq:palette-density-alternative}
 C_0=k\quad\text{or}\quad C_0\text{ has zero upper Banach density}.
\end{equation}
For this purpose the upper Banach density of $C\subset k$ is the
supremum of $M(1_C)$ over translation-invariant means $M$ on bounded
functions on the discrete additive group $k$. Such means exist because
this group is abelian. If this density is zero, the density of $C$ along
every Folner sequence tends to zero: a nonzero upper limit would yield
an invariant mean with positive value by taking a weak limit of the
averaging functionals.

First, a run of $r$ consecutive points of $C_0$ in a direction $u_0$
extends to the adjacent point on either side. Otherwise that adjacent
point, lying outside $C_0$, would have the forbidden run immediately
ahead in direction $u_0$ or $-u_0$. Iteration fills the entire integer
progression.

Suppose now that $C_0$ has positive upper Banach density. To prove it
thick, fix any finite list $g_1,\ldots,g_d\in k$, and choose an invariant
mean $M$ with $M(1_{C_0})>\eta>0$. The finite multidimensional
Szemeredi theorem supplies a box size $N_0$ such that every subset of
$\{1,\ldots,N_0\}^d$ of density at least $\eta$ contains an affine
homothetic copy of $\{0,\ldots,r-1\}^d$, with positive integer dilation
$\ell\leq N_0$. This is the finite-box form of the density theorem
of \cite[Theorem~B]{FurstenbergKatznelson}.
Put $L_0=\operatorname{lcm}(1,\ldots,N_0)$ and average, with respect to
$M$, the density of the pullback of $C_0$ under
\[
 (m_1,\ldots,m_d)\longmapsto
 x+\sum_{j=1}^d m_jg_j/L_0.
\]
Translation invariance makes this average $M(1_{C_0})>\eta$, so one
translate has pullback density at least $\eta$. Injectivity of this
box map is not required. The resulting homothetic $r$-box lies in
$C_0$, with steps $\ell g_j/L_0$. Extend progressions successively along
each coordinate axis. This fills the whole integer grid with these
steps. Because $\ell$ divides $L_0$, that grid contains a translate of
$\{0,g_1,\ldots,g_d\}$. Thus $C_0$ is thick. If any $x$ were outside
$C_0$, then $C_0-x$ would also be thick, contrary to the preceding
restriction. This proves Equation~\ref{eq:palette-density-alternative}.

Now let the palette be random with law $\mu$. For a fixed Folner
sequence, Equation~\ref{eq:palette-density-alternative} makes the
densities of $C_0$ converge pointwise to $1_{\{C_0=k\}}$. Uniform
coordinate marginals and dominated convergence give
\[
 \mathbb P(C_0=k)=\frac{|\mathcal F\setminus\mathcal S|}{|\mathcal F|}.
\]
For each $s$, the event $C_0=k$ is contained in $s\in C_0$, and the
latter has exactly this same probability. Therefore
\begin{equation}\label{eq:palette-indicator-translation}
 1_{\mathcal S}(c_{\tau_sh})=1_{\mathcal S}(c_h)
 \quad\text{almost surely for every }s\in k,\ h\in\mathcal H.
\end{equation}
Countability makes these identities simultaneous for all $s,h$.
The palette rules also give, for every prefix $h$, invariance of this
indicator under left rotation by any element of $\ker\phi$ and under
conjugation by any element of $U(k)$, since $\mathcal S$ is
conjugacy-invariant. These are exact finite-coordinate identities, so
they hold throughout $\mathcal C$.

We have reached a concrete restriction on a common full-measure set of
palettes: membership of a color in $\mathcal S$ is unchanged under
scalar additive translations, kernel left rotations, and all unitary
conjugations, at every prefix. It remains to show that these
transformations already force invariance under all left colors.

\subsection{Generating left rotations by squares}

Let $\mathcal K$ be the subgroup of $\mathcal H$ generated by the three
classes of transformations just listed. Equation~\ref{eq:palette-indicator-translation}
and the simultaneous prefix identities show that
\begin{equation}\label{eq:palette-indicator-generated}
 1_{\mathcal S}(c_{ah})=1_{\mathcal S}(c_h)
 \qquad(a\in\mathcal K,\ h\in\mathcal H)
\end{equation}
on one set of full measure. This assertion is obtained by composing
finite-coordinate identities; it does not assume that $\mu$ is
invariant under the whole group $\mathcal K$.

Fix $\xi\in U(k)$. The commutative algebra $E=L(\xi)$ is a field, since
$D$ is division, and it is stable under $*$ because $\xi^*=\xi^{-1}$.
Let $E_0$ be its fixed field, so $E=LE_0$ and
$[E:E_0]=2$. Write $E^1$ for its norm-one group to $E_0$.
Choose $n_0\in E^1\cap\ker\phi$ whose finite Cayley coordinate
$x_0\in E_0$ generates $E_0$ over $k$. Such a choice is possible:
the rational norm-one torus is Zariski dense by its Cayley
parametrization, the power map on that torus is dominant, and powers
of an exponent killing $\mathcal F$ lie in $\ker\phi$. The requirement
that $x_0$ be finite and primitive is a nonempty Zariski-open condition.

Since there are only finitely many colors, choose $t_0\in k$ for which
the set
\[
 I=\{s\in k:\phi(\tau_sn_0)=\phi(\tau_{t_0}n_0)\}
\]
is infinite. For each $s\in I$, the word
\begin{equation}\label{eq:palette-affine-word}
 H_s=L_{n_0}^{-1}\tau_s^{-1}
 L_{(\tau_sn_0)(\tau_{t_0}n_0)^{-1}}
 \tau_{t_0}L_{n_0}
\end{equation}
belongs to $\mathcal K$. Both indicated left rotations have kernel
parameters. The word successively sends $1$ to $n_0$, to
$\tau_{t_0}n_0$, to $\tau_sn_0$, to $n_0$, and back to $1$.
It is represented by a fractional matrix over $E$; fixing $1$ in the
unitary chart means fixing infinity in the additive chart. Its latter
matrix is therefore upper triangular, say
\[
 \begin{pmatrix}\alpha&\gamma\\0&\beta\end{pmatrix},
 \qquad\alpha,\beta\in E^\times.
\]
On the commutative $E_0$-chart its affine multiplier is
\begin{equation}\label{eq:palette-affine-multiplier}
 a_s=\frac{\alpha}{\beta}
     =\frac{(x_0+t_0)^2-i^2}{(x_0+s)^2-i^2}\in E_0^\times.
\end{equation}
To check the multiplier, use
$q'(x)/q(x)=-2i/(x^2-i^2)$ in
Equation~\ref{eq:palette-affine-word}. The derivative at $u=1$ is
$((x_0+s)^2-i^2)/((x_0+t_0)^2-i^2)$; the affine multiplier at infinity
is its reciprocal. All denominators are nonzero because
$x_0+s\in E_0$ whereas $i\notin E_0$.

This commutative calculation produces transformations on the full
noncommutative chart, not merely on $E_0$. Conjugating $T_{s'}$ by the
upper triangular matrix gives the matrix
\[
 \begin{pmatrix}1&\alpha s'\beta^{-1}\\0&1\end{pmatrix}
 =\begin{pmatrix}1&s'a_s\\0&1\end{pmatrix}
 \qquad(s'\in k),
\]
since $s'$ is central. Thus $\mathcal K$ contains translations by every
$k$-multiple of $a_s$, and hence by their $k$-linear span.

That span is all of $E_0$. Indeed suppose a nonzero $k$-linear
functional on $E_0$ annihilated all the $a_s$ with $s\in I$.
Extend scalars to an algebraic closure and write this functional as a
linear combination of the distinct embeddings of $E_0$, which may be
extended while fixing $L$. Put $x_j$ for the corresponding distinct
conjugates of $x_0$. We obtain a linear combination, zero for infinitely
many $s$, of the rational functions
\begin{equation}\label{eq:palette-reciprocal-quadratics}
 \frac{(x_j+t_0)^2-i^2}{(x_j+s)^2-i^2}.
\end{equation}
It must vanish identically. These functions are linearly independent.
Their poles are $-x_j+i$ and $-x_j-i$. Poles from distinct denominators
can coincide only when the corresponding centers differ by $2i$ or
$-2i$. The resulting finite components form chains: each center has at
most one successor and predecessor, and characteristic zero excludes
cycles. An endpoint pole belongs to just one denominator, so its
coefficient in any linear relation is zero. Remove that denominator
and repeat along the chain. All numerators in
Equation~\ref{eq:palette-reciprocal-quadratics} are nonzero, so every
coefficient vanishes, the required contradiction. Consequently
$\mathcal K$ contains all $E_0$-translations.

Left rotation by $-1$ belongs to $\mathcal K$, because $-1$ is a central
kernel scalar. In the additive chart it is $x\mapsto i^2/x$, represented
by an antidiagonal matrix over $k$. It conjugates all upper
$E_0$-unipotents to all lower $E_0$-unipotents. These generate
$\SL_2(E_0)$, so $\mathcal K$ contains the corresponding fractional
transformations of the full chart. In particular it contains the
transformation whose matrix in the unitary chart is
$\diag(\xi,\xi^{-1})$: its conjugate by the Cayley change of basis is
\[
 \frac12
 \begin{pmatrix}
 \xi+\xi^{-1}&i(\xi-\xi^{-1})\\
 (\xi-\xi^{-1})/i&\xi+\xi^{-1}
 \end{pmatrix}\in\SL_2(E_0).
\]
Its action is $u\mapsto\xi u\xi$. Composing with the allowed conjugation
$u\mapsto\xi u\xi^{-1}$ gives $u\mapsto\xi^2u$. Hence
\begin{equation}\label{eq:palette-left-squares}
 L_{\xi^2}\in\mathcal K\qquad(\xi\in U(k)).
\end{equation}

\begin{proof}[Proof of Theorem~\ref{thm:no-simple-colors}]
All the preceding constructions were made under the assumption that
$\phi$ exists. Choose a palette satisfying the simultaneous identities
in Equation~\ref{eq:palette-indicator-generated}. Such palettes have
probability one by Proposition~\ref{prop:palette-measure} and the
recurrence argument. Equations~\ref{eq:palette-indicator-generated}
and~\ref{eq:palette-left-squares}, together with the exact left-rotation
palette rule, make membership in $\mathcal S$ invariant under left
multiplication by every square in $\mathcal F$, at all prefixes.
The subgroup generated by all squares is characteristic and nontrivial
in the nonabelian simple group $\mathcal F$: a group in which every
square is $1$ is abelian. Thus the squares generate $\mathcal F$.
Starting at the identity coordinate of the chosen palette, their left
multiplications run through every color. The membership indicator would
therefore have the same value on all of $\mathcal F$, contrary to
$\mathcal S$ being nonempty and proper. This contradiction proves the
theorem, subject to Lemma~\ref{lem:finite-obstruction}, whose proof follows.
\end{proof}
==== END The-Margulis-Platonov-conjecture-over-number-fields//sections/palettes.tex ====

==== BEGIN The-Margulis-Platonov-conjecture-over-number-fields//sections/unitary-induction.tex ====
\section{The remaining unitary groups}
\label{sec:unitary-induction}

We now pass from the odd-degree division case to arbitrary outer forms of
type~$A$.  The induction is on the \emph{reduced degree}, simultaneously
over all number fields.  Thus a special unitary group of an algebra of
degree~$m$ over a finite extension of~$k$ is an admissible induction input
whenever $m$ is smaller than the degree currently under consideration.
Restriction of scalars does not change this convention.

\begin{theorem}\label{thm:outer-a}
The Margulis--Platonov assertion holds for every simply connected
absolutely almost simple group of outer type~$A$ over a number field.
\end{theorem}

Write $L/k$ for a quadratic extension, $D$ for a central simple
$L$-algebra of degree~$n$, and $*$ for a unitary involution on~$D$.
Set $S=\SU(D,*)$ and $U=\U(D,*)$.  We continue to write $i\in L^\times$
for a fixed element with $i^*=-i$.  By the established isotropic cases
and Proposition~\ref{prop:finite-defect}, it suffices to assume that $S$
is $k$-anisotropic and prove $V_0/R=1$, where $R=S(k)^e$ is an abstract
power subgroup.  In this section a group element is said to be
\emph{killed} when its image in the finite group $S(k)/R$ is trivial.
Lemma~\ref{lem:restriction-mp} will be used for smaller special groups
and their restrictions of scalars.  Its local power criterion applies
to their images in~$S$ as well.

The degree-two case is an established inner type~$A_1$ case of
Theorem~\ref{thm:known-mp}.  For odd division degree,
Proposition~\ref{prop:color-dichotomy} and
Theorems~\ref{thm:no-characters} and~\ref{thm:no-simple-colors} give
$V_0/R=1$.  Three arguments remain: a ternary split-algebra case, the
odd matrix-algebra case, and the even-degree induction step.

\subsection{Uniform local power roots}

Some of the smaller subgroups below vary with the element being
factored.  The following observation allows their local neighborhoods
to be chosen in the fixed ambient algebra before those subgroups are
constructed.

\begin{lemma}\label{lem:unitary-uniform-roots}
Fix a completion $k_v$, a finite-dimensional matrix realization of~$D$,
and an integer $e\geq1$.  There is a neighborhood of~$1$ with the
following property.  An element in that neighborhood which belongs to
one of the unitary or special unitary subgroups arising from a
$*$-stable subalgebra of~$D$, or from an invariant summand of its
Hermitian module, has an $e$-th root in the same subgroup.  The
neighborhood can be chosen uniformly for all such subalgebras and
summands of the bounded degrees occurring here.
\end{lemma}

\begin{proof}
In a sufficiently small matrix neighborhood, the convergent series
$(1+X)^{1/e}$ defines the root close to~$1$.  A finite-dimensional local
subalgebra is closed, so the series stays in every subalgebra containing
$X$.  Uniqueness of the root close to~$1$ gives
$(a^{1/e})^*=(a^*)^{1/e}=(a^{1/e})^{-1}$ when $a^*=a^{-1}$.
The same series is the identity on a summand where $a$ is the identity.

For a special group, the reduced determinant of the root has $e$-th
power~$1$.  All its reduced eigenvalues occur among the ambient
eigenvalues, and all are close to~$1$ over the local algebraic closure
when the ambient matrix is sufficiently close to~$1$.  There are at
most $n$ eigenvalues in each reduced determinant.  Shrinking the fixed
neighborhood therefore makes all the relevant determinants lie in a
neighborhood containing no $e$-th root of unity other than~$1$.  This
proves the special condition uniformly, including after restriction of
scalars.
\end{proof}

\subsection{Ternary Hermitian forms}

\begin{proposition}\label{prop:unitary-ternary}
If $D=M_3(L)$, then $V_0/R=1$.
\end{proposition}

\begin{proof}
At every finite place a ternary Hermitian space is isotropic, so $A$ is
empty and $V_0=S(k)$.  We may assume that the global form is anisotropic.
Use the convention that its pairing $\langle\ ,\ \rangle$ is linear in
the first variable.  Scale the form so that a fixed vector $x$ has norm
$1$, and choose $e_2\perp x$ with norm $\kappa\ne0$.  Let $\Delta$ be
the determinant of the full form in fixed volume coordinates.

Choose a finite set $B$ containing the archimedean places, all places
where $-\Delta$ or $-\kappa$ is not a local norm from~$L$, and the finite
rank-zero places of the special stabilizer of~$x$.  The first two norm
conditions exclude only finitely many finite places: outside the
ramification and the divisors of the constants, units are norms from
an unramified quadratic extension.  Enlarge $B$ for fixed models as
necessary.  We shall factor a representative of a prescribed coset
into three binary special transformations, all locally small at~$B$.

For a representative $u$, put
\[
 z=ux,\qquad \alpha=\langle x,z\rangle,\qquad
 \delta=1-N_{L/k}(\alpha),\qquad
 k_0=2-\alpha-\bar\alpha.
\]
We require $\delta k_0(\delta-k_0)\ne0$.  In the nondegenerate plane
spanned by $x,z$, there is a unique vector $y_*$ whose pairing with each
of $x,z$ is~$1$; inversion of its two-by-two Gram matrix gives
\[
 \langle y_*,y_*\rangle=k_0/\delta.
\]
Choose the vector $v\perp x,z$ with $\det(x,z,v)=1$.  The determinant
of the Gram matrix in this basis gives
$\langle v,v\rangle=\Delta/\delta$.  Consequently a vector
$y=a y_*+b v$ has norm~$1$, has both marked pairings equal to~$a$, and
satisfies $1-N_{L/k}(a)=\kappa N_{L/k}(c)$ precisely when
\begin{equation}\label{eq:unitary-three-norms}
 t=N_{L/k}(a),\qquad
 \frac{\delta-k_0t}{\Delta}=N_{L/k}(b),\qquad
 \frac{1-t}{\kappa}=N_{L/k}(c).
\end{equation}
We insist that $a,b,c$ be nonzero.

We first produce local models at the places in~$B$.  Choose
$y=a x+c e_2$ close to~$x$, with $a,c\ne0$ and norm~$1$; the local norm
map is a submersion at~$1$, so this is possible with $a$ close to~$1$
and $c$ small.  The special isometry $w_1$ of the plane $\langle x,y\rangle$
sending $x$ to~$y$, extended by the identity, is close to~$1$.
Choose $\rho$ close to~$1$ in the special stabilizer of~$y$, and set
\[
 z=\rho x,\qquad
 w_2=\rho w_1^{-1}\rho^{-1},\qquad u_0=w_2w_1.
\]
Then $w_2y=z$, $u_0x=z$, and
$\langle y,x\rangle=\langle y,z\rangle=a$.
We can also arrange linear independence of $x,y,z$ and the three
inequalities imposed above.  Indeed the projection of $x$ onto
$y^\perp$ is nonisotropic, and its generic image under the special
stabilizer gives linear independence.  The scalar inequalities are
nonempty algebraic conditions: taking minus the identity on
$y^\perp$ gives $\alpha=2N_{L/k}(a)-1$, and a generic choice of~$a$
avoids their zeros.  Local points near~$1$ are Zariski dense, so these
nonempty algebraic openings can be met while retaining all the
smallness conditions.

At these models Equation~\eqref{eq:unitary-three-norms} has all three
norm arguments nonzero.  Since a quadratic norm map on nonzero
elements is a local submersion, solutions persist for nearby $u$, even
with $t$ fixed.  Proposition~\ref{prop:finite-defect} now supplies a
representative of the prescribed coset sufficiently close to these
models at~$B$.  At a place outside~$B$, choose $c\ne0$ small and put
$t=1-\kappa N_{L/k}(c)$.  The first factor is then close to~$1$; the
second is close to a nonzero norm because
\[
 \delta-k_0=-N_{L/k}(1-\alpha)
 \quad\hbox{and}\quad -\Delta\in N_{L/k}(L_v^\times).
\]
Thus Equation~\eqref{eq:unitary-three-norms} has the required local
solutions everywhere, with the prescribed ones at~$B$.

Here is the global step, including the passage from a product of
norms to the three separate norms.  Write
\[
 f_1(t)=t,\qquad f_2(t)=(\delta-k_0t)/\Delta,\qquad
 f_3(t)=(1-t)/\kappa.
\]
Their roots $0,\delta/k_0,1$ are distinct.  A smooth projective model
of
\begin{equation}\label{eq:unitary-chatelet}
 N_{L/k}(\xi)=f_1(t)f_2(t)f_3(t)
\end{equation}
is a Ch\^atelet surface.  The theorem of
Colliot-Th\'el\`ene--Sansuc--Swinnerton-Dyer identifies the closure of
its rational points with its Brauer--Manin set
\cite{CTSansucSD}; see also \cite[Section~5.2]{CT1992}.
For a split cubic, its
Brauer group modulo constants is generated by the quaternion symbols
of the quadratic extension and the linear factors
\cite{CTSansucSD}.  The local points just constructed are orthogonal
to these classes: each $f_j(t)$ is individually a norm, so a symbol
written with $f_j$ evaluates to zero.  Replacing $f_j$ by its monic
linear factor only adds a constant Brauer class, and the sum of the
local invariants of that constant is zero by reciprocity.  These
choices give an adelic point; integral choices are available at all
but finitely many places.

Enlarge the finite approximation set to contain $B$, the ramified
places of $L/k$, and all divisors of the leading coefficients, nonzero
roots, and pairwise root differences of the three linear factors.  Apply the
Ch\^atelet theorem to approximate the chosen local points there, on
the affine open where $f_1f_2f_3\ne0$.  At a remaining nonsplit place
the extension is unramified and all these nonzero constants are units.  If
$t$ is integral, at most one $f_j(t)$ has positive valuation.  If $t$
is not integral, all three $f_j(t)$ have valuation $\ord_v(t)$.  Since
their product is a norm by Equation~\eqref{eq:unitary-chatelet}, the
nonzero valuation in the first case, or the common valuation in the
second case, is even.  Units are norms in an unramified quadratic
extension.  Thus every $f_j(t)$ is separately a local norm at every
place, including the approximated places.  The quadratic Hasse norm
theorem \cite[Chapter~VIII, Theorem~3.1]{MilneCFT} gives global $a,b,c$ in
Equation~\eqref{eq:unitary-three-norms}.  Its norm torsors are rational
conics once soluble, so weak approximation gives the prescribed
approximations to $a,b,c$ at~$B$.

For the resulting $y$, both planes $\langle x,y\rangle$ and
$\langle z,y\rangle$ have Gram determinant
$1-N_{L/k}(a)=\kappa N_{L/k}(c)\ne0$.  Their special groups have no
finite rank-zero place outside~$B$, since $-\kappa$ is a local norm
there.  In a nondegenerate binary Hermitian plane, an isometry between
marked norm-one vectors has a unique special extension: extend it to
the orthogonal line and adjust the determinant on that line.  In
local orthogonal coordinates this extension varies smoothly with the
plane and marked vectors.  Hence the two resulting transformations
$w_1,w_2$ approximate the prescribed local transformations, and the
remaining factor $(w_2w_1)^{-1}u$ fixes~$x$ and is close to~$1$ at~$B$.
Lemma~\ref{lem:unitary-uniform-roots}, followed by the established
$A_1$ case and Lemma~\ref{lem:restriction-mp}, kills all three factors.
This kills the prescribed coset.
\end{proof}

\subsection{Odd matrix degree}

Reduction to Hermitian dimension two is an established strategy in the
unitary problem; see Tomanov \cite[p.~895]{Tomanov1992}.
Here the induction parameter is the reduced degree over all number fields.
Suppose $n$ is odd and $D=M_l(\Delta_0)$, where $\Delta_0$ is a division
algebra of degree $d$ and $l>1$.  Then $l\geq3$ and $n=ld$.  The
classification of unitary involutions realizes $S$ as the special
group of a Hermitian $l$-space over a unitary division algebra
\cite[Theorem~4.2(2)]{KMRT}.  Anisotropy implies that every subspace is nondegenerate.

The group is generated by its special transformations on
two-dimensional $\Delta_0$-subspaces.  To see this, choose a nonzero
vector $x$ and an isometry~$u$.  In a plane containing $x,ux$, Witt's
extension theorem gives an isometry sending $x$ to~$ux$.  Its
determinant can be adjusted on the orthogonal line in that plane.
Here the unitary group of a nondegenerate $\Delta_0$-line maps onto
$L^1$ by reduced determinant.  At a nonsplit finite place, the
existence of a unitary involution makes the Brauer corestriction of
the algebra zero.  For a quadratic extension of nonarchimedean local
fields, corestriction preserves the local Brauer invariant, so the
algebra itself is split.  The line therefore becomes an ordinary
Hermitian $d$-space, and its unitary determinant is onto by varying
one orthogonal coordinate.  At a nonsplit real place the extension
is $\bbC/\bbR$, the algebra is again split, and the same argument
works for every Hermitian signature.  At split finite places use
reduced norm surjectivity.  At split real places the division degree is
odd, so there is no quaternionic sign restriction.  Globally, for
$d>1$, each determinant fiber is a torsor under the simply connected
special group of that line; Theorem~\ref{thm:foundations-hasse} gives
its rational point.  For $d=1$ surjectivity is immediate.  Dividing by the resulting
special plane move fixes~$x$, and repetition in its orthogonal
complement proves the generation assertion.

If $d>1$, these plane groups have reduced degree $2d<n$.  They have no
finite rank-zero places.  At a split place this follows from their
two-dimensional module over the local division algebra; at a
nonsplit finite place the algebra is split and the Hermitian
dimension $2d\geq6$ is isotropic.  The induction hypothesis therefore
kills their rational points.  If $d=1$, put each plane move inside a
three-dimensional subspace and apply
Proposition~\ref{prop:unitary-ternary}.  This proves the odd
matrix-algebra case.

\subsection{A quadratic Hermitian subfield in even degree}

Assume now that $n=2m$ with $m\geq2$.  We first construct a fixed
quadratic subfield $S_0\subset D$, disjoint from~$L$, on which $*$ is
the identity.  The centralizer of~$S_0$ will supply the smaller special
unitary group in the induction.

\begin{lemma}\label{lem:unitary-hermitian-quadratic}
There is a quadratic field $S_0=k(s)\subset D$, disjoint from~$L$,
with $s^*=s$ and $s^2=r\in k^\times$, such that the two geometric
eigenspaces of~$s$ have dimension~$m$.
\end{lemma}

\begin{proof}
Choose $r\in N_{L/k}(L^\times)$ with the following local conditions.
At the finitely many exceptional nonsplit places, require $r$ to be a
square.  At a place split in~$L$ where the local index of~$D$ does not
divide~$m$, require $r$ to be a nonsquare, including the corresponding
real sign condition when needed.  At one additional finite place
split in~$L$, require $r$ to be a nonsquare.  This last condition
ensures that $S_0$ is a field and differs from~$L$.  The requirements
are compatible: at split places the norm map is surjective, and a
square can be approximated by norms at nonsplit places.  Weak
approximation in~$L$, followed by the norm, supplies~$r$.

We verify local embeddings of $k_v[s]/(s^2-r)$ with the stated
multiplicities and involution.  At split places, a quadratic field
extension reduces the local division index by its factor of two,
whenever this is required to make the index divide~$m$.  This is
exactly the local embedding criterion for a degree-two etale algebra
in a central simple algebra of degree~$2m$.  In the split quadratic
case one uses two blocks of degree~$m$.  The two components over~$L$
are paired by the involution.

At a nonsplit place where $S_0$ splits, decompose the Hermitian space
as an orthogonal sum of two $m$-spaces.  At every other nonsplit place
we may, by the choice of the exceptional set, assume that $L/k$ is
unramified and the Hermitian form is unimodular.  Its determinant is
a norm.  If $S_0$ locally equals~$L$, the form is hyperbolic and dual
isotropic $m$-spaces give the embedding.  If the two quadratic fields
are distinct, use the orthogonal sum of $m$ copies of one half the
trace to~$L$ of the standard Hermitian line over $LS_0/S_0$.  In the
basis $1,s$ its determinant is~$r$, a norm, so the local
classification of Hermitian forms identifies this form with the
given one.  These facts about algebras and local Hermitian forms are
the usual local classification and embedding theorems
\cite{Reiner,KMRT}.

Let $X$ be the variety of such embeddings, equivalently the images
of~$s$ with $s^*=s$, $s^2=r$, and the fixed geometric multiplicities.
Over the algebraic closure it is a single orbit under~$S$, with
connected stabilizer
\[
 \mathrm{S}(\GL_m\times\GL_m).
\]
The units on $\overline X$ are constant, and, because the ambient
group is simply connected, $\Pic(\overline X)$ is the character
lattice of this stabilizer, of rank one
\cite[Proposition~6.10, equation~(6.10.1)]{Sansuc}. The Hochschild--Serre edge
map gives an injection
$\Br_a X\longrightarrow H^1(k,\Pic(\overline X))$, where
$\Br_a X=\Br_1X/\Br_0X$ and $\Br_0X=\im(\Br k\to\Br X)$;
its kernel before taking this quotient
is precisely the image of the constant classes.  This map commutes
with localization. Define
\[
 B(X)=\ker\!\left(\Br_a X\longrightarrow
                       \prod_v\Br_a X_{k_v}\right).
\]
These are the algebraic Brauer classes locally trivial modulo constants.
Hence $B(X)$ injects into
$\Sha^1(k,\Pic(\overline X))$.  This last group vanishes.  Indeed a
rank-one lattice has trivial or sign Galois action.  The trivial case
has $H^1(k,\bbZ)=0$; in the sign case any nonzero class factors
through the quadratic sign quotient and is detected at an inert
unramified place by Chebotarev.  Inflation adds no classes since a
continuous homomorphism from a profinite group to~$\bbZ$ is zero.

We have local points everywhere, and integral points at almost all
places by Lang's theorem \cite[Theorem~1 and its corollary]{Lang1956}
and smooth lifting. Borovoi's
Hasse-principle theorem for homogeneous spaces with connected
stabilizer therefore applies: its obstruction is the pairing with
the locally trivial algebraic Brauer group just shown to vanish
\cite[Theorem~2.2]{Borovoi}.  Thus $X(k)\ne\varnothing$, as required.
\end{proof}

Fix this $s$ and put
\[
 J_0=k(j),\qquad j=is,\qquad C_s=C_D(s).
\]
The field $LS_0$ is biquadratic over~$k$, with third quadratic field
$J_0$.  The algebra $C_s$ has degree~$m$ over~$LS_0$, and its
involution is unitary over~$S_0$.  We shall factor a representative~$u$
into a quaternion norm-one element and an element of~$\U(C_s)$.

\subsection{The factorization and its local norm conditions}

For $u\in S$, define
\begin{equation}\label{eq:unitary-factor-data}
 \begin{aligned}
 s'&=usu^{-1},& t&=s's^{-1},&
 p&=(u+sus^{-1})/2,\\
 z&=(1+t)/2=up^*,&
 h&=zz^*=pp^*=(2+t+t^{-1})/4.
 \end{aligned}
\end{equation}
These are identities in~$D$.  For example, expanding $4pp^*$ gives
$2+usu^{-1}s^{-1}+sus^{-1}u^{-1}$.  We have $p\in C_s$, and both $s$
and $s'$ invert~$t$ by conjugation.  Hence $h$ commutes with both of
them.

Over a splitting field, use the two $s$-eigenspaces to write
\[
 u=\begin{pmatrix}A&B'\\ C&D'\end{pmatrix}.
\]
The two diagonal blocks of~$h$ have the same characteristic polynomial
$\Psi$.  On the open where the blocks are invertible, the first is
\begin{equation}\label{eq:unitary-compression}
 h_+=A(u^{-1})_{++}
     =(1-B'(D')^{-1}CA^{-1})^{-1}.
\end{equation}
The analogous expression for the second block is conjugate to this
one, since the two products of the invertible off-diagonal block
maps are conjugate.  The equality of characteristic polynomials then
holds everywhere by polynomial continuation.  Exchange of the
$s$-labels preserves $\Psi$, and $h=h^*$ preserves it under the
unitary involution; thus $\Psi\in k[T]$.

Set $d_0=\Nrd_{C_s}(p^*)$.  Then $d_0\in J_0$: conjugation of
$LS_0/S_0$ takes it to $\Nrd_{C_s}(p)$, and exchange of the two
$s$-labels does the same, by the complementary-minor identities for
$u$ and $\det u=1$.  Their product therefore fixes~$d_0$, which is the
condition that it belong to~$J_0$.

We will choose $u$ so that $F=k(h)$ is a field of degree~$m$ and
$h(1-h)\ne0$.  When $m\geq3$, we also require that its normal closure
have group~$S_m$ even after adjoining~$LS_0$.  Taking determinants of
$pp^*=h$ gives
\begin{equation}\label{eq:unitary-norm-compatibility}
 N_{F/k}(h)=N_{J_0/k}(d_0).
\end{equation}
In~$D$, the elements
\[
 j=is,\qquad b=i\bigl(s'-(2h-1)s\bigr)
\]
anticommute and satisfy
\begin{equation}\label{eq:unitary-quaternion}
 j^2=i^2r,\qquad b^2=4j^2h(1-h).
\end{equation}
Indeed $ss'+s's=2r(2h-1)$, from which both assertions follow by
expansion.  Since both squares are nonzero, the quaternion algebra
they define over~$F$ is central simple, and its homomorphism into~$D$
is injective.  Denote its image by~$Q$.  The involution acts on~$Q$ as
quaternion conjugation.  Moreover $t=(is')j^{-1}$, so $z\in Q$ and
its quaternion norm is~$h$.  Scalar extension gives
$LQ=C_D(F)$, a central simple algebra of degree two over~$LF$.
Consequently $\Res_{F/k}\SL_1(Q)$ maps into~$S$.

Suppose that we find $a\in J_0F$ satisfying
\begin{equation}\label{eq:unitary-simultaneous-norms}
 N_{J_0F/F}(a)=h,
 \qquad N_{J_0F/J_0}(a)=d_0\quad\text{if }m\geq3.
\end{equation}
The subfield $J_0F$ is contained in both $C_s$ and~$Q$, and $*$ acts
on it as the involution over~$F$.  We obtain
\begin{equation}\label{eq:unitary-factorization}
 u=wq,\qquad w=za^{-1}\in\SL_1(Q),\qquad
 q=a(p^*)^{-1}\in\U(C_s)\cap S.
\end{equation}
The last membership follows because $wq=u$ and $w$ is unitary of
ambient determinant one.  For $m\geq3$, the second norm condition
also gives $\Nrd_{C_s}(q)=1$, so $q\in\SU(C_s)$.  We now construct
$u,a$ with the local control needed to kill these factors.

Choose a finite set $B$ containing all archimedean and fixed
exceptional places, the places below the rank-zero places of
$\SU(C_s)$ over~$S_0$, and the anisotropic finite places of~$S$.
At these places ask that $u$ be close to~$1$.  Additional places
split in~$LS_0$ allow us to prescribe all needed Frobenius cycle
types of~$F$.  The block matrices
\begin{equation}\label{eq:unitary-near-identity-model}
 u=\begin{pmatrix}I&\varepsilon I\\
                  \varepsilon X&I+\varepsilon^2X\end{pmatrix},
 \qquad h_+=I+\varepsilon^2X
\end{equation}
have determinant one and are arbitrarily close to~$1$ for small
$\varepsilon$.  Taking $X$ with any prescribed separable residue
factorization gives the desired local unramified algebra for~$F$.
These are open conditions on~$u$.  Prescribing representatives of
every conjugacy class of~$S_m$ at places split in~$LS_0$ ensures that
the Galois subgroup over~$LS_0$ is~$S_m$: a proper subgroup misses a
conjugacy class, by the derangement theorem for its coset action.
For $m=2$ a single irreducible quadratic residue condition gives the
required disjointness.  Include these auxiliary places in~$B$.

Near~$1$, the norm equations have local solutions near~$1$.  Start
with the analytic square root $a_0=h^{1/2}\in F\otimes k_v$; its norm
from $J_0F$ to~$F$ is~$h$.  For $m\geq3$, the ratio
\[
 c_0=d_0/N_{J_0F/J_0}(a_0)
\]
lies in $J_0^1$, by Equation~\eqref{eq:unitary-norm-compatibility},
and is close to~$1$.  Choose its $m$-th root $c\in J_0^1$ close to~$1$
and replace $a_0$ by~$ca_0$.  This solves both norm equations.  All
reduced eigenvalues involved are ambient eigenvalues of matrices
near~$1$, so $a,w,q$ can be made uniformly small in the sense of
Lemma~\ref{lem:unitary-uniform-roots}.

\subsection{Controlling every other place}

It remains to ensure local solvability of
Equation~\eqref{eq:unitary-simultaneous-norms} and splitting of~$Q$
away from the finitely many places at which the factors are made
small.  To apply Proposition~\ref{prop:power-approximation}, we first
specify the allowed local conditions.

Let $\Omega\subset S$ be the fixed Zariski open on which $\Psi$ is
separable and $h(1-h)$ is invertible.  For each place outside~$B$,
let $\mathcal U_v$ be the union of $S(k_v)\setminus\Omega(k_v)$,
the locus in $\Omega(k_v)$ on which $Q$ is split and the required
norm equations are soluble, and a sufficiently small generic
identity opening $I_v\subset\Omega(k_v)$ on which the analytic-root
construction supplies norm solutions with both factors $w,q$ local
$e$-th powers.  The preceding analytic construction makes $I_v$
nonempty.  Thus a point of $I_v$ is allowed even when its quaternion
algebra is not split.

The integral and pole tests below will give the split-$Q$ condition
for points in~$\Omega$.  Including the complement of~$\Omega$ in
$\mathcal U_v$ lets these tests meet the universal integral-point
condition of Proposition~\ref{prop:power-approximation} without adding
a discriminant divisor to its codimension-two exclusion.  We will
ensure that the final rational representative lies in~$\Omega$ by
one of the auxiliary local openings already prescribed.  After the
tests, we specify which exceptional places use the small opening
$I_v$ and which use the split-$Q$ condition.

At integral reduction, consider the four block-determinant
hypersurfaces
\[
 \det A=0,\quad\det B'=0,\quad\det C=0,\quad\det D'=0.
\]
Exclude their pairwise intersections.  If all four determinants are
nonzero and $m\geq3$, also require that $\Psi$ have at least one
simple root over the algebraic closure of the residue field.  If
exactly one block determinant vanishes, require, for $m\geq3$, that
$0$ be a simple root of~$\Psi$ for a diagonal block, or $1$ a simple
root for an off-diagonal block.  The latter roots follow respectively
from the block and complementary-minor determinant identities for
$h$ and~$1-h$.

All excluded loci have geometric codimension at least two.  This can
be checked first on~$\GL_{2m}$: the four block determinants define
distinct irreducible hypersurfaces.  On the open where they are all
invertible, the map
$u\mapsto B'(D')^{-1}CA^{-1}$ is a submersion, as is seen by varying
$B'$ with the other three blocks fixed.  The transformation
$M\mapsto(1-M)^{-1}$ in Equation~\eqref{eq:unitary-compression}
preserves eigenvalue multiplicities on this open.  For $m\geq3$, the locus of
monic degree-$m$ polynomials with no simple root has dimension at most
$\lfloor m/2\rfloor$ in coefficient space and thus codimension at
least two.  The characteristic-polynomial map on matrices is flat:
each fiber has dimension $m^2-m$, by the decomposition into finitely
many Jordan types and their centralizer dimensions.  Thus the same
codimension bound holds for its inverse image.  On each single block
hypersurface the requisite simple root holds on a nonempty open, as
one sees from independent two-by-two transformations between paired
eigenlines, with only one of the relevant scalar blocks vanishing.
Finally, scalar multiplication changes none of the zero or root
conditions, so passage from $\GL_{2m}$ to $\SL_{2m}$ preserves these
codimensions.  The union is Galois stable and spreads to the chosen
integral models after finitely many exceptions.

Consider first an integral place outside the fixed exceptions where
$J_0$ is inert.  Frobenius interchanges the two diagonal block
divisors and also the two off-diagonal block divisors.  To check
this, exchange of the $s$-labels swaps each pair, and the unitary
diagram action does the same: in adapted dual bases it is inverse
transpose, and complementary minors swap the paired determinants.
Exactly one of these actions occurs when the third quadratic field
$J_0$ is inert.  A rational residue point therefore cannot lie on
exactly one divisor.  All four determinants are nonzero.  Hence $h$
and $1-h$ are units in every local component of~$F$.  The quadratic
algebra $J_0F/F$ is unramified or split, so these units are norms.
In particular Equation~\eqref{eq:unitary-quaternion} shows that $Q$
splits, and one can choose unit preimages of~$h$ for the first norm
equation.

For $m\geq3$ the simple residue root of~$\Psi$ supplies an
unramified local factor $E/k_v$ of~$F$.  It need not be a degree-one
factor.  Write $K=k_v$ and $J=J_0\otimes_k K$, an unramified
quadratic field in the case under consideration.  If $a_0$ denotes
the chosen unit preimages of~$h$, the required correction is
\[
 \eta=d_0/N_{JF/J}(a_0)\in\mathcal O_J^1,
\]
by Equation~\eqref{eq:unitary-norm-compatibility}.  Hilbert~90 writes
$\eta=y/\bar y$ with $y\in J^\times$.  Since $J/K$ is unramified,
multiplication by a power of a $K$-uniformizer makes $y$ a unit
without changing this quotient.  The unramified etale algebra
$JE/J$ has surjective norm on units.  Choose a unit $x\in JE$ with
$N_{JE/J}(x)=y$, and put $\beta=x/x^*$.  Then
\[
 N_{JE/E}(\beta)=1,\qquad N_{JE/J}(\beta)=\eta.
\]
Extend $\beta$ by~$1$ on the other factors of $JF$ and replace
$a_0$ by $a_0\beta$.  This corrects the determinant while preserving
the norm to~$F$, for either parity of $[E:K]$.

If $J_0$ splits, then $Q$ splits automatically.  Write
$a=(a_+,a_-)$ and $d_0=(d_+,d_-)$.  Choose
$N_{F/K}(a_+)=d_+$ and set $a_-=h/a_+$; compatibility gives
$N_{F/K}(a_-)=d_-$.  Thus one component must have a prescribed norm
from~$F$.
When all block determinants are nonzero, the prescribed value is a
unit, so an unramified factor supplies it by ordinary unit norm
surjectivity.  If one determinant vanishes, the prescribed simple
residue root is literally $0$ or~$1$, so Hensel's lemma supplies a
degree-one factor of~$F$.  Its norm map supplies any prescribed value,
including a nonunit.  This proves all the local assertions at the
permitted integral reductions.  When $m=2$, no determinant norm is
being imposed, so the splitting and first-norm assertions already
suffice.

We must check the single poles permitted by
Proposition~\ref{prop:power-approximation}.  Enlarge its fixed
splitting field to split $L,S_0,D$ and all the fixed data.  At a
single pole the place splits in this field, and
\[
 u=g_L\diag(t_{\rm p}^{ed_1},\ldots,t_{\rm p}^{ed_{2m}})g_R,
 \qquad d_1<\cdots<d_{2m},\quad \ord_v(t_{\rm p})>0,
\]
with integral side factors.  For $m\geq3$ impose total splitting of
$\Psi$ by requiring finitely many nonzero leading minors.  Explicitly,
if $c_b$ is its $b$-th elementary coefficient, exterior-power
expansion of the compression in
Equation~\eqref{eq:unitary-compression} gives
\begin{equation}\label{eq:unitary-pole-valuations}
 \ord_v(c_b)=e\ord_v(t_{\rm p})
       \sum_{a=1}^b(d_a-d_{2m+1-a}),\qquad 1\leq b\leq m.
\end{equation}
For completeness, the least exponent uses the first $b$ exponent
indices in the positive exterior power and the last $b$ in its
inverse.  Its coefficient is the product of the corresponding
opposite minors in the two rank-$m$ projections
\[
 g_RP_+g_R^{-1},\qquad g_L^{-1}P_+g_L,
\]
where $P_+$ is projection onto one $s$-eigenspace.  All other terms
have larger valuation.  The indicated minors are nonzero for a
generic conjugate of a rank-$m$ projection; require them all to be
units on the pole hyperplane.  These are nonempty simultaneous open
conditions.  Their choices can be made in the staged order required
by Proposition~\ref{prop:power-approximation}.  With all other torus
parameters equal to~$1$, the boundary of the factor with conjugator
$k_j$ has $g_L=xk_j$ and $g_R=k_j^{-1}$.  First choose the basic
conjugators so that every right-projection minor for $k_j^{-1}$ is
nonzero, along with the basic-list dominance conditions.  Then choose
$x$ in the intersection of the nonempty left-minor opens for the
finitely many $xk_j$.  All coordinate conjugates give only finitely
many such dense open conditions.  For an appended factor, with $x$
already fixed, its conjugator $k$ can be chosen in the intersection
of the right-minor open and the inverse translate by $x$ of the
left-minor open.  This intersection is again dense open and meets
the required local approximation openings.  Setting each new
parameter to~$1$ preserves all the old boundary tests.  This proves
nonemptiness of every single-pole test without changing the earlier
choices.

The consecutive slopes in Equation~\eqref{eq:unitary-pole-valuations}
are strictly increasing.  Every Newton-polygon segment has horizontal
length one, so $\Psi$ splits completely over~$k_v$.  Since $J_0$
also splits here, all required norm equations are soluble and~$Q$
splits.  For $m=2$, splitting of~$J_0$ alone suffices.

The tests establish the required membership in~$\mathcal U_v$ at
permitted integral reductions and single poles.  It remains to
retain global genericity and to choose the openings at the finite
exceptional places.  The final rational representative lies
in~$\Omega$ because one of the existing auxiliary local openings is
contained in~$\Omega(k_v)$.  In
Equation~\eqref{eq:unitary-near-identity-model}, choose $X$ with
separable characteristic polynomial and with both $X$ and
$I+\varepsilon^2X$ invertible.  Then $h_+=I+\varepsilon^2X$ has
distinct eigenvalues different from $0,1$, and these conditions
persist in its local opening.  This argument uses the local matrix
itself, not the reduction of~$\Psi$, which can have repeated roots
when $\varepsilon$ is small.
After the initial representative $x$ is chosen, let $B_1$ be its
finite set of denominator places outside~$B$.  At these places
specifically choose the opening $I_v\subset\mathcal U_v$.
This requires no splitting assumption; the block model in
Equation~\eqref{eq:unitary-near-identity-model} was needed only to
prescribe Frobenius types at the auxiliary split places.  The finitely
many later model exceptions are assigned unit-parameter
reductions with all four block determinants nonzero and separable
$\Psi$; such opens are nonempty over sufficiently large residue
fields, with small fields already included among the fixed
exceptions, and thus use the split-$Q$ branch as well.  Consequently
the small branch is used only at the controlled finite set
$B^+=B\cup B_1$.  The construction produces a representative of
the required coset for which $Q$ is split outside~$B^+$.
At every place Equation~\eqref{eq:unitary-simultaneous-norms} is
soluble with the prescribed near-identity solutions at~$B^+$.

For $m\geq3$, the simultaneous equations form a torsor under the
norm torus of Lemma~\ref{lem:norm-torus}, with quadratic field~$J_0$
and degree-$m$ field~$F$.  The full symmetric action over~$LS_0$
implies the joint $S_m\times C_2$ hypothesis for $F,J_0$.  That lemma
gives a global~$a$ with the local approximations.  When $m=2$, the
quadratic Hasse norm theorem over~$F$
\cite[Chapter~VIII, Theorem~3.1]{MilneCFT} and weak approximation on its
norm torsor give~$a$.  In either case the quaternion factor~$w$ in
Equation~\eqref{eq:unitary-factorization} is killed: all its
rank-zero places lie over the controlled set, where it is a local
$e$-th power by Lemma~\ref{lem:unitary-uniform-roots}.  If $m\geq3$,
the factor $q$ lies in the smaller group $\SU(C_s)$ over~$S_0$ and is
small at every rank-zero place of that fixed group.  Induction and
Lemma~\ref{lem:restriction-mp} kill~$q$ as well.

\subsection{The determinant correction in degree four}

It remains to treat $m=2$.  The preceding construction has killed~$w$
and produced $q\in\U(C_s)\cap S$, small at the fixed set~$B$, but its
determinant in~$C_s$ may not be~$1$.  Write
\[
 r_0=\Nrd_{C_s}(q).
\]
Both conjugations of $LS_0$ over $S_0$ and over~$L$ invert~$r_0$:
the first by unitarity, the second by ambient determinant one.  Thus
$r_0\in J_0^1$.  Taking $q$ sufficiently close to~$1$ at~$B$ ensures
$r_0\ne-1$.  Put
\[
 d=(1+r_0)/2\in J_0^\times,\qquad r_0=d/\bar d,
\]
where the bar is the nontrivial automorphism of~$J_0/k$.  The element
$d$ is close to~$1$ at~$B$.

We will remove this determinant by two elements in smaller binary
special groups.  The following construction gives the freedom needed
to keep their varying rank-zero places under control.  For any finite
set of places outside the fixed exceptions, one can choose a
quadratic field $F_i=k(f_i)\subset C_s$, disjoint from~$LS_0$, with
$f_i^*=f_i$, such that $F_i$ splits at these places and the binary
group $\SU(C_D(F_i),*)$ is split over both local components of~$F_i$.
Here and below $B$ includes all fixed exceptions of the structures
defined from~$s$.

To prove the construction, let $B_s$ be the quaternion algebra over
$S_0$ obtained by descent from~$C_s$: its defining semilinear
involution is the composition of $*$ and standard quaternion
conjugation.  These involutions commute, and
$B_s\otimes_{S_0}LS_0=C_s$.  For a trace-zero element $y\in B_s$,
the element $f=i^{-1}y$ satisfies $f^*=f$.  Prescribe
$f^2=c\in k^\times$, so $y^2=i^2c$.  A quadratic etale algebra
embeds into $B_s$ if and only if it is a field at every ramified
place of~$B_s$, by the local embedding criterion and Brauer
invariants.  We may impose this by choosing $i^2c$ nonsquare there.
At a finite place of~$k$ the kernel of the square-class map to a
quadratic component of~$S_0$ has at most two elements; in the split
case it has one.  Thus the forbidden square classes do not exhaust
the local square classes.  At real places the requirement is simply
a sign.  These ramified places are above~$B$.  At the prescribed
places outside~$B$ require $c$ to be a square, and at one further
place split completely in~$LS_0$ require it to be a nonsquare.
Weak approximation supplies~$c$.  The last condition ensures that
$F_i$ is a field disjoint from~$LS_0$.

The embedding of $S_0(\sqrt{i^2c})$ into~$B_s$ can approximate any
prescribed local embeddings: they are conjugate, and weak
approximation in the open affine variety $B_s^\times$ approximates
the conjugators.  At a specified place split in~$L$, ordinary split
block embeddings make the two binary centralizers split.  At a good
unramified nonsplit place where $S_0$ splits, decompose both
two-dimensional $s$-blocks into orthogonal lines with unit norm
values.  Give each $f_i$-eigenspace one line from each block.  Its
Hermitian determinant is a unit, and hence a norm, so each binary
space is hyperbolic.  If $S_0$ does not split there, it locally equals
$L$; the two $s$-blocks are dual totally isotropic spaces.  Choose
paired line decompositions and assign one hyperbolic plane to each
$f_i$-eigenvalue.  In each case $f_i$ is Hermitian and traceless in
$C_s$, and the desired local property persists on an open set of
embeddings.  This proves the construction.

For such a field $F_i$, the compositum $LS_0F_i$ is a maximal
subfield of~$C_s$, and there is an inclusion
\begin{equation}\label{eq:unitary-binary-norm-tori}
 (J_0F_i/F_i)^1
 \ \subset\ \SU(C_D(F_i),*)\cap\U(C_s).
\end{equation}
Indeed the determinant in the binary algebra $C_D(F_i)$ is the norm
from $LS_0F_i$ to~$LF_i$.  On $J_0F_i$ its nontrivial automorphism
exchanges the $s$-labels, which is the norm-one condition displayed
on the left.  The determinant of the same element in~$C_s$ is its
norm from $J_0F_i$ to~$J_0$.

Choose $F_1$ first.  Outside~$B$, all but finitely many places have
both of the following properties: $d$ is a local norm from
$J_0F_1$ to~$J_0$ at every place above the given $k$-place, and the
binary group in Equation~\eqref{eq:unitary-binary-norm-tori} is
isotropic at every place of~$F_1$ above it.  This follows from
unramified unit norm surjectivity and good reduction.  Call these
the places covered by~$F_1$.  Choose $F_2$ by the preceding
construction to split, with split binary groups in both components,
at every remaining place outside~$B$.  Every such remaining place
is covered by~$F_2$, since the local norm from a split quadratic
algebra is surjective.

Solve
\begin{equation}\label{eq:unitary-multinorm-correction}
 N_{J_0F_1/J_0}(x_1)N_{J_0F_2/J_0}(x_2)=d.
\end{equation}
This equation satisfies the Hasse principle and weak approximation
on its locus where the variables are invertible.  Indeed putting
$y_2=x_2^{-1}$ turns it into the nonzero locus of the nonsingular
four-variable quadratic equation
\[
 N_{J_0F_1/J_0}(x_1)-dN_{J_0F_2/J_0}(y_2)=0.
\]
Hasse--Minkowski \cite[Chapter~VIII, Theorem~3.5(b)]{MilneCFT}
gives a rational isotropic vector whenever there
are local ones.  The corresponding projective quadric is rational
once it has a point, and its affine cone minus the vertex has weak
approximation.  Removing the two norm-zero loci preserves the
needed approximation to local points in this open set.

The requisite local points can be chosen with additional control.
At~$B$ take $x_1=\sqrt d$ and $x_2=1$, near~$1$; the square root is
in the local $J_0$-algebra.  Outside~$B$, use one covering field for
the entire $k$-place to carry the norm~$d$, and take the other
variable to be~$1$ at every component above that place.  In
particular, at any rank-zero place of one of the two binary groups,
its own variable can be taken to be~$1$: that field cannot be the
covering field there.  These rank-zero places form a finite set.
Use weak approximation in
Equation~\eqref{eq:unitary-multinorm-correction} at their underlying
places and at~$B$, obtaining global invertible~$x_1,x_2$ with these
local approximations.

Set $b_i=x_i/(x_i^*)$.  By
Equation~\eqref{eq:unitary-binary-norm-tori}, each $b_i$ belongs to
the indicated binary special group, of reduced degree two over
$F_i$, and
\[
 \Nrd_{C_s}(b_1)\Nrd_{C_s}(b_2)
   =\frac{d}{\bar d}=r_0.
\]
At every rank-zero place of its binary group, $b_i$ is as close to~$1$
as necessary.  The $A_1$ case, local power lifting, and
Lemma~\ref{lem:restriction-mp} therefore kill~$b_1,b_2$.  The element
$(b_1b_2)^{-1}q$ has determinant one in~$C_s$ and remains close to~$1$
at~$B$.  Its special group over~$S_0$ has degree two and all its
rank-zero places lie above~$B$, so it too is killed.  Together with
the already killed factor~$w$, this kills~$u$.

The even induction step is complete.  The degree-two base, the
odd-degree division result, the odd matrix argument, and this step
prove $V_0/R=1$ in every reduced degree over every number field.
Proposition~\ref{prop:finite-defect} now proves
Theorem~\ref{thm:outer-a}.
==== END The-Margulis-Platonov-conjecture-over-number-fields//sections/unitary-induction.tex ====