==== TABLE OF CONTENTS ====
The-Blockwise-Alperin-Weight-Conjecture//figures/field-tower.tex
The-Blockwise-Alperin-Weight-Conjecture//figures/proof-interfaces.tex
The-Blockwise-Alperin-Weight-Conjecture//paper.tex
The-Blockwise-Alperin-Weight-Conjecture//sections/consequences.tex
The-Blockwise-Alperin-Weight-Conjecture//sections/differentials.tex
The-Blockwise-Alperin-Weight-Conjecture//sections/foundations.tex
The-Blockwise-Alperin-Weight-Conjecture//sections/group.tex
The-Blockwise-Alperin-Weight-Conjecture//sections/intro.tex
The-Blockwise-Alperin-Weight-Conjecture//sections/specialization.tex
The-Blockwise-Alperin-Weight-Conjecture//sections/twist.tex
The-Blockwise-Alperin-Weight-Conjecture//sections/valuations.tex
The-Blockwise-Alperin-Weight-Conjecture//references.bib
===========================

==== BEGIN The-Blockwise-Alperin-Weight-Conjecture//figures/field-tower.tex ====
\begin{figure}[!ht]
\centering
\begin{tikzpicture}[
  every node/.style={font=\small},
  field/.style={inner sep=4pt},
  inclusion/.style={-{Stealth[length=4pt]},semithick}
]
\begin{scope}[xshift=-3.45cm]
  \node[font=\small\bfseries] at (0,4.65) {Characteristic zero};
  \node[field] (kb) at (0,0) {$K_b$};
  \node[field] (ks) at (-1.45,1.35) {$K_s$};
  \node[field] (kbp) at (1.45,1.35) {$K_b'$};
  \node[field] (l) at (-1.45,2.8) {$L$};
  \node[field] (lp) at (0,4) {$L'=LK_b'$};
  \draw[inclusion] (kb)--(ks);
  \draw[inclusion] (kb)--(kbp);
  \node[font=\footnotesize,align=left] at (1.85,.35)
    {unramified\\degree $n$};
  \draw[inclusion] (ks)--(l);
  \draw[inclusion] (l)--(lp);
  \draw[inclusion] (kbp)--(lp);
\end{scope}
\begin{scope}[xshift=3.45cm]
  \node[font=\small\bfseries] at (0,4.65) {Residue fields};
  \node[field] (f0) at (0,0) {$F_0$};
  \node[field] (fp) at (1.45,1.35) {$F'$};
  \node[field] (f) at (-1.45,2.8) {$F$};
  \node[field] (s) at (0,4) {$S=FF'$};
  \draw[inclusion] (f0)--(fp);
  \draw[inclusion] (f0)--(f);
  \draw[inclusion] (f)--(s);
  \draw[inclusion] (fp)--(s);
\end{scope}
\node[font=\footnotesize,align=center] at (0,-.85)
 {$\operatorname{res}(K_b)=\operatorname{res}(K_s)=F_0,
   \qquad [F:F_0]=[L:K_s],\qquad [L:K_s]_pm\bigl(h+\log_p m\bigr).
 \label{eq:genus-rounding-threshold}
\end{equation}
Let $F/F_0$ be finite with $[F:F_0]_p\le p^h$, and let $Z$ be the
normalization of $E_F$ in a finite purely inseparable extension of its
function field of degree $d\le m$.  Choose a compositum $S=FF'$.
The curve $Z_S$ is integral and regular.  Let $r_i$ be the ramification
index of $Z_S\to E_S$ at the unique point over $t_i$.
With $\nu(Z)$ computed over $F$, one has
\begin{equation}
 \frac{\nu(Z)}d\ge\sum_{i=1}^n\left(1-\frac1{r_i}\right).
 \label{eq:inseparable-genus-bound}
\end{equation}
The bound is uniform in $F$, $Z$, and the tuple-class multiset used
later in the counting argument.
\end{theorem}

The proof has two parts. Absolute differentials give the same
inequality with an error of at most $h+\log_p d$. After multiplication
by $d$, the difference between the two sides is a difference of
divisor degrees over $F$, hence a multiple of $p^{s-h}$. The strict
threshold then forces this difference to be nonnegative. We develop
the differential identities first, together with the separable
ramification bound needed later in \Cref{sec:specialization}.

\subsection{Divisor degrees, duality, and separable ramification}

The required identities hold over more general fields than the
twisted-curve construction. Throughout this and the next subsection,
let $\kappa$ be any perfect field of characteristic $p$, let
$S/\kappa$ be finitely generated, and put
$\rho=\operatorname{trdeg}_\kappa S$. Curves and degrees in these two
subsections are over $S$.

The classical theory of inseparable genus change begins with Tate
\cite{Tate1952}; Schr\"oer \cite{Schroer2009} gives a modern account
for geometrically integral curves. Here absolute differentials retain
arithmetic genus without a geometric-reducedness hypothesis and also
measure ramification at the marked points.

\begin{lemma}[Degrees over the stated ground field]
\label{lem:curve-degrees}
For an invertible sheaf $\mathcal L$ on $C$,
\begin{equation}
 \chi(C,\mathcal L)=\deg_S\mathcal L+\chi(C).
 \label{eq:curve-euler-degree}
\end{equation}
A finite dominant morphism $f:C\to C'$ of such curves is finite flat.  If
its function-field degree is $\delta$, then
\[
 \deg_S f^*D=\delta\deg_S D
 \quad\text{and}\quad
 \deg_S f_*D'=\deg_S D'
\]
for divisors $D$ on $C'$ and $D'$ on $C$.  These assertions, and Euler
characteristics, add over disjoint unions of regular integral curves.
After a finite separable extension $S_1/S$, divisor and line-bundle degrees
and coherent Euler characteristics have the same numerical values over
$S_1$ as before extension over $S$.
\end{lemma}

\begin{proof}
For a closed point $x$ put $\deg_S(x)=[\kappa(x):S]$.  For any divisor $D$
the quotient in
\[
 0\longrightarrow\mathcal O_C(D)
 \longrightarrow\mathcal O_C(D+x)
 \longrightarrow\mathcal Q_x\longrightarrow0
\]
has length one over the DVR $\mathcal O_{C,x}$, so its dimension over $S$
is $[\kappa(x):S]$.  Finite-support sheaves have no higher cohomology.
Adding and subtracting closed points proves
$\chi(\mathcal O_C(D))=\chi(C)+\deg_S D$.  A rational trivialization of
$\mathcal L$ identifies it with $\mathcal O_C(D)$ for a divisor $D$, giving
\eqref{eq:curve-euler-degree}.  Applied to a principal divisor, the same
formula gives degree zero, so this degree depends only on $\mathcal L$.

Locally on $C'$, the finite module $f_*\mathcal O_C$ is torsion-free over
a DVR and therefore free, of generic rank $\delta$.
For a closed point $x\in C'$ and a uniformizer $t$ there, reduction of
this free module modulo $t$ gives
\begin{equation}
 \sum_{y\mapsto x}e_y[\kappa(y):\kappa(x)]=\delta,
 \label{eq:local-degree-formula}
\end{equation}
where $e_y$ is the valuation of $t$ in $\mathcal O_{C,y}$.
Indeed the summand at $y$ has a filtration of length $e_y$ with
successive quotients $\kappa(y)$.  Multiplication by
$[\kappa(x):S]$ proves the pullback degree formula.  The pushforward
formula follows from
$f_*y=[\kappa(y):\kappa(x)]x$ and the tower formula for residue degrees.

Finite separable scalar extension preserves regularity, and a regular
curve after such extension is a disjoint union of its integral
components.  Flat base change for coherent cohomology
in \Cref{foundation:cohomology} gives
\[
 H^i(C_{S_1},\mathcal G_{S_1})
   =H^i(C,\mathcal G)\otimes_S S_1.
\]
Thus the dimensions over the respective fields, and hence Euler
characteristics, agree.  Applying
\eqref{eq:curve-euler-degree} proves the assertion for line bundles and
their Cartier divisors.  In particular it also applies to a reduced
finite divisor: separable scalar extension preserves its reducedness.
\end{proof}

\begin{proposition}[The absolute determinant computes arithmetic genus]
\label{prop:absolute-genus}
The absolute cotangent sheaf $\Omega_{C/\kappa}$ is locally free of rank
$\rho+1$, and
\begin{equation}
 \deg_S\det\Omega_{C/\kappa}=\nu(C).
 \label{eq:absolute-determinant-genus}
\end{equation}
\end{proposition}

\begin{proof}
An affine coordinate ring of $C$ is a localization of a finitely
generated integral $\kappa$-algebra: choose a finitely generated domain
with fraction field $S$, clear the finitely many coefficients in a
presentation over $S$, and take the contraction of the defining prime
ideal before localizing.  Its local rings are consequently essentially
of finite type over $\kappa$.  The regularity criterion over a perfect
field in \Cref{foundation:regularity} says that these regular local rings
are local rings at smooth points of the corresponding
$\kappa$-variety.  Their absolute differentials are free of rank
\[
 \operatorname{trdeg}_\kappa\kappa(C)=\rho+1.
\]
The sheaf is coherent: the relative differentials over $S$ are coherent,
and $\Omega_{S/\kappa}$ is finite-dimensional, so the cotangent sequence
gives a finite presentation locally.  This proves the asserted local
freeness without making a smoothness assumption over $S$.

Choose a projective embedding $i:C\hookrightarrow\mathbb P^b_S$ and let
$\mathcal I$ be its ideal sheaf.  This is a regular immersion of
codimension $b-1$.  Here is a local verification.  Let $R$ be an ambient
regular local ring, $I$ the local ideal, and $R/I$ the regular local ring
of $C$.  Their dimensions differ by $b-1$, by the dimension formula at
the common residue field.  Since both are regular, the kernel of
\[
 \mathfrak m_R/\mathfrak m_R^2
   \longrightarrow
 \mathfrak m_{R/I}/\mathfrak m_{R/I}^2
\]
has dimension $b-1$.  Choose $b-1$ elements of $I$ lifting a basis of this
kernel.  They form part of a regular system of parameters of $R$ and
cut out a regular local quotient of dimension $\dim(R/I)$.
The remaining ideal defining $R/I$ in that quotient is prime and must
be zero: a nonzero prime in a finite-dimensional local domain strictly
lowers the quotient dimension.  Thus the chosen elements generate $I$.
It follows that $\mathcal I/\mathcal I^2$ is locally free of rank $b-1$.

The absolute conormal sequence is in fact short exact:
\begin{equation}
 0\longrightarrow\mathcal I/\mathcal I^2
 \longrightarrow i^*\Omega_{\mathbb P^b_S/\kappa}
 \longrightarrow\Omega_{C/\kappa}\longrightarrow0.
 \label{eq:absolute-conormal}
\end{equation}
To check the left end, the kernel of the last arrow is locally free,
since the last module is locally free.  Its rank is
$(b+\rho)-(1+\rho)=b-1$.  The conormal sequence surjects
$\mathcal I/\mathcal I^2$ onto this kernel.  A surjection between locally
free modules of the same rank is an isomorphism, proving
\eqref{eq:absolute-conormal}.

The projective space is obtained from $\mathbb P^b_\kappa$ by extending
constants.  Its absolute cotangents therefore split into relative
cotangents over $S$ and the constant bundle of $\Omega_{S/\kappa}$.
Taking determinants in \eqref{eq:absolute-conormal} gives
\[
 \det\Omega_{C/\kappa}
  \simeq\mathcal L_c\otimes_S\det\Omega_{S/\kappa},
 \qquad
 \mathcal L_c=
 \mathcal O_C(-b-1)\otimes
       \det(\mathcal I/\mathcal I^2)^\vee.
\]
The last tensor factor is a constant line and has degree zero.

We compute the degree of $\mathcal L_c$ using the
projective-space duality in \Cref{foundation:duality}.
A Koszul resolution for the local regular sequence defining $C$ shows
that
\[
 \mathcal E xt^v_{\mathbb P^b_S}
       (i_*\mathcal O_C,\mathcal O(-b-1))
 =\begin{cases}
   i_*\mathcal L_c,&v=b-1,\\
   0,&v\ne b-1.
  \end{cases}
\]
After dualizing the Koszul resolution, its top cohomology
is the dual of the top exterior power of the free module of equations,
restricted to $C$.  A change of those equations acts on this exterior
power by the determinant of their change on $\mathcal I/\mathcal I^2$.
Thus the local top cohomology modules glue to
$\det(\mathcal I/\mathcal I^2)^\vee\otimes\mathcal O_C(-b-1)$.

The local-to-global Ext spectral sequence has only this one nonzero
row.  Projective-space duality consequently gives, for $r=0,1$,
\[
 H^r(C,\mathcal O_C)^\vee
  \simeq
 \operatorname{Ext}^{b-r}_{\mathbb P^b_S}
       (i_*\mathcal O_C,\mathcal O(-b-1))
  \simeq H^{1-r}(C,\mathcal L_c).
\]
Hence $\chi(C,\mathcal L_c)=-\chi(C)$.  Applying
\eqref{eq:curve-euler-degree} yields
$\deg_S\mathcal L_c=-2\chi(C)$, which proves
\eqref{eq:absolute-determinant-genus}.
\end{proof}

\begin{lemma}[A separable determinant bound, including the wild term]
\label{lem:separable-different}
Let $f:C\to C'$ be a finite dominant, generically separable morphism of
regular integral projective curves over $S$, of degree $\delta$.
There is an effective divisor $R_f$ on $C$ such that
\[
 \nu(C)=\delta\nu(C')+\deg_S R_f.
\]
If $y\in C$ has ramification index $a_y$ over $f(y)$, then
\begin{equation}
 \operatorname{ord}_y R_f\ge a_y-1+\mathbf 1_{p\mid a_y}.
 \label{eq:separable-different-bound}
\end{equation}
No separability of $\kappa(y)/\kappa(f(y))$ is required.
The formula and the inequalities can be added componentwise after a
finite separable extension of $S$.
\end{lemma}

\begin{proof}
The morphism
$f^*\Omega_{C'/\kappa}\to\Omega_{C/\kappa}$ is generically an isomorphism:
at the function fields a finite separable extension has no relative
differentials, and both absolute differential spaces have dimension
$\rho+1$.  Its determinant is therefore a nonzero regular morphism of
line bundles.  The divisor of this morphism is effective; call it $R_f$.
Its degree is the difference of the two determinant degrees, which is
$\nu(C)-\delta\nu(C')$ by
\Cref{lem:curve-degrees,prop:absolute-genus}.

Write $x=f(y)$ and choose a uniformizer $t$ of the DVR
$B=\mathcal O_{C',x}$.  Its absolute differential is primitive in
$\Omega_{B/\kappa}$, meaning that it is part of a $B$-basis.  Indeed the
conormal sequence for the residue field is
\[
 (t)/(t^2)\longrightarrow
 \Omega_{B/\kappa}\otimes_B\kappa(x)
 \longrightarrow\Omega_{\kappa(x)/\kappa}\longrightarrow0.
\]
The middle term has dimension $\rho+1$ and the right term dimension
$\rho$, since $\kappa(x)$ is finite over $S$ and
\Cref{foundation:fields} computes absolute differential dimension over
the perfect field $\kappa$.  Thus the first map has nonzero
one-dimensional image, generated by $dt$.  Consequently $dt$ is
nonzero modulo the maximal ideal, and it extends to a basis of the
free module $\Omega_{B/\kappa}$.

In $A=\mathcal O_{C,y}$ write $t=u\tau^{a_y}$, where $u$ is a unit and
$\tau$ a uniformizer.  Absolute differentiation gives
\[
 dt=\tau^{a_y}\,du+
       a_yu\tau^{a_y-1}\,d\tau.
\]
All coordinates of this column of the differential matrix are
divisible by $\tau^{a_y-1}$; when $p\mid a_y$, the second term is zero,
so they are divisible by $\tau^{a_y}$.  Expanding the determinant in
this column proves \eqref{eq:separable-different-bound}.  Every step
uses absolute differentials over $\kappa$, so an inseparable extension
of residue fields causes no change to the argument.
\end{proof}

\subsection{The differential calculation for a height-one map}

We have established the determinant formula for arithmetic genus and
the separable ramification bound. The remaining ingredient is an
exact genus identity for a purely inseparable map of exponent one;
successive such maps will handle arbitrary purely inseparable covers.

Write $T_C=(\Omega_{C/\kappa})^\vee$ for the absolute tangent bundle.
It has rank $\rho+1$, even when the relative tangent sheaf over $S$ has
different behavior.

The calculation below is a curve-level form of the canonical-bundle
formula for a $p$-closed foliation; compare
\cite[Section~I.1, Proposition~1.1 and Equations~(1.2)--(1.3)]{Ekedahl1988}.
We prove the version needed here directly from local presentations,
using absolute differentials over the perfect constants. In this way
the kernel, the Frobenius-power cokernel, and all inseparable residue
extensions remain explicit.

\begin{lemma}[Height-one kernels and cokernels]
\label{lem:height-one}
Let $\pi:C\to C'$ be a finite dominant morphism of regular integral
projective curves over $S$ such that
$\kappa(C)^p\subseteq\kappa(C')$.  Put
$q=[\kappa(C):\kappa(C')]=p^a$ and
\[
 \mathcal F=\ker(T_C\longrightarrow\pi^*T_{C'}),
 \qquad
 \mathcal Q=\operatorname{coker}(T_C\longrightarrow\pi^*T_{C'}).
\]
Both $\mathcal F$ and $\mathcal Q$ are locally free of rank $a$, and
\begin{equation}
 \det\mathcal Q\simeq(\det\mathcal F)^{\otimes p}.
 \label{eq:height-one-cokernel}
\end{equation}
Consequently
\begin{equation}
 \nu(C)=q\nu(C')+(p-1)\deg_S\mathcal F.
 \label{eq:height-one-genus}
\end{equation}
\end{lemma}

\begin{proof}
\textit{The integral presentation.}
On an affine open of $C'$, every element of the finite upstairs
coordinate ring has its $p$th power in the downstairs ring: the power
lies in the downstairs fraction field and is integral, so normality
puts it in that ring.  There is therefore a unique prime upstairs over
each downstairs prime, since membership of an element is determined
by membership of its $p$th power.  At a closed point we may consequently
write $B\subset A$ for the downstairs and upstairs DVRs, with $A$ still
finite over $B$.  It is finite free of rank $q$ and satisfies
$A^p\subset B$.

Let $\tau$ be an upstairs uniformizer, let $\epsilon$ be the
ramification index, and put $f=[\kappa(A):\kappa(B)]$.
The element $\tau^p\in B$ has upstairs valuation $p$; all nonzero
elements of $B$ have upstairs valuation in $\epsilon\mathbb Z$.
Thus $\epsilon\mid p$, so $\epsilon$ is $1$ or $p$.
Reduction of the rank-$q$ free module modulo a downstairs uniformizer
gives $q=\epsilon f$ by \eqref{eq:local-degree-formula}.
The residue extension has exponent at most one.  Successively
adjoining a generator outside the preceding residue field gives a
degree-$p$ extension each time.  Choose residue generators
$\bar b_1,\ldots,\bar b_r$ in this manner, so their monomials with
exponents in $\{0,\ldots,p-1\}$ form a $\kappa(B)$-basis of $\kappa(A)$
and $f=p^r$.  Lift them to $b_1,\ldots,b_r\in A$.  If $\epsilon=p$,
append $b_{r+1}=\tau$; if $\epsilon=1$, append nothing.  In both cases
there are exactly $a$ generators, and $c_\ell=b_\ell^p$ belongs to $B$.

For a downstairs uniformizer $t$, the ideal $tA$ is
$\tau^\epsilon A$.  Filter $A/tA$ by the powers of $\tau$ from $0$ to
$\epsilon$.  Each successive quotient is $\kappa(A)$, and the lifted
residue monomials give its basis after multiplication by the
corresponding power of $\tau$.  It follows that the $q$ monomials
\[
 b_1^{i_1}\cdots b_a^{i_a},\qquad 0\le i_\ell1$.

\textit{The Frobenius tower and its recurrence.}
Let $M_e=S(E)$ and $M_z=S(Z)$, and set
\[
 M_j=M_eM_z^{p^j}\quad(0\le j\le b),\qquad
 M_0=M_z,\qquad M_b=M_e,
\]
where $b$ is the first index for which the last equality holds.
Let $Y_j$ be the normalization of $E_S$ in $M_j$, and write
\[
 \gamma_j:Y_j\longrightarrow E_S,\qquad
 d_j=[M_j:M_e],\qquad
 \pi_j:Y_j\longrightarrow Y_{j+1},\qquad
 q_j=d_j/d_{j+1}=p^{m_j}.
\]
In particular $Y_0=Z_S$, $d_0=d$, $Y_b=E_S$, and $d_b=1$.
The defining fields satisfy
\begin{equation}
 M_{j+1}=M_eM_j^p.
 \label{eq:frobenius-tower-field}
\end{equation}
Thus each $\pi_j$ has exponent at most one and
\Cref{lem:height-one} applies.

Put $T_j=(\Omega_{Y_j/\kappa})^\vee$ and
\[
 \mathcal F_j=\ker(T_j\longrightarrow\pi_j^*T_{j+1}).
\]
This is also the kernel of the composite tangent map
$T_j\to\gamma_j^*T_{E_S}$.  To see this, at the function field a
$\kappa$-derivation kills $M_e$ if and only if it kills
$M_eM_j^p=M_{j+1}$, since every derivation kills $p$th powers.
The two kernels are therefore equal generically.  They are equal as
sheaves: a section whose image vanishes generically has zero image
everywhere in either of the locally free, hence torsion-free, targets.

The genus identity involves $\deg_S\mathcal F_j$. We will estimate
this degree through the image of the tangent map to $E_S$.
Choose a nonzero invariant differential $\eta$ on $A_0$.
The constant-curve identification of $E_S$ gives the splitting
\[
 \Omega_{E_S/\kappa}
   \simeq\mathcal O_{E_S}\eta
      \oplus(\mathcal O_{E_S}\otimes_S\Omega_{S/\kappa}).
\]
Thus its dual is a trivial bundle of rank $n+1$. In this trivial
target we can use the marked points to impose linear vanishing
conditions on the tangent-map image.

Let $\mathcal H_j$ be the image in $\gamma_j^*T_{E_S}$, and put
$b_j^\circ=-\deg_S\mathcal H_j$.
It is a torsion-free coherent sheaf on a regular curve, hence locally
free.  Its rank is $n+1-m_j$, since
$\operatorname{trdeg}_\kappa S=n$.
The exact sequence with kernel $\mathcal F_j$ and image $\mathcal H_j$,
together with \eqref{eq:height-one-genus}, gives
\[
 \nu(Y_j)=q_j\nu(Y_{j+1})+(p-1)\deg_S\mathcal F_j,
 \qquad
 b_j^\circ=\nu(Y_j)+\deg_S\mathcal F_j.
\]
Eliminating $\deg_S\mathcal F_j$ and using $d_j=q_jd_{j+1}$ yields
\begin{equation}
 \frac{\nu(Y_j)}{d_j}
  =\frac1p\frac{\nu(Y_{j+1})}{d_{j+1}}
     +\left(1-\frac1p\right)\frac{b_j^\circ}{d_j}.
 \label{eq:genus-recurrence}
\end{equation}
The terminal term $\nu(Y_b)$ is zero because $E_S$ has genus one.
It remains to give a lower bound for
$b_j^\circ/d_j=-\deg_S\mathcal H_j/d_j$.

\textit{Vanishing at a ramified label.}
For each $j$ let $S_j$ be the set of labels at which $\gamma_j$ has
ramification index greater than one.  For $i\in S_j$ write
$D_{j,i}=\gamma_j^*(t_i)$ and
\[
 v_i=(1,-t_i^*\eta)\in S\oplus\Omega_{S/\kappa}.
\]
Regard $v_i$ as a constant linear functional on
$\gamma_j^*T_{E_S}$. We claim that its restriction to $\mathcal H_j$
vanishes along the entire divisor $D_{j,i}$; equivalently, its values
lie in $\mathcal O_{Y_j}(-D_{j,i})$.

Let $a$ be a local parameter defining the rational point $t_i$ on
$E_S$.  Restriction of absolute differentials along $t_i$ sends
$(\lambda,\beta)$ in the displayed splitting to
$\lambda t_i^*\eta+\beta$.  Its kernel is the line spanned by $v_i$.
Since $a$ restricts to zero, $da$ belongs to this kernel modulo $(a)$.
Moreover its relative component is nonzero: $t_i$ is an $S$-rational
point on a smooth curve over $S$, and $a$ generates its relative
cotangent space.  Hence
\[
 da\equiv\lambda_i v_i\pmod{a\Omega_{E_S/\kappa}}
 \quad\text{for some }\lambda_i\in S^\times.
\]
At the unique point upstairs write $a=u\tau^r$, with $u$ a unit.
The nontrivial purely inseparable index $r$ is divisible by $p$;
therefore
\[
 d(\gamma_j^*a)=\tau^r\,du=(\gamma_j^*a)\,u^{-1}du.
\]
It lies in the ideal of the \emph{full} divisor $D_{j,i}$ times
$\Omega_{Y_j/\kappa}$.  Pulling back the preceding congruence and
dividing by the unit $\lambda_i$ shows that the pullback of $v_i$ lies
in that same ideal times $\Omega_{Y_j/\kappa}$.
To relate this differential to the claimed functional, denote the
tangent map by $d\gamma_j:T_j\to\gamma_j^*T_{E_S}$ and write
$\gamma_j^*v_i$ for the image of the pulled-back covector in
$\Omega_{Y_j/\kappa}$. For a local section $\theta$ of $T_j$, duality
gives
\[
 v_i(d\gamma_j(\theta))=(\gamma_j^*v_i)(\theta).
\]
The divisibility just proved makes this value a section of
$\mathcal O_{Y_j}(-D_{j,i})$. Since $\mathcal H_j$ is the image of
$d\gamma_j$, the claim follows. The vanishing is along
$\gamma_j^*(t_i)$ with its full multiplicity, not just its reduced
support.

\textit{From independent vanishing conditions to a degree bound.}
By \Cref{lem:label-rank}, among the $v_i$ for $i\in S_j$ there are
$c\ge |S_j|-h$ independent covectors over $S$.

Complete the $c$ chosen covectors to a constant basis of the dual of
the target trivial tangent bundle.  The full-divisor constraint gives
an injection
\begin{equation}
 \mathcal H_j\hookrightarrow
 \mathcal O_{Y_j}^{n+1-c}\oplus
       \bigoplus_{i\ \mathrm{chosen}}\mathcal O_{Y_j}(-D_{j,i}).
 \label{eq:label-image-injection}
\end{equation}
Let $R=n+1-m_j$ be the rank of $\mathcal H_j$.
The $R$th exterior power of the bundle on the right is a direct sum
of line bundles, each the tensor product of a choice of $R$ summands.
The induced
map from $\det\mathcal H_j$ is nonzero, so it has a nonzero component
in at least one of these lines. Only $n+1-c$ of the available
summands are trivial. The chosen line must therefore contain at least
$R-(n+1-c)=c-m_j$ negative divisor summands when this number is
positive, and at least zero otherwise.
Each $D_{j,i}$ has degree $d_j$ by finite flatness and rationality of
$t_i$. If the nonzero component uses $k$ negative summands, its target
line has degree $-kd_j$.  A nonzero regular map from one line bundle
to another has an effective zero divisor, and consequently
\[
 \deg_S\mathcal H_j\le-kd_j,
 \qquad
 \frac{b_j^\circ}{d_j}\ge c-m_j\ge |S_j|-h-m_j.
\]
The argument also covers rank zero, with $\det(0)=\mathcal O$.

\textit{The weighted estimate.}
Write $r_i=p^{u_i}$.  Every height-one step has local index at most
$p$, as proved in \Cref{lem:height-one}. The composite
$Y_0\to Y_j$ consequently has ramification index at most $p^j$ at
this label. In particular, $u_i\leq b$. Since ramification indices
multiply, the remaining map
$Y_j\to E_S$ has index at least $p^{u_i-j}$ whenever $j-N.
\]
A multiple of $N$ strictly greater than $-N$ is nonnegative.  This
proves \eqref{eq:inseparable-genus-bound}.
The sufficient threshold \eqref{eq:genus-rounding-threshold} contains
no datum from $F$, $Z$, or a tuple-class multiset.  If a threshold
independent of the prime is desired, the stronger condition
\[
 2^{s-h}>m(h+\log_2m)
\]
suffices for every $p\ge2$.  This proves the stated uniformity.
\end{proof}
==== END The-Blockwise-Alperin-Weight-Conjecture//sections/differentials.tex ====

==== BEGIN The-Blockwise-Alperin-Weight-Conjecture//sections/foundations.tex ====
\section{Classical foundations used}\label{sec:foundations}

We state the classical inputs to make the scope of the proof explicit. We also use elementary linear algebra, field and group theory, and the basic definitions and functorial formalism of schemes, sheaves, divisors, and K\"ahler differentials. The counting, index, inseparable-genus, valued-field, and specialization assertions particular to this proof are established in the later sections. In particular, no weight-counting theorem or conjectural reduction is an input.

\begin{foundation}[Finite groups]\label{foundation:groups}
For a finite group $X$, its $p$-subgroups lie in Sylow subgroups of order $|X|_p$. A nontrivial finite $p$-group has nontrivial center and an element of order $p$, and each of its maximal proper subgroups is normal. We use the orbit--stabilizer formula and the usual fixed-point and conjugation actions.
\end{foundation}

\begin{foundation}[Finite-dimensional algebras and characters]\label{foundation:algebra}
The Jacobson radical of a finite-dimensional algebra is nilpotent. Over an algebraically closed field, its semisimple quotient is a product of full matrix algebras, one for each simple module up to isomorphism. A finite-dimensional commutative algebra over an algebraically closed field is a product of local algebras with that field as residue, indexed by its primitive idempotents. Finite group algebras in characteristic zero are semisimple. Over a splitting field the simple endomorphism algebras are the scalar field, and an ordinary irreducible character satisfies
\[
 \sum_{x\in X}\chi(x)\chi(x^{-1})=|X|.
\]
Idempotents lift uniquely from the residue algebra of a finite commutative algebra over a complete discrete valuation ring. Nakayama's lemma applies to finite modules over any local ring.
\end{foundation}

\begin{foundation}[Commutative algebra]\label{foundation:commalg}
A finite-type algebra over a Noetherian ring is Noetherian. Integral extensions satisfy lying over. Closed points of a variety over a field are dense and have finite residue fields over the ground field. For an integral variety, the local dimension at a point plus the transcendence degree of its residue field is the transcendence degree of the function field. A regular local ring is a normal domain; a part of a minimal system of generators of its maximal ideal is a regular sequence and its quotient is regular of the correspondingly smaller dimension. A one-dimensional Noetherian normal local domain is a discrete valuation ring, as is a Noetherian local domain with nonzero principal maximal ideal. Finite torsion-free modules over discrete valuation rings are free. Completion of a discrete valuation ring retains its residue field and uniformizer.
\end{foundation}

\begin{foundation}[Fields and absolute field differentials]\label{foundation:fields}
If a finite group $D$ acts faithfully by automorphisms on a field $M$, then $M/M^D$ is Galois of degree $|D|$. Finite separable field extensions are simple. A finite field extension has a maximal separable subextension, and the remaining extension is purely inseparable; the number of embeddings in an algebraic closure is the separable degree. Multiplicative groups of finite fields are cyclic. Algebraically closed fields are classified by their characteristic and transcendence degree. If $S$ is a finitely generated field of transcendence degree $\rho$ over a perfect field $\kappa$, then
\[
 \dim_S\Omega_{S/\kappa}=\rho;
 \qquad [S:S^p]=p^\rho\quad\text{in characteristic }p.
\]
\end{foundation}

\begin{foundation}[Valuations over a complete discrete base]\label{foundation:valuation}
A valuation extends to an algebraic field extension. A complete discrete valuation on a field extends uniquely to any finite field extension; its valuation ring is the integral closure of the original ring, is finite over it, and is again a complete discrete valuation ring. The fields to which we apply the latter assertion initially have characteristic zero. No corresponding finiteness assertion for an arbitrary nondiscrete valuation ring is assumed.
See \cite[Tag 00IA]{Stacks} for prolongation and
\cite[Tags 032W, 09EM and 0325]{Stacks} for the complete
discrete-base finiteness assertions.
\end{foundation}

\begin{foundation}[Descent and finite models]\label{foundation:descent}
A projective scheme with a semilinear descent datum under a finite Galois field extension descends if equipped with a compatible linearized ample line bundle. The same statement holds for a finite \emph{\'etale} faithfully flat Galois extension of rings and a relatively ample line bundle. Equivariant morphisms and invariant closed subschemes descend along with the scheme. Flatness, smoothness, finiteness, \emph{\'etaleness}, and relative ampleness in these settings are checked after the faithfully flat extension. Geometric integrality of a finite-type scheme over a field can be checked after an extension of that field. A finite diagram of projective schemes and morphisms of finite presentation over a directed union of fields is defined over a finite stage, including its maps to a base scheme already defined there and the identities among those maps. One may realize projective descent by descent of semilinear modules and graded algebras followed by $\Proj$.
For the projective and module descent assertions, see \cite[Tags 0CCJ, 0CDR and 0D1V]{Stacks}.
The five locality assertions are respectively
\cite[Tags 02L2, 02VL, 02LA, 02VN and 0D3C]{Stacks}; geometric
integrality is covered by \cite[Tags 038K, 0384 and 054P]{Stacks}.
The finite-stage assertion follows from \cite[Tags 01ZM and 01ZP]{Stacks},
including chosen projective embeddings among the finite data.
\end{foundation}

\begin{foundation}[Normalization]\label{foundation:normalization}
The normalization of an integral variety over a field in a finite extension of its function field is finite. The normalization of a finite-type $\mathbb Z$-domain in its fraction field is finite. In the integral closure of a normal domain in a finite Galois extension of its fraction field, the Galois group acts transitively on the primes above any fixed prime of the base.
The normalization finiteness assertions follow from the Nagata property in \cite[Tag 0335]{Stacks}.
\end{foundation}

\begin{foundation}[Regularity, smoothness, and properness]\label{foundation:regularity}
Finite separable scalar extension preserves regularity and reducedness. Smooth varieties over a field are regular. A regular local ring essentially of finite type over a perfect field $\kappa$ is smooth at the corresponding point of a finite-type $\kappa$-model; its module of absolute differentials is free of rank the transcendence degree of the function field over $\kappa$. We use the valuative criterion for properness, including the unique extension of maps from fraction fields of valuation rings, and that proper maps are closed.
For regularity and smoothness over a perfect field, see \cite[Tags 00TV and 0B8X]{Stacks}.
\end{foundation}

\begin{foundation}[Proper cohomology and ample section rings]\label{foundation:cohomology}
Over a Noetherian base, the higher direct images of coherent sheaves under a proper morphism are coherent. Over an affine base their cohomology commutes with flat affine base change. For a projective scheme over an affine Noetherian base and a relatively ample line bundle, sufficiently high powers give projective embeddings, and sufficiently high twists annihilate higher coherent cohomology. The ample section ring is finitely generated and its $\Proj$ recovers the scheme. Finite pullback preserves ampleness; tensor products of relatively ample line bundles are relatively ample, and a line bundle is relatively ample if a positive power is. Formation of $\Proj$ commutes with base change. On a projective curve over a field, coherent cohomology vanishes in degrees greater than one, and for sheaves of finite support in degrees greater than zero. Global functions on a geometrically integral projective variety over a field equal the ground field.
For proper coherence, flat cohomology base change, and ample twists, respectively, see \cite[Tags 02O5, 02KH and 0B5T]{Stacks}.
For Proj recovery and the ampleness rules, see
\cite[Tags 0EKI, 0892, 0890 and 02NN]{Stacks}.
\end{foundation}

\begin{foundation}[Differentials, adjunction, and duality]\label{foundation:duality}
We use the transitivity sequence
\[
 \Omega_{V/U}\otimes_VW\longrightarrow\Omega_{W/U}
 \longrightarrow\Omega_{W/V}\longrightarrow0
\]
and the conormal sequence for a closed immersion. Let $S$ be a field. On $\mathbb P^b_S$,
$\det\Omega_{\mathbb P^b_S/S}=\mathcal O(-b-1)$. For a coherent sheaf $\mathcal G$ there, Serre duality identifies
\[
 H^i(\mathcal G)^\vee
 \simeq\operatorname{Ext}^{b-i}
       (\mathcal G,\mathcal O(-b-1)).
\]
We use the local-to-global Ext spectral sequence and the Koszul resolution of a quotient by a regular sequence. These inputs will yield the required absolute determinant-degree formula even when the curve is not smooth over $S$.
For the projective-space dualizing complex and the adjunction formalism, see \cite[Tag 0A9W and Section~48.15]{Stacks}.
The local-to-global Ext sequence is \cite[Tag 0BQP]{Stacks}.
\end{foundation}

\begin{foundation}[Elliptic curves]\label{foundation:elliptic}
A smooth Weierstrass cubic with its point at infinity is an elliptic curve. A generalized Weierstrass equation over a discrete valuation ring with unit discriminant gives a smooth elliptic scheme, polarized by three times its origin. Its relative cotangent line is trivial with invariant differential $\eta$, and multiplication by an integer $a$ pulls $\eta$ back to $a\eta$. For a positive integer $a$, multiplication by $a$ on an elliptic curve or elliptic scheme is finite locally free of degree $a^2$.
For the relative group law, see
\cite[II, Proposition~2.7]{DeligneRapoport1973}; the unit-discriminant
Weierstrass family and the invariant cotangent description are given in
\cite[Tags 072S and 047I]{Stacks}. See also
\cite[Chapter~III]{Silverman2009} for the Weierstrass formulation.
Multiplication is finite flat and surjective by
\cite[Proposition~20.7]{MilneAV2022}; over the Noetherian bases here it
is finite locally free by \cite[Tag 02KB]{Stacks}.
\end{foundation}

\begin{foundation}[Complex curves and covers]\label{foundation:complex}
Finite \'etale algebraic covers of a smooth complex algebraic curve
correspond to finite unramified topological covers of its analytification,
including morphisms. For a punctured projective curve, normalization of
the original projective curve in the resulting algebraic function field
gives the finite compact cover. Normal complex curves are smooth, and the
local punctured covering completes by a power map, so monodromy cycle
lengths give ramification indices. Topological covers are classified by
fundamental-group actions on finite fibers. The fundamental group of a
genus-one surface punctured at $n$ points has generators
$x,y,u_1,\ldots,u_n$ with the single relation
\[
 [x,y]u_1\cdots u_n=1.
\]
For a finite map of smooth proper curves over an algebraically closed field of characteristic zero, Riemann--Hurwitz states
\[
 2g(C)-2=\deg(f)\bigl(2g(C')-2\bigr)
              +\sum_{P\in C}(e_P-1).
\]
For the algebraic/topological cover correspondence, see \cite[Theorem~3.4]{MilneLEC2008}; for Riemann--Hurwitz, see \cite[Section~53.12, Tag 0C1B]{Stacks}.
\end{foundation}

\begin{foundation}[Noetherian connectedness]\label{foundation:connectedness}
A proper morphism over a locally Noetherian base whose structure-sheaf direct image is canonically the structure sheaf of the base has geometrically connected fibers. This is the connected-fiber assertion of Stein factorization. We apply it only after constructing a Noetherian normal model, not directly over the nondiscrete valuation ring.
See \cite[Tag 03H0]{Stacks}.
\end{foundation}

\subsection{Coefficient fields and scalar extension}\label{sec:coefficients}

We justify the coefficient choice in \Cref{sec:introduction} without
requiring a mixed-characteristic lift of every algebraically closed
field. Put $\kappa=\overline{\mathbb F}_p$ and choose an embedding
$\kappa\hookrightarrow k$. For any finite group $X$, the ideal
$\rad(\kappa X)\otimes_\kappa k$ is nilpotent, and the quotient of
$kX$ by this ideal is the scalar extension of the split matrix product
$\kappa X/\rad(\kappa X)$. That quotient is again a split semisimple
algebra. The ideal is therefore exactly $\rad(kX)$, and the simple
modules correspond under scalar extension.

The center also commutes with this extension: it is the kernel of the
finitely many linear maps $a\mapsto ax-xa$, $x\in X$, and field
extension preserves kernels. Each local factor of $Z(\kappa X)$ has
nilpotent radical and residue $\kappa$; after extension it still has
nilpotent radical, now with residue $k$, so remains local. Thus
primitive blocks correspond and their simple-module counts agree.
Coefficient restriction, restriction to centralizers in the Brauer
map, and the quotient maps for normal subgroups all commute with
scalar extension. The residue characters of the local central factors
extend as well. Consequently the central-character block assignment
in \eqref{eq:block-assignment} is preserved, for every subgroup and
subquotient involved in a weight. It suffices to prove the numerical
identity over $\kappa$.

To construct the coefficient ring, start with $\mathbb Z_p$, lift a
tower of finite separable extensions of its residue field $\mathbb F_p$
that exhausts $\kappa$, and complete the union. The construction at the
start of the proof of \Cref{prop:mixed-lift} shows that this gives a
complete discrete valuation ring of characteristic zero with residue
$\kappa$.
In an algebraic closure of its fraction field, choose matrix
realizations of every ordinary irreducible representation of every
subgroup of $G$. There are only finitely many such representations and
matrix entries. Adjoining their entries gives a finite field extension
that splits all those subgroups. Its valuation ring is again a complete
DVR by \Cref{foundation:valuation}; its finite residue extension is
still $\kappa$, which is algebraically closed. This gives the system
$(K,\mathcal O,\kappa)$ used in \Cref{sec:group}. Quotient
representations inflate, so the same coefficients split the required
subquotients. These coefficient fields are separate from the auxiliary
geometric fields constructed later.

The remaining sections prove the blockwise chain identity, the marked
high-index twist, the genus estimate with exact rounding, the required
valued-field assertions, and the specialization model. The foundations
above supply their classical inputs, not the desired weight equality.
==== END The-Blockwise-Alperin-Weight-Conjecture//sections/foundations.tex ====

==== BEGIN The-Blockwise-Alperin-Weight-Conjecture//sections/group.tex ====
\section{From block counts to generating tuples}\label{sec:group}

Fix a prime $p$. Write $O_p(X)$ for the largest normal $p$-subgroup
of a finite group $X$, and put $[x,y]=xyx^{-1}y^{-1}$. A
$p$-element includes the identity; a $p'$-element has order prime to
$p$; and a $p$-singular element has order divisible by $p$.
The geometric sections will establish the following precise counting
statement. Its uniformity is part of the assertion.

\begin{theorem}[Uniform divisibility for generating tuples]
\label{thm:tuple-divisibility}
Let $J$ be a finite group with $O_p(J)=1$. For $s\geq0$, $n=p^s$,
and a multiset
$\mathcal M$ of $n$ conjugacy classes of $p$-singular elements of $J$,
let $N_{J,s}(\mathcal M)$ be the number, modulo simultaneous
$J$-conjugation, of ordered tuples
\[
 (x,y,u_1,\ldots,u_n)\in J^{n+2}
\]
satisfying
\begin{equation}\label{eq:tuple-relation}
 [x,y]u_1\cdots u_n=1,\qquad
 \langle x,y,u_1,\ldots,u_n\rangle=J,
 \qquad \{[u_1],\ldots,[u_n]\}=\mathcal M.
\end{equation}
For every integer $h\geq1$, there is an integer $s_0=s_0(p,J,h)$
such that
\[
 p^h\mid N_{J,s}(\mathcal M)
 \qquad(s\geq s_0)
\]
for every such multiset $\mathcal M$.
\end{theorem}

The entries $u_i$ remain ordered: prescribing their multiset of classes
does not quotient by permutations. Empty tuple sets have count zero,
so the statement includes $J=1$ and groups of order prime to $p$.
The proof of \Cref{thm:tuple-divisibility} is completed after
\Cref{prop:no-small-cover}. This section proves that it implies the
block equality, using \Cref{foundation:groups,foundation:algebra}
and proving the required counting reductions explicitly.

\subsection{Coefficient projection and the weight transform}

For each ambient finite group $X$, choose a splitting coefficient system
$(K,\mathcal O,k)$ as in \Cref{sec:introduction,sec:coefficients}, with
$X$ in place of $G$. Hold this system fixed throughout the calculations
with $X$ and its subquotients.
This system also splits all subquotients that occur below: a subgroup of a
quotient lifts to a subgroup by taking its preimage, and a
representation of a quotient inflates to that preimage. These
observations apply after any number of subgroup and quotient steps.
Ordinary irreducibles may therefore be realized over $K$. Their
identification with complex irreducibles, when needed, follows by
first splitting the group algebra over $\overline{\mathbb Q}$ and
then extending scalars to algebraically closed characteristic-zero
fields.

For a central idempotent $E\in kX$, define
\begin{align*}
 l(X,E)&=\#\{\text{simple $kX$-modules selected by $E$}\},\\
 k_{\mathrm{ord}}(X,E)&=\#\{\text{ordinary irreducibles selected by the lift of $E$}\},\\
 z(X,E)&=\#\{\chi\text{ selected by that lift}:\chi(1)_p=|X|_p\}.
\end{align*}
Here and throughout, modules and characters are counted up to
isomorphism. The class sums form a basis of $Z(\mathcal O X)$, and
their reductions form a basis of $Z(kX)$. Idempotent lifting over the
complete DVR thus gives a unique central idempotent lifting $E$.
Every integral central element acts on an ordinary irreducible by
a scalar in $\mathcal O$: the scalar lies in $K$ by splitting and
satisfies a monic integral equation by finiteness, and $\mathcal O$
is integrally closed. Reduction of this scalar defines a character
of $Z(kX)$. The Artinian local decomposition of that center shows
that it is precisely $\lambda_b$, where $b$ is the block containing
the ordinary irreducible.

For $H\leq X$, let $s_H$ denote coefficient restriction from
$Z(kX)$ to $Z(kH)$, or its integral analogue when appropriate.
For a $p$-subgroup $Q$, let $\operatorname{Br}_Q$ denote restriction
of coefficients from the $Q$-invariants $(kX)^Q$ to $kC_X(Q)$.

\begin{lemma}[Normalizer projection]\label{lem:normalizer-projection}
The map $\operatorname{Br}_Q:(kX)^Q\longrightarrow kC_X(Q)$ is a
unital algebra homomorphism. Suppose that
\[
 Q\leq H\leq N_X(Q),\qquad C_X(Q)\leq H.
\]
Writing $\pi_Q:kH\longrightarrow k[H/Q]$ for the quotient map, one
has, for $v\in Z(kX)$,
\begin{equation}\label{eq:normalizer-projection}
 \pi_Q(s_H(v))=\pi_Q(\operatorname{Br}_Q(v)),
 \qquad
 \lambda_b(s_H(v))=\lambda_b(\operatorname{Br}_Q(v))
\end{equation}
for every block $b$ of $H$. In particular $\lambda_b\circ s_H$ is
the central character of a unique block of $X$.
\end{lemma}

\begin{proof}
To compute the coefficient of $g\in C_X(Q)$ in a product of two
$Q$-invariant elements, let $Q$ conjugate the pairs $(a,b)$ with
$ab=g$. Coefficient products are constant on the orbits. Every
nontrivial orbit has size divisible by $p$, and the fixed pairs are
exactly those with $a,b\in C_X(Q)$. This proves multiplicativity;
the identity is preserved.

Each coset $hQ$ is stable under conjugation by $Q$, because
$h\in N_X(Q)$. Applying the same orbit cancellation to the sum
of coefficients over that coset proves the first identity in
\eqref{eq:normalizer-projection}. Also
$\operatorname{Br}_Q(v)\in Z(kH)$, since its support lies in
$C_X(Q)\leq H$ and it is invariant under $H$.

Every nonzero module for a finite $p$-group over $k$ has nonzero
invariants. For completeness, choose a central element $z$ of order
$p$; the nonzero kernel of $z-1$, whose $p$th power is zero, is a
module for the smaller group $Q/\langle z\rangle$, and induction
applies. If the module is simple for $H$, its $Q$-invariants are
$H$-stable, and hence the whole module. Thus every simple $kH$-module
factors through $H/Q$, proving the second identity. The composition
with $\operatorname{Br}_Q$ is a unital algebra character. Characters
of the finite commutative algebra $Z(kX)$ are its local residue
characters, one for each block, giving uniqueness.
\end{proof}

For $H=N_X(Q)$ put
\[
 \overline E_Q
   =\pi_Q(s_H(E))
   =\pi_Q(\operatorname{Br}_Q(E))\in Z(k[H/Q]).
\]
This is an idempotent. Its selected defect-zero quotient characters
are exactly the weights assigned to the blocks appearing in $E$.
Indeed, the lift of $\operatorname{Br}_Q(E)$ maps to the lift of
$\overline E_Q$ by uniqueness of central idempotent lifting. Its
action on the inflation of a quotient character is therefore one
exactly when the latter is selected by $\overline E_Q$. If $b$ is
the block of this inflation, the same test is
\[
 \lambda_b(\operatorname{Br}_Q(E))
   =\lambda_b(s_H(E))=1.
\]
By \Cref{lem:normalizer-projection}, this says that the induced block
is one of the blocks in $E$, with the coefficient-projection
definition in \eqref{eq:block-assignment}. Once $Q$ is fixed up to $X$-conjugacy,
there is no further identification of its quotient characters:
$N_X(Q)$ acts on $N_X(Q)/Q$ by inner automorphisms.

For a conjugation-invariant function $g$ of such pairs $(X,E)$,
define
\begin{equation}\label{eq:weight-transform}
 (Tg)(X,E)=
 \sum_{\substack{10$, the binomial
formula gives
\[
 \operatorname{val}((1+a)^p-1)
 \geq
 \min\{\operatorname{val}(p)+\operatorname{val}(a),
                    p\operatorname{val}(a)\}.
\]
After normalizing the discrete valuation to integer values, each
iteration increases the positive valuation by at least one, unless
the difference has already become zero. These powers tend to one.
This argument applies also to $p=2$ and requires no finiteness of
the residue field.

For positive-length chains the limiting trace therefore counts
all ordinary irreducibles selected by $E_{\mathcal C}$. At length
zero it omits the semisimple blocks, each of which contributes
exactly one defect-zero character by
\Cref{lem:defect-zero-detection}. There are finitely many chains
and irreducibles, so limits may be summed, proving
\eqref{eq:trace-limit}.
\end{proof}

The next identity follows from Frobenius's commutator-counting formula
and is the weighted genus-one case of the surface formula in
\cite[Theorem~2.1 and Equation~(3.1)]{Klug2025}. Its single
denominator is important when chain orbits are replaced by actual
chains, so we verify the normalization directly.

\begin{lemma}[Trace normalization]\label{lem:tuple-trace}
For $H\leq X$ and $w=s_H(\widehat v)$,
\begin{equation}\label{eq:tuple-trace}
 \operatorname{Tr}(w^n\mid V(KH))
 =\frac1{|H|}
   \sum_{\substack{x,y,u_1,\ldots,u_n\in H\\
                   [x,y]u_1\cdots u_n=1}}
       \prod_{i=1}^n\alpha_{u_i}.
\end{equation}
\end{lemma}

\begin{proof}
The sum on the right before division is
$\tau(d_Hw^n)$. The regular trace of any element of $KH$ is
$|H|$ times its identity coefficient. On the other hand,
\eqref{eq:ordinary-commutator-scalar} and the regular decomposition
give
\begin{align*}
 |H|\tau(d_Hw^n)
  &=\sum_{\chi\in\operatorname{Irr}(H)}
       \chi(1)^2\frac{|H|^2}{\chi(1)^2}\lambda_\chi(w)^n\\
  &=|H|^2\operatorname{Tr}(w^n\mid V(KH)).
\end{align*}
Dividing proves exactly the single denominator in
\eqref{eq:tuple-trace}.
\end{proof}

For a subgroup $J\leq X$, define an integer on actual chains,
without quotienting by conjugation,
\begin{equation}\label{eq:chain-weight}
 a_X(J)=
 \sum_{\substack{\mathcal C\text{ a chain in }X\\
                  J\leq H_{\mathcal C}}}(-1)^{|\mathcal C|}.
\end{equation}

\begin{lemma}[Cancellation by a normal $p$-subgroup]
\label{lem:pcore-cancellation}
If $O_p(J)\ne1$, then $a_X(J)=0$.
\end{lemma}

\begin{proof}
Put $P=O_p(J)$. In any chain normalized by $J$, each $Q_i$ is
normalized by $P\leq J$, so $Q_iP$ is a $p$-subgroup. Both
factors are normalized by $J$, making their product
$J$-normalized. Let $i$ be the last index such that $P\nleq Q_i$;
it exists since $Q_0=1$ and $P\ne1$. Toggle $Q_iP$ immediately
after $Q_i$. It strictly contains $Q_i$ and is contained in all
later members. Insertion and deletion preserve the last member
not containing $P$, so this is an involution. It is defined also
for the length-zero chain and for terminal insertion or deletion.
It preserves the property of being normalized by $J$ and reverses
the sign. The sum of actual chains in \eqref{eq:chain-weight}
therefore vanishes.
\end{proof}

\begin{proposition}[Reduction to uniform tuple divisibility]
\label{prop:group-reduction}
If \Cref{thm:tuple-divisibility} holds for every finite group with
trivial $p$-core, then for every finite group $X$ and central
idempotent $E\in kX$ one has $l(X,E)=((1+T)z)(X,E)$.
In particular the block equality in \Cref{thm:main} follows, with
the stated block assignment and with $Q=1$ included.
\end{proposition}

\begin{proof}
The orbit of a chain $\mathcal C$ has size
$|X|/|H_{\mathcal C}|$. Consequently
\Cref{lem:tuple-trace} changes the orbit sum
\eqref{eq:chain-trace} into the following sum over actual tuples
and actual chains:
\begin{equation}\label{eq:trace-by-generated-group}
 \Phi_n=\frac1{|X|}
 \sum_{\substack{x,y,u_1,\ldots,u_n\in X\\
                 [x,y]u_1\cdots u_n=1}}
 a_X\bigl(\langle x,y,u_1,\ldots,u_n\rangle\bigr)
 \prod_{i=1}^n\alpha_{u_i}.
\end{equation}
To verify the factor, each orbit contribution has prefactor
$1/|H_{\mathcal C}|$; replacing its representative by all
$|X|/|H_{\mathcal C}|$ conjugates gives the common prefactor
$1/|X|$. For a fixed tuple the admissible chains are exactly those
normalized by its generated subgroup. This is precisely
$a_X(J)$.

By \Cref{lem:pcore-cancellation}, only generated subgroups
$J$ with $O_p(J)=1$ contribute. By the support of $\widehat v$,
only tuples with every $u_i$ $p$-singular contribute. Fix such a
subgroup $J$ and group its generating tuples according to the
multiset $\mathcal M$ of $J$-conjugacy classes of the $u_i$.
The product $\prod_i\alpha_{u_i}$ is constant on this multiset:
the coefficients are constant on $X$-classes, hence on $J$-classes,
and their product is unchanged by reordering. Denote this integral
product by $\alpha(\mathcal M)$. A generating tuple has simultaneous
conjugation stabilizer $Z(J)$, since an element centralizing all
its entries centralizes the group they generate. Thus its number
of actual tuples is $|J|/|Z(J)|$ times its orbit count. For
$n=p^s$, the numerator in \eqref{eq:trace-by-generated-group}
is therefore
\begin{equation}\label{eq:uniform-numerator}
 \sum_{\substack{J\leq X\\O_p(J)=1}}
 a_X(J)\frac{|J|}{|Z(J)|}
 \sum_{\mathcal M}
   \alpha(\mathcal M)N_{J,s}(\mathcal M).
\end{equation}

Fix any $h\geq1$. By \Cref{thm:tuple-divisibility}, each summand
of the inner sum belongs to $p^h\mathcal O$ for all sufficiently
large $s$, with a threshold independent of $\mathcal M$. Although
the number of multisets grows with $s$, the sum is finite for each
$s$ and remains in that same ideal. There are only finitely many
subgroups $J\leq X$, so take the largest of their thresholds.
The integer factors in \eqref{eq:uniform-numerator} preserve
divisibility. We conclude that its numerator tends to zero in the
valuation topology. The fixed denominator $|X|$ does not change
this conclusion, so $\Phi_{p^s}\longrightarrow0$.

Combine this with \Cref{lem:trace-limit} and
\Cref{lem:alternating-subsections}. The resulting equality is
\eqref{eq:chain-inversion}; its integer terms agree in the
characteristic-zero field $K$, and hence agree as integers.
The same reasoning applies to every pair and to the splitting
subquotients arising in the transform. \Cref{lem:chain-transform}
now gives $l=(1+T)z$. The weight interpretation of
\eqref{eq:weight-transform} proves the asserted block equality.
\end{proof}
==== END The-Blockwise-Alperin-Weight-Conjecture//sections/group.tex ====

==== BEGIN The-Blockwise-Alperin-Weight-Conjecture//sections/intro.tex ====
\section{Introduction}\label{sec:introduction}

The local--global principle in modular representation theory seeks to recover
information about representations of a finite group from normalizers of its
$p$-subgroups. Alperin's weight conjecture predicts an exact numerical form
of this principle, block by block: simple modules are counted by defect-zero
ordinary characters of normalizer quotients. Alperin formulated the
conjecture at the 1986 Arcata conference; the original account appeared in
the proceedings in 1987 \cite{Alperin1987,FLZ2023}.

Let $p$ be a prime and $G$ a finite group. For an algebraically closed field
$k$ of characteristic $p$, a block is a primitive central idempotent of
$kG$. For a block $B$, let $l(B)$ be the number of simple $kG$-modules
belonging to $B$, equivalently the number of its irreducible Brauer
characters. For character calculations we use a splitting $p$-modular
system $(K,\mathcal O,k)$: $\mathcal O$ is a complete discrete valuation
ring with characteristic-zero fraction field $K$ and residue field $k$,
and $K$ splits every subgroup of $G$. A block also denotes its unique
central lift to $\mathcal O G$. In \Cref{sec:coefficients} we justify
reduction to $k=\overline{\mathbb F}_p$ and construct such a system;
the resulting counts and block assignments are unchanged over any
algebraically closed extension of this field.

For a positive integer $a$, write $a_p$ for its largest power-of-$p$ divisor. A \emph{$p$-weight} of $G$ is a pair $(Q,\phi)$, where $Q\leq G$ is a possibly trivial $p$-subgroup and
\[
 \phi\in\Irr(N_G(Q)/Q),\qquad
 \phi(1)_p=|N_G(Q)/Q|_p.
\]
Thus $\phi$ is an ordinary irreducible character of defect zero in the quotient. The block assigned to this weight is defined using the block $b$ of the inflated character in $H=N_G(Q)$.

We specify the block-assignment convention. For $H\leq G$, coefficient restriction is the linear map
\[
 s_H:Z(kG)\longrightarrow Z(kH),\qquad
 \sum_{g\in G}a_g g\longmapsto\sum_{h\in H}a_hh.
\]
If $c$ is a block of a finite group, its central character $\lambda_c$ is the residue character of the corresponding local factor of the center. In particular it is the scalar action of the center on any simple module in $c$. We say that $b^G=B$ when
\begin{equation}\label{eq:block-assignment}
 \lambda_b\circ s_H=\lambda_B.
\end{equation}
This is the central-character definition of Brauer block induction \cite[p.~338]{Carlson1971}; see also the corrigendum \cite{Carlson1973}. In the normalizer situations arising from weights, the composite is indeed a central character; we prove this directly in \Cref{sec:group}. Let $W_p(B)$ be the set of simultaneous $G$-conjugacy classes of weights assigned to $B$.

\begin{theorem}\label{thm:main}
For every prime $p$, every finite group $G$, and every $p$-block $B$ of $G$,
\[
                  l(B)=|W_p(B)|.
\]
\end{theorem}

\Cref{thm:main} resolves the numerical blockwise Alperin weight conjecture
positively, with no restriction on the defect groups, the prime, or the
finite group. The case $Q=1$ is part of the assertion. The conclusion is
an equality of cardinalities; a canonical or equivariant correspondence
is not required for this numerical statement.

\subsection*{Local--global consequences}

The numerical equality also gives the following established consequences
of the weight conjecture. Write $\IBr_p(H)$ for the set of irreducible
$p$-Brauer characters of a finite group $H$.

\begin{corollary}[Local--global and principal-block bounds]\label{cor:block-consequences}
Let $p$ be a prime, $G$ a finite group, $B$ a $p$-block with defect group
$D$, and $b$ its Brauer correspondent in $N_G(D)$. Then
\[
 l(B)\geq l(b),\qquad
 D\text{ abelian}\Longrightarrow l(B)=l(b).
\]
For $P\in\operatorname{Syl}_p(G)$, one also has
\[
 |\IBr_p(G)|\geq|\IBr_p(N_G(P))|.
\]
Let $B_0$ be the principal $p$-block of $G$, and let $k_p(G)$ count the
conjugacy classes of $p$-power-order elements, including the identity.
Then
\[
 \begin{aligned}
 k_p(G)=3&\Longrightarrow l(B_0)\geq (p-1)/2,\\
 p\mid |G|&\Longrightarrow l(B_0)\geq 2\sqrt{p-1}+1-k_p(G).
 \end{aligned}
\]
\end{corollary}

The local block bounds are Alperin's Consequences~1--2
\cite{Alperin1987}; the principal-block bounds are due to Hung,
Sambale, and Tiep \cite[Propositions~3.1--3.2]{HST2024}.
Their weight-conjecture hypotheses are supplied by \Cref{thm:main}.
The complete derivation, including the passage to the block-free
inequality, appears in \Cref{sec:consequences}.

The block-free inequality for $p=2$ was already proved by
Malle--Navarro--Tiep \cite[Theorem~B]{MNT2023}; the local inequality for
maximal-defect $2$-blocks was proved by Feng--Mart\'inez--Rossi
\cite[Theorem~D]{FMR2025}.

The sharper classification of character-count and defect data, and the
functorial formulation of the theorem, are given in
\Cref{sec:consequences}. We first explain the historical setting and
the proof of the local--global equality itself.

\subsection*{Prior work and the proof's ingredients}

Alperin's formulation extends the notion of weights from the modular
representation theory of finite groups of Lie type in defining
characteristic to arbitrary finite groups \cite{AlperinFong1990}.
Early large-family cases include Alperin and Fong's work on symmetric
groups and on finite general linear groups in odd modular characteristic
\cite{AlperinFong1990}. Kn\"orr and Robinson expressed the conjecture
as alternating sums over chains of $p$-subgroups
\cite{KnorrRobinson1989}. Navarro and Tiep reduced the non-blockwise
conjecture to conditions on finite simple groups \cite{NavarroTiep2011},
and Sp\"ath obtained a blockwise reduction \cite{Spath2013}. These
reductions organize the problem around local correspondences for simple
groups. Among the resulting advances, Feng, Li, and Zhang proved the
blockwise conjecture for finite special linear and special unitary groups
and for finite groups with abelian Sylow $3$-subgroups \cite{FLZ2023}.
Feng, Malle, and Zhang subsequently studied generic weights for finite
reductive groups under explicit hypotheses \cite{FMZ2026}.

Other recent progress includes the inductive blockwise condition for
$F_4(q)$ at odd primes not dividing $q$, proved by An, Hiss, and L\"ubeck
\cite{AnHissLubeck2024}, and Hu and Zhou's results on inertial blocks,
including an alternative proof for $2$-blocks with abelian defect
\cite{HuZhou2025}. Huang and Y\i lmaz develop chain, Galois, and
functorial reformulations \cite{HuangYilmaz2026}. A recent preprint of
Zhang claims the inductive condition for types $\mathsf B$, $\mathsf C$
and sporadic groups \cite{Zhang2026BC}; those classes do not exhaust
the scope of \Cref{thm:main}. Our proof instead passes from a direct
block-algebra calculation to covers of curves; it does not use a
classification of finite simple groups or an inductive weight condition.

The algebraic part uses classical constructions with a specific counting
purpose. The subgroup-chain argument is related to the Kn\"orr--Robinson
reformulation. Powers modulo commutators belong to K\"ulshammer's theory
of generalized Reynolds ideals \cite{HHKM2005}, and conjugation averaging
is the group-algebra Higman trace \cite{Higman1955,LorenzTokoly2011}.
The sum of the $p$-elements detects defect-zero blocks as in Tsushima's
work \cite{Tsushima1971}.
The passage from central traces to tuples is a weighted genus-one form
of Frobenius's character-counting formula; see \cite{Klug2025} for the
surface-group formulation. We prove the exact versions needed here,
including the block projections and normalization of the tuple count.
The further assertion is uniform divisibility as the number of punctures
increases, not merely a character formula for a fixed surface.

Chen's relation between Nielsen orbit counts and degrees of moduli of
elliptic covers \cite{Chen2024} informed the geometric divisibility
viewpoint. The many-marked, high-index construction and the uniform
puncture-divisibility theorem required here are developed in this paper,
not supplied by Chen's theorem.

The geometry combines several established ideas with that divisibility
problem. High-index genus-one curves are classical objects in period--index
theory; Clark and Lacy prove arbitrary-index existence over infinite fields
finitely generated over their prime fields \cite{ClarkLacy2019}. Our
construction also prescribes marked points giving independent absolute
field differentials. The genus calculation belongs to the theory of inseparable genus
change initiated by Tate \cite{Tate1952}; see Schr\"oer
\cite{Schroer2009} for a modern treatment. Its height-one determinant
identity is a curve-level form of the canonical-bundle formula for
$p$-closed foliations \cite{Ekedahl1988}. Here the calculation must keep
additional constants and inseparable residue degrees, and the large index
turns a bounded-error estimate into an exact inequality. Finally,
maximal-immediate-extension methods of Krull and Kaplansky
\cite{Krull1932,Kaplansky1942,Poonen1993} provide a suitable valued field,
while the center-valued obstruction studied by D\`ebes and Douai
\cite{DebesDouai1997} explains the arithmetic descent issue for marked
covers. We give the required constructions and descent argument directly.

\subsection*{The mechanism of the proof}

Chain inversion first reduces the weight equality to an identity between
a defect-zero character count and an alternating sum of simple-module
counts in subgroup normalizers. A calculation with powers modulo
commutators, followed by cancellation of chains, replaces those simple
counts by ordinary-character counts. We then construct a central
group-algebra element supported on $p$-singular elements, with residue
scalars zero or one chosen to remove the defect-zero contribution from
the alternating sum. Lift it by class sums to characteristic zero and restrict
it to each chain normalizer. The alternating sum of traces of the
$p^s$th powers converges to the difference between the alternating
ordinary-character sum and the defect-zero count.
Expanding the traces gives weighted tuple counts. A further chain
cancellation leaves only tuples generating subgroups with no nontrivial
normal $p$-subgroup. The complete reduction is
\Cref{prop:group-reduction}.

Here is the counting problem that remains. Fix a finite group $J$ with
$O_p(J)=1$, where $O_p(J)$ is its largest normal $p$-subgroup, and set
$n=p^s$. We count ordered tuples $(x,y,u_1,\ldots,u_n)\in J^{n+2}$
satisfying
\[
 [x,y]u_1\cdots u_n=1,\qquad
 \langle x,y,u_1,\ldots,u_n\rangle=J.
\]
Here $[x,y]=xyx^{-1}y^{-1}$, and each puncture entry $u_i$ is
$p$-singular: its order is divisible by $p$. Prescribe a multiset of
$n$ conjugacy classes for the $u_i$, and identify tuples only by
simultaneous inner conjugation in $J$, not by permutations of their
positions. For every fixed integer $h\geq1$, we must prove that all
these counts are divisible by $p^h$ for sufficiently large $s$
(\Cref{thm:tuple-divisibility}). The threshold may depend on $p,J,h$,
but not on the multiset. This uniformity makes each weighted trace sum
tend to zero even though the number of multisets grows with $s$.

The geometric construction supplies two properties on the same marked
genus-one curve $E/F_0$ in characteristic $p$. First, every divisor
degree on $E$ is divisible by $p^s$. Second, after a cyclic splitting
extension it becomes a constant elliptic curve with independently
varying marked points. More precisely, put
$\kappa=\overline{\mathbb F}_p$, choose an ordinary elliptic curve
$A_0/\kappa$, and take $F'=\kappa(A_0^n)$. The extension $F'/F_0$
is cyclic Galois of degree $n$ and splits $E$. Over $F'$, the marked points are the
generic projection points $t_i\in A_0(F')$. If $\eta$ is a nonzero
invariant differential on $A_0$, then the pullbacks $t_i^*\eta$
form a basis of $\Omega_{F'/\kappa}$. Their independence is in this
field-differential space, not in the one-dimensional space of regular
relative differentials on the elliptic curve. After a finite extension
$F/F_0$ with $[F:F_0]_p\leq p^h$, every divisor degree on $E_F$
is still divisible by $p^{s-h}$, and at most $h$ independent
differential directions are lost over the compositum $FF'$
(\Cref{prop:twist-index,cor:index-base-change,lem:label-rank}).

For a purely inseparable cover of this curve, a calculation along
successive maps of exponent one gives a genus bound with an error controlled
by $h$ and the cover degree. The index property makes this bound exact:
after multiplication by the cover degree, the difference between the
genus term and the ramification expression is a multiple of $p^{s-h}$.
The differential calculation bounds that difference below by a negative
constant independent of $s$. Once $p^{s-h}$ exceeds the constant's
magnitude, the difference must be nonnegative. This is the rounding
step in \Cref{thm:inseparable-genus}. The resulting estimate is useful
independently of the block-algebra reduction: it applies to regular
curves over imperfect fields, retains inseparable residue degrees and
extra constants, and does not require geometric reducedness.

We lift the marked genus-one curve to mixed characteristic and pass to a
spherically complete valued field $K_s$ with value group $\mathbb Q$
and residue field $F_0$. Over an algebraic closure, the generating
tuples are precisely the monodromy data of connected $J$-covers of the
punctured genus-one curve, with the $J$-action specified. Simultaneous
inner conjugacy changes only the chosen fiber point; it does not forget
the ordering of the punctures. The lift contains all $|J|$th roots of
unity. Galois therefore preserves the local conjugacy classes as it
permutes the marked points, so it acts on the finite cover set with
the prescribed multiset. This action factors through a finite Galois
group $\Gamma$; take a Sylow $p$-subgroup of $\Gamma$.

An orbit of size less than $p^h$ would give a cover defined over an
extension $L/K_s$ with $[L:K_s]_pm\bigl(h+\log_p m\bigr).
\end{equation}
This is a sufficient threshold for \Cref{thm:inseparable-genus}.
The relatively ample effective divisor $D$ has degree $n$ on each fiber
and is disjoint from the marked sections after their splitting extension.

\begin{proposition}\label{prop:no-small-cover}
Suppose $O_p(J)=1$ and \eqref{eq:specialization-threshold} holds. There is no
finite extension $L/K_s$ with $[L:K_s]_p0\}=\mathfrak m_R A_j.
\end{equation}
These assertions also hold on any $L$-linear subspace of $V_j$.
At this point reduction means the map of rational functions
\[
 A_j\longrightarrow\prod_{w\in\mathcal V}K(w),
 \qquad a\longmapsto(\bar a_w)_w.
\]
Its kernel is the positive ball $\mathfrak m_R A_j$, so its image
has dimension $\dim_L V_j$ over $F$. No model of $Y$ has been used.
The next lemma locates these residue-field elements in the section
spaces of the separately normalized curves $C_w$.

\begin{lemma}\label{lem:section-specialization}
For all sufficiently large $j$, simultaneous residue gives an injection
\begin{equation}\label{eq:section-reduction}
 A_j/\mathfrak m_R A_j
 \lhook\joinrel\longrightarrow
 \bigoplus_{w\in\mathcal V}
 H^0(C_w,\mathcal O_{C_w}(jD_w)).
\end{equation}
These maps respect products. If a section vanishes on the reduced
inverse images of a selected set of marked points, its reductions
vanish on the corresponding reduced inverse images on the $C_w$,
provided the selections are defined over the respective ground fields.
The vanishing assertion may be checked after splitting the marked
points over $L'$ and $S$.
\end{lemma}

\begin{proof}
Let $a\in A_j$. Form the polynomial using every element of $J$,
including any repeated conjugates:
\begin{equation}\label{eq:conjugate-section-polynomial}
 P_a(T)=\prod_{g\in J}(T-g(a))
       =T^m+c_1T^{m-1}+\cdots+c_m.
\end{equation}
The coefficients are $J$-invariant rational functions, hence belong to
$L(Y)^J=L(X)$. At a fixed $w\in\mathcal V$ every
$g(a)$ is integral, because $g$ permutes $\mathcal V$. Thus each $c_k$
has nonnegative Gauss value. Moreover
\[
 c_k\in H^0(X,\mathcal O_X(kjD_X)).
\]
To see the pole bound, each conjugate has poles bounded by $jD_Y$, so
the elementary symmetric function of degree $k$ has poles bounded by
$kjD_Y$. Since the coefficient is a rational function on $X$, its
pole bound on $X$ follows from that on its finite surjective pullback.

By \Cref{lem:gauss-section-rule}, $\bar c_k$ has poles bounded by
$kjD_F$. The residue $\bar a_w\in K(w)$ satisfies the reduction of
\eqref{eq:conjugate-section-polynomial}. Let $t$ be a local equation for
$jD_F$ at any point of $E_F$. Then $t\bar a_w$ satisfies the monic
equation whose coefficient of degree $m-k$ is $t^k\bar c_k$.
These coefficients are regular there. At every point of $C_w$ above
this neighborhood, normality therefore makes $t\bar a_w$ regular.
This proves the required pole bound and hence
\eqref{eq:section-reduction}; injectivity was established in
\eqref{eq:section-lattice}. Residue maps themselves respect multiplication.

For the last assertion, work at a selected marked point after splitting.
Every $g(a)$ vanishes at every point of its reduced inverse fiber,
because this fiber is $J$-invariant. The non-leading coefficients
$c_k$ consequently vanish at the marked point downstairs. There are
no poles there, since $D_X$ avoids all the marked points.
By \eqref{eq:gauss-evaluation}, every $\bar c_k$ vanishes at $t_i$.
The reduced monic equation at any point above $t_i$ is then
$\bar a_w^m=0$ in its residue field, so $\bar a_w$ vanishes there.
\end{proof}

\subsection{Residue degrees and the inseparable quotient}

For $j$ sufficiently large, the line-bundle Euler-characteristic formula
and ample vanishing turn \eqref{eq:section-reduction} into
\begin{equation}\label{eq:full-section-comparison}
 jmn+\chi(Y)
 \leq jn\sum_{w\in\mathcal V}m_w
       +\sum_{w\in\mathcal V}\chi(C_w).
\end{equation}
Here $\deg D_X=\deg D_F=n$, and finite pullback multiplies divisor
degree by the map degree. Letting $j$ grow gives
$m\leq\sum_wm_w$. On the other hand,
\Cref{lem:simultaneous-residues}, applied to the Gauss valuation on
$L(X)$ and these distinct prolongations to the degree-$m$ extension
$L(Y)$, gives the reverse inequality. Its hypotheses hold: the value
group is $\mathbb Q$ and the residue field $F(E_F)$ is infinite.
We have proved
\begin{equation}\label{eq:residue-degree-equality}
 \sum_{w\in\mathcal V}m_w=m.
\end{equation}

Let $\Delta_w\leq J$ be the stabilizer of $w$, and let $I_w$ be the
kernel of its action on $K(w)$. The orbit has $m/|\Delta_w|$ members,
all of the same residue degree. Therefore
\begin{equation}\label{eq:decomposition-residue-degree}
 m_w=|\Delta_w|.
\end{equation}
In particular this equality follows from the section comparison, before
any assertion about the degree of an inseparable quotient.

\begin{lemma}\label{lem:inseparable-residue-quotient}
The invariant field $M_w=K(w)^{\Delta_w}$ is purely inseparable over
$F(E_F)$, of degree
\begin{equation}\label{eq:inertia-inseparable-degree}
 d=[M_w:F(E_F)]=|I_w|.
\end{equation}
In particular $d$ is a power of $p$ and $d\leq m$. If $Z_w$ denotes the
normalization of $E_F$ in $M_w$, the map $C_w\to Z_w$ is generically
separable Galois of degree $m_w/d$, with group $\Delta_w/I_w$.
All these towers are isomorphic over $E_F$ as $w$ varies in $\mathcal V$.
\end{lemma}

\begin{proof}
Take $x\in K(w)^{\Delta_w}$. Simultaneous approximation from
\Cref{lem:simultaneous-residues} gives an element $a\in L(Y)$ integral
at every member of $\mathcal V$, with residue $x$ at $w$ and zero at
the other members. The coefficients of the conjugate polynomial
$P_a(T)$ in \eqref{eq:conjugate-section-polynomial} have nonnegative
Gauss value. At $w$, its reduction is
\begin{equation}\label{eq:invariant-residue-polynomial}
 \overline{P_a}(T)
 =T^{m-|\Delta_w|}(T-x)^{|\Delta_w|}
 \in F(E_F)[T].
\end{equation}
Indeed, elements of $\Delta_w$ act on the residue $x$ trivially, and
each other conjugate draws its residue from a different member of
the orbit, where the prescribed residue is zero.

Write $|\Delta_w|=p^v q$ with $p\nmid q$. In characteristic $p$,
\[
 (T-x)^{|\Delta_w|}=(T^{p^v}-x^{p^v})^q.
\]
The coefficient of $T^{m-p^v}$ in
\eqref{eq:invariant-residue-polynomial} is therefore
$-q x^{p^v}$. The scalar $q$ is nonzero in $F(E_F)$, so
$x^{p^v}\in F(E_F)$. This also covers $x=0$, for which the coefficient
is zero. Thus $M_w/F(E_F)$ is purely inseparable.

The effective automorphism group on $K(w)$ is $\Delta_w/I_w$.
Finite fixed-field theory gives
\[
 [K(w):M_w]=|\Delta_w/I_w|.
\]
Combining this with \eqref{eq:decomposition-residue-degree} and the
tower formula proves \eqref{eq:inertia-inseparable-degree}.
The assertions about the normalizations and their separable function
fields follow. Conjugation by $J$ supplies the asserted isomorphisms
between the towers, so $d$ is independent of $w$.
\end{proof}

\subsection{Ramification on the residue curves}

Pass temporarily to the finite separable splitting field $S/F$.
The curve $Z_{w,S}$ remains integral: its finite map to $E_S$ is
surjective and radicial, while separable scalar extension preserves
regularity and hence reducedness. The curve $C_{w,S}$ is regular and
is a disjoint union of integral components. Each component dominates
$Z_{w,S}$: the finite flat map $C_{w,S}\to Z_{w,S}$ restricts on a
component to a nonzero finite locally free summand, of positive rank
on the connected target. The components are consequently the
normalizations in their respective function fields. These uses of
normalization and separable scalar extension are included in
\Cref{foundation:normalization,foundation:regularity}.

Let $z_i$ be the unique point of $Z_{w,S}$ above $t_i$, and write $r_i$
for its ramification index over $t_i$. Write $a_i$ for the ramification
index of $C_{w,S}\to Z_{w,S}$ above $z_i$. We justify that $a_i$ is
independent of the point above $z_i$, even when $C_{w,S}$ is disconnected.
Put $G_w=\Delta_w/I_w$ and $M_{w,S}=S(Z_{w,S})$. The generic algebra
of this map is
\[
 K(w)\otimes_{M_w}M_{w,S}=\prod_{\lambda=1}^t K_\lambda.
\]
It is a product of fields because $K(w)/M_w$ is finite separable.
The group $G_w$ acts on the first tensor factor, and the invariant
algebra is $M_{w,S}$: taking invariants is taking a kernel of linear
maps, which commutes with this scalar extension. An orbit of the
field factors gives an invariant idempotent. Since $M_{w,S}$ is a
field, there can be only one orbit.

The total dimension over $M_{w,S}$ is $|G_w|$, and transitivity makes
all factor degrees equal to $|G_w|/t$. The stabilizer of a factor has
this same order. It acts faithfully on that factor, since the map
from $K(w)$ into the factor is injective. Thus the factor extension
is Galois with that stabilizer as its group. Transitivity on primes
in each Galois normalization gives transitivity within each component
above $z_i$; the action on the factors joins these componentwise
orbits. We obtain transitivity on the entire fiber of $C_{w,S}$.
Ramification indices are preserved by this action. The isomorphisms
between the towers for different $w$ give the same $r_i,a_i$ throughout
the orbit.

Multiplicities in the tower multiply, so every point $P$ of $C_{w,S}$
above $t_i$ has index $r_i a_i$ over $t_i$. Finite flatness gives
\[
 \sum_{P\in C_{w,S},\ P\mid t_i}
    r_i a_i\,[\kappa(P):S]=m_w.
\]
Summing over $w$ and using \eqref{eq:residue-degree-equality} yields
\begin{equation}\label{eq:reduced-special-fiber-degree}
 \sum_{w\in\mathcal V}\ 
 \sum_{P\in C_{w,S},\ P\mid t_i}[\kappa(P):S]
 =\frac{m}{r_i a_i}.
\end{equation}
In particular all residue-field degrees, including inseparable ones,
are retained in the degree of the reduced fiber.

Apply \Cref{lem:separable-different} to $C_{w,S}\to Z_{w,S}$,
component by component. At a point above $z_i$, the determinant
vanishing is at least
$a_i-1+\mathbf 1_{p\mid a_i}$. The total separable degree of the
components over $Z_{w,S}$ is $m_w/d$. The preceding local-degree
calculation gives $m_w/(r_i a_i)$ for the sum of the full residue-field
degrees above $t_i$. Hence, for each $w$,
\[
 \nu(C_w)\geq\frac{m_w}{d}\nu(Z_w)
 +\sum_{i=1}^n\frac{m_w}{r_i a_i}
       \bigl(a_i-1+\mathbf 1_{p\mid a_i}\bigr).
\]
Euler characteristics and divisor degrees have the same numerical
values over $F$ as after the separable extension to $S$, so this
inequality is written in the original normalization over $F$.
The hypotheses of
\Cref{thm:inseparable-genus} hold for $Z_w/E_F$:
$[F:F_0]_p\frac1{e_i}\right\}.
\]
At these labels, the degree $m/(r_i a_i)$ of the reduced special fiber
exceeds the degree $m/e_i$ of the reduced generic fiber. Imposing
vanishing there will make the section comparison detect this excess.
This selection descends on both sides. The generic index $e_i$ is
constant on the label orbits of $\operatorname{Gal}(L'/L)$, and the
special indices $r_i,a_i$ are constant on the label orbits of
$\operatorname{Gal}(S/F)$. These two permutation actions agree by
\Cref{lem:residue-permutations}. Hence the same subset of labels
defines a divisor over $L$ on the generic base and a divisor over $F$
on the special base.

Let $T_Y$ be the reduced inverse image on $Y$ of this selected divisor,
and let $T_w$ be its counterpart on $C_w$. These are effective divisors
on regular curves over their indicated fields. Their degrees may be
computed after the finite separable splitting extensions, which preserve
reducedness. The generic Galois cover and
\eqref{eq:reduced-special-fiber-degree} give
\begin{equation}\label{eq:selected-divisor-degrees}
 \deg_L T_Y=\sum_{i\in\mathcal I}\frac m{e_i},\qquad
 \sum_{w\in\mathcal V}\deg_F T_w
    =\sum_{i\in\mathcal I}\frac m{r_i a_i}.
\end{equation}

Restrict the norm to
$H^0(Y,\mathcal O_Y(jD_Y-T_Y))$. Its integral ball again has a
full-dimensional reduction by \eqref{eq:section-lattice}, and
\Cref{lem:section-specialization} places that reduction in
\[
 \bigoplus_{w\in\mathcal V}
 H^0(C_w,\mathcal O_{C_w}(jD_w-T_w)).
\]
For sufficiently large $j$, ample vanishing and
\eqref{eq:residue-degree-equality} therefore give
\[
 jmn+\chi(Y)-\deg_L T_Y
 \leq jmn+\sum_w\chi(C_w)-\sum_w\deg_F T_w.
\]
After rearranging and using \eqref{eq:selected-divisor-degrees}, this is
\begin{equation}\label{eq:vanishing-comparison}
 \frac{\nu(Y)}m
 \geq \frac{\sum_{w\in\mathcal V}\nu(C_w)}m
 +2\sum_{i=1}^n
   \max\left\{\frac1{r_i a_i}-\frac1{e_i},0\right\}.
\end{equation}

The three comparisons now force equality. To display their exact
interaction, put
\[
 q_i=\frac1{r_i a_i},\qquad b_i=\frac1{e_i},\qquad
 \delta_i=\mathbf 1_{p\mid a_i}.
\]
The contribution of label $i$ in \eqref{eq:residue-genus} is
$1-q_i+\delta_iq_i$. Combining that inequality with
\eqref{eq:vanishing-comparison} and subtracting
\eqref{eq:generic-hurwitz} gives
\begin{align}
 0
 &\geq \sum_{i=1}^n
       \bigl(b_i-q_i+2\max\{q_i-b_i,0\}+\delta_iq_i\bigr)
       \notag\\
 &=\sum_{i=1}^n\bigl(|q_i-b_i|+\delta_iq_i\bigr)
 \geq0.\label{eq:genus-nonnegative-excess}
\end{align}
Every summand vanishes, and every intervening inequality is an equality.
In particular,
\begin{equation}\label{eq:sharp-specialization}
 r_i a_i=e_i,\qquad p\nmid a_i\quad(1\leq i\leq n),
 \qquad d=|I_w|>1,
 \qquad \chi(Y)=\sum_{w\in\mathcal V}\chi(C_w).
\end{equation}
For the strict inequality $d>1$, use $p\mid e_i$ and $p\nmid a_i$:
then $r_i>1$, and $r_i$ divides the purely inseparable degree $d$ by
finite flatness at $t_i$. Also $q_i=b_i$ makes the extra term in
\eqref{eq:vanishing-comparison} zero, so equality there gives precisely
the displayed Euler-characteristic equality.

\subsection{A finitely presented model over the valuation ring}

\begin{proposition}\label{prop:specialization-model}
The degree equality \eqref{eq:residue-degree-equality} and the
Euler-characteristic equality in \eqref{eq:sharp-specialization}
produce a flat projective scheme $\mathcal Y$ of finite presentation
over $R$, with
\[
 \mathcal Y_L\simeq Y,\qquad
 \mathcal Y_F\simeq\coprod_{w\in\mathcal V}C_w.
\]
\end{proposition}

\begin{proof}
Form the graded $R$-algebra
\[
 A=\bigoplus_{j\geq0}A_j.
\]
Multiplication is defined because each $w$ is a valuation:
$A_iA_j\subseteq A_{i+j}$. Each piece is finite free by
\eqref{eq:section-lattice}. Moreover $A_0=R$, since $Y$ is projective
and geometrically integral and hence $H^0(Y,\mathcal O_Y)=L$.
Put
\[
 C=\coprod_{w\in\mathcal V}C_w,
 \qquad B_j=\bigoplus_{w\in\mathcal V}
 H^0(C_w,\mathcal O_{C_w}(jD_w)),
 \qquad B=\bigoplus_{j\geq0}B_j.
\]
The two equalities in the proposition make the source and target
dimensions in \eqref{eq:section-reduction} equal for every sufficiently
large $j$. Thus there is an integer $j_0$ such that
\begin{equation}\label{eq:full-high-degree-reduction}
 A_j/\mathfrak m_R A_j\simeq B_j
 \qquad(j\geq j_0),
\end{equation}
compatibly with products. Equality in each high degree, rather than
only equality of leading terms, is where the Euler-characteristic
equality is used.

We first choose a Veronese regrading for which the reduced algebra is
generated in degree one. The divisor on $C$ given by the $D_w$ is ample,
so $B$ is finitely generated by \Cref{foundation:cohomology}. Choose
positive-degree homogeneous generators $b_1,\ldots,b_t$ over $B_0$,
of degrees $d_1,\ldots,d_t$. Let $W$ be a common multiple of these
positive degrees. In a monomial in the $b_i$, remove pure powers
$b_i^{W/d_i}$, each of weight $W$, until the remaining exponent of
$b_i$ is less than $W/d_i$. The weight of the remainder is less than
$tW$. Choose $N=qW\geq j_0$ with $q\geq t$.

If the original monomial has degree $kN$, its remainder has weight
$cW$ for an integer $0\leq ci),\qquad
 \delta_i\leq\gamma_i<\delta_{i+1},\qquad
 \gamma_1<\gamma_2<\cdots .
\end{equation}
The balls $B(a_i,\gamma_i)$ are nested, lie inside $B(a_i,\delta_i)$,
and still have empty intersection.

Extend the valuation to an algebraic closure of $K_s$.
Call $b$ a \emph{pseudo-limit} of this sequence if
$v(b-a_i)=\gamma_i$ for all sufficiently large $i$.
No element of $K_s$ is a pseudo-limit, since a pseudo-limit belongs to
every ball $B(a_i,\gamma_i)$.
For any point $b$ which is not a pseudo-limit, the values $v(b-a_i)$
are eventually constant and smaller than $\gamma_i$.
To see this, if $v(b-a_i)<\gamma_i$ at one stage, the strong triangle
law and \eqref{eq:pseudoconvergent-centers} give the same value at every
later stage.  If $v(b-a_i)>\gamma_i$, the next value is $\gamma_i$, which
is smaller than $\gamma_{i+1}$, reducing to the preceding case.
Otherwise equality holds eventually, as required for a pseudo-limit.

Suppose first that an algebraic pseudo-limit $\alpha$ exists, and
choose one of smallest degree $d$ over $K_s$.  Every root $\beta$ of a
nonzero polynomial $f\in K_s[T]$ with $\deg fv(x)$. It follows that $v(c)=v(x)$, so no new values
occur. If $v(x)=0$, the same inequality says that $x$ and $c$ have
the same residue, so no new residue-field elements occur either.
Thus $K_s(\alpha)/K_s$ is a proper immediate extension, a contradiction.

Suppose instead that no algebraic pseudo-limit exists.  Factoring any
nonzero polynomial over the fixed algebraic closure shows that its
values at $a_i$ are eventually constant and finite; the same holds for
nonzero rational functions.  Define, on $K_s(T)$,
\[
                 w(f)=\text{the eventual value of }v(f(a_i)),
                 \qquad w(0)=+\infty.
\]
Evaluation proves multiplicativity and the strong triangle inequality,
so $w$ is a valuation extending that of $K_s$.
For fixed $f\ne0$ and $j>i\gg0$, every linear factor in its numerator
and denominator satisfies
$(a_j-\beta)/(a_i-\beta)\equiv1\pmod{\mathfrak m_{\overline{K_s}}}$,
where $\beta$ is its root. Indeed $v(a_i-\beta)<\gamma_i=v(a_j-a_i)$.
Consequently
\[
                    v(f(a_j)-f(a_i))>v(f(a_i))=w(f).
\]
For a fixed sufficiently large $i$, the left side has an eventual
value as $j$ increases, since it evaluates the rational function
$f(T)-f(a_i)$.  Thus
$w(f-f(a_i))>w(f)$ unless the difference is zero, when the assertion
is immediate. Thus every nonzero element $f$ of $K_s(T)$ admits a
scalar $f(a_i)$ with this strict approximation inequality. As in the
algebraic case, this shows that neither the value group nor the residue
field changes. This is again a proper immediate extension,
contradicting maximality.  Both cases being impossible, $K_s$ is
spherically complete.
\end{proof}

\begin{lemma}\label{lem:orthogonal-bases}
Let $K_*$ be a spherically complete field with value group $\mathbb Q$,
valuation ring $R_*$, maximal ideal $\mathfrak m_*$, and residue field
$k_*$.  Let $V$ be a finite-dimensional $K_*$-vector space with a
valuation norm $\lambda:V\to\mathbb Q\cup\{+\infty\}$: it is finite
away from zero, has $\lambda(0)=+\infty$, satisfies the strong triangle inequality, and obeys
$\lambda(ax)=v(a)+\lambda(x)$ for $a\ne0$.
There is a basis $e_1,\ldots,e_d$ of norm zero such that
\begin{equation}\label{eq:orthogonal-basis}
             \lambda\left(\sum_i a_i e_i\right)=\min_i v(a_i).
\end{equation}
In particular its integral and positive balls are
\begin{equation}\label{eq:orthogonal-lattice}
 V_{\geq0}=\bigoplus_i R_*e_i,\qquad
 V_{>0}=\bigoplus_i\mathfrak m_*e_i
             =\mathfrak m_*V_{\geq0},
 \qquad \dim_{k_*}(V_{\geq0}/V_{>0})=d.
\end{equation}
The normed space $V$ is spherically complete.
\end{lemma}

\begin{proof}
A subspace with an orthogonal basis is spherically complete: its closed
balls are products of closed balls in the finitely many coordinates,
with the radii shifted by the norms of the basis vectors.  Each
coordinate nest has a common point.

Suppose an orthogonal basis has already been constructed for a subspace
$U$, and choose $x\notin U$.  For every value $\gamma$ achieved by
$\lambda(x-u)$, $u\in U$, consider
\[
                  B_\gamma=\{u\in U:\lambda(x-u)\geq\gamma\}.
\]
This is a nonempty closed ball in $U$: if $u_\gamma$ belongs to it,
then it equals $\{u:\lambda(u-u_\gamma)\geq\gamma\}$.
These balls form a nest, so they have a common point $u_0$.
The finite value $\ell=\lambda(x-u_0)$ is at least every achieved value
and itself is achieved.  Thus $u_0$ is a best approximation to $x$.
Put $y=x-u_0$.  For $u\in U$, maximality of $\ell$ gives
$\lambda(y+u)\leq\ell$.  If $\lambda(u)<\ell$, the strong triangle
law gives $\lambda(y+u)=\lambda(u)$; if $\lambda(u)\geq\ell$, it gives
$\lambda(y+u)\geq\ell$, hence equality.  Therefore
\[
                       \lambda(y+u)=\min\{\ell,\lambda(u)\}.
\]
Homogeneity shows that adjoining $y$ preserves orthogonality.
Induction gives an orthogonal basis of $V$.  Since every basis norm is
in the base value group, scale its vectors to norm zero.
Formulae \eqref{eq:orthogonal-basis} and \eqref{eq:orthogonal-lattice}
follow, and the same coordinate argument proves spherical completeness
of $V$.
\end{proof}

\begin{proof}[Finite-extension assertions of \Cref{prop:spherical-base}]
For any algebraic extension of a field with value group $\mathbb Q$,
every nonzero algebraic element has value in $\mathbb Q$.
Indeed, in a polynomial relation $\sum_i a_i x^i=0$, at least two
nonzero terms must have the same minimum value, so
\[
                       (i-j)v(x)=v(a_j)-v(a_i)\in\mathbb Q
                       \quad(i\ne j).
\]
Divisibility of $\mathbb Q$ and torsion-freeness of ordered value groups
imply $v(x)\in\mathbb Q$.
Apply \Cref{lem:orthogonal-bases} to a finite extension $L/K_s$ with any
extended valuation, viewed as a normed vector space over $K_s$.
Its integral and positive balls are exactly its valuation ring and
maximal ideal.  Formula \eqref{eq:orthogonal-lattice} proves
\eqref{eq:full-residue-degree}, including the full inseparable degree
of the residue extension.  Orthogonal coordinates also show that $L$
is spherically complete.  The same argument applies to every finite
extension of $L$.
\end{proof}

The preceding argument gives equality of field and residue degrees for
finite extensions of the spherical base. We also need a statement for
valued fields that are not assumed spherically complete, where several
prolongations can occur. The next lemma supplies the residue-degree bound
used for curve function fields in \Cref{sec:specialization} and completes
the uniqueness assertion for the spherical base.

\begin{lemma}[Simultaneous residue approximation]\label{lem:simultaneous-residues}
Let $T$ be a nontrivially valued field with value group $\mathbb Q$ and
infinite residue field $k_T$.  Let $U/T$ be finite and let
$w_1,\ldots,w_r$ be distinct prolongations of its valuation, with residue
fields $k_1,\ldots,k_r$.  Then
\[
             \bigcap_{i=1}^r\mathcal O_{w_i}\longrightarrow
                         \prod_{i=1}^r k_i
\]
is surjective, and
\begin{equation}\label{eq:residue-degree-bound}
                         \sum_{i=1}^r[k_i:k_T]\leq[U:T].
\end{equation}
\end{lemma}

\begin{proof}
The preceding polynomial-relation argument puts all the values in
$\mathbb Q$, with their normalization fixed by the valuation on $T$.
For $i\ne j$, distinctness gives an element whose two values differ.
Inverting it if necessary and multiplying by an element of $T$ of a
suitable rational value produces $z\in U$ with
$w_i(z)>0>w_j(z)$.
Choose a unit $c\in T$ such that, at every $w_\ell$ for which
$w_\ell(z)=0$, the residue of $1+cz$ is nonzero.
Each such condition excludes at most one value for the residue of $c$
in $k_T^\times$, so the infinitude of $k_T$ permits this choice.
Then
\[
                         q_{ij}=\frac{1}{1+cz}
\]
is integral at every $w_\ell$, has residue $1$ at $i$, and has residue
$0$ at $j$.  Indeed, a positive value of $z$ makes the denominator a
unit with residue $1$; a negative value of $z$ gives the denominator
that negative value; and value zero is covered by the choice of $c$.

Put $q_i=\prod_{j\ne i}q_{ij}$, with $q_i=1$ if $r=1$.
It has residue $1$ at $i$ and positive value at all other places.
For a prescribed residue $b_i\in k_i$, choose any lift $\widetilde b_i$
integral at $i$. If the lift is nonzero, choose $N_i$ so that
\[
 N_iw_\ell(q_i)+w_\ell(\widetilde b_i)>0
 \qquad(\ell\ne i).
\]
There are only finitely many conditions, and every $w_\ell(q_i)$ in
them is positive. At $i$, the factor $q_i^{N_i}$ has residue one.
Thus $q_i^{N_i}\widetilde b_i$ is integral everywhere, has residue
$b_i$ at $i$, and has residue zero at every other place. A zero lift
contributes the zero term. Summing these terms realizes any tuple $(b_i)_i$.

For completeness, each $[k_i:k_T]$ is finite and at most $[U:T]$ by the
single-place lifting argument already used in the proof of
\Cref{prop:mixed-lift}.  Choose a $k_T$-basis of each $k_i$, and use the
surjectivity just proved to lift its elements with zero residues in the
other coordinates.  These lifts are linearly independent over $T$.
For if they satisfied a nonzero relation, divide it by a coefficient
of minimum value.  Every coefficient then is integral and at least one
is a unit.  Reducing in a coordinate which contains such a unit
contradicts the chosen residue basis there.  Their total number is
therefore at most $[U:T]$, proving \eqref{eq:residue-degree-bound}.
\end{proof}

\begin{proof}[Completion of the proof of \Cref{prop:spherical-base}]
The residue of $K_s$, and that of every finite extension $L/K_s$, is
infinite.  If two distinct valuations extended the valuation of $L$ to
a finite extension $U/L$, the finite-extension argument above would
give residue degree $[U:L]$ for each.  This contradicts
\eqref{eq:residue-degree-bound}.  Existence of prolongations is part of
\Cref{foundation:valuation}; uniqueness on finite extensions implies
uniqueness on arbitrary algebraic extensions by restriction to finite
subextensions.

Finally $|K_s|\leq\mathfrak c$ by the cardinality argument, while
$\mathbb Q_p\subset K_s$ gives $|K_s|\geq\mathfrak c$.
Its algebraic closure has the same cardinality.
An uncountable algebraically closed field of characteristic zero has
cardinality equal to its transcendence degree over $\mathbb Q$: the
field generated by a transcendence basis, and then its algebraic
closure, both have cardinality the maximum of that basis cardinality
and $\aleph_0$.  Both $\overline{K_s}$ and $\mathbb C$ therefore have
transcendence degree $\mathfrak c$ over $\mathbb Q$.
The algebraically closed field classification in
\Cref{foundation:fields} gives the asserted abstract isomorphism.
No compatibility with either field's topology is asserted or used.
\end{proof}

\subsection{Marked covers and a small orbit}

Choose an embedding $K_b'\hookrightarrow\overline{K_s}$ over $K_b$ and
an abstract isomorphism $\overline{K_s}\simeq\mathbb C$.
Write $X_s=\mathcal X_{K_s}$. The embedding carries the sections
$\xi_i$ on the split model $\mathcal X_{R_b'}$ to the specified
geometric marked points on $X_s$.
A \emph{marked $J$-cover} of $\mathcal X_{\overline{K_s}}$ will mean a connected
smooth projective curve $Y$ with a finite map to $\mathcal X_{\overline{K_s}}$
which is Galois of group $J$, together with a specified left action of
$J$ by deck transformations.  Isomorphisms are over the fixed base
curve and commute with every element of this specified action.

\begin{lemma}\label{lem:marked-cover-classification}
After choosing oriented handle and puncture generators on the complex
punctured curve, isomorphism classes of marked $J$-covers branched only at
$\xi_1,\ldots,\xi_n$ correspond bijectively to
\[
 \left\{(x,y,u_1,\ldots,u_n)\in J^{n+2}:
 \begin{array}{l}
 [x,y]u_1\cdots u_n=1,\\
 \langle x,y,u_1,\ldots,u_n\rangle=J
 \end{array}\right\}/J,
\]
where $J$ acts by simultaneous inner conjugation.
The inertia order at $\xi_i$ is $|u_i|$.
For a prescribed multiset $\mathcal C$ of $n$ $J$-conjugacy classes of
$p$-singular elements, let $\mathcal T_{\mathcal C}$ be the finite set
of these marked-cover classes having that multiset of local classes.
Then $\mathcal T_{\mathcal C}$ is preserved by
$\operatorname{Gal}(\overline{K_s}/K_s)$.
\end{lemma}

Thus $|\mathcal T_{\mathcal C}|=N_{J,s}(\mathcal C)$ in the notation
of \Cref{thm:tuple-divisibility}. The labels $\xi_1,\ldots,\xi_n$
remain ordered: prescribing their class multiset does not divide out
by permutations of the labels.

\begin{proof}
Under the chosen complex identification the base is a smooth
Weierstrass cubic. In characteristic zero, completing the square gives
an equation $y^2=f_3(x)$ with three distinct roots. Its double cover of
the projective line branches simply at those roots and at infinity.
Cut the sphere along two disjoint arcs joining pairs of these four
points and glue the two sheets crosswise. The resulting compact surface
is a torus. This identifies the topological type of our particular
base without a general analytic--algebraic genus comparison.

Remove the $n$ marked points. Its complex fundamental group has
generators and relation
$[x,y]u_1\cdots u_n=1$ as in \Cref{foundation:complex}.
Choose a point in a fiber of a marked cover and identify the fiber with
the regular left $J$-set.  Path lifting commutes with the left deck
action, so monodromy acts by right translations.  With path multiplication
chosen in the order of successive traversal, these translations specify
a homomorphism from the fundamental group to $J$.  The cover is
connected exactly when the monodromy action is transitive, which for
this regular action means that the homomorphism is surjective.
Changing the chosen fiber point conjugates this homomorphism by one
element of $J$.  Conversely, a surjective homomorphism defines such a
fiber action and therefore a connected unramified cover.
An equivariant fiber bijection is a right translation, and compatibility
with monodromy says exactly that the two homomorphisms are simultaneously
conjugate.  Thus the quotient is by inner conjugation; keeping the
specified $J$-action does not identify covers through outer automorphisms
of $J$.

Riemann existence in \Cref{foundation:complex} algebraizes the
unramified cover of this punctured algebraic curve, with its morphisms
and specified deck action. Normalize the original projective base in
the resulting function field. Finiteness of normalization gives a
projective algebraic cover; normality makes its source smooth in
characteristic zero. The deck action and isomorphisms extend uniquely
across normalization. In a local power-map chart $z\mapsto z^e$, the
ramification index is $e$, the order of the associated monodromy
element. These extensions give precisely the branched covers and
isomorphisms in the statement.

We describe the local conjugacy class algebraically in order to check
Galois invariance.  For each divisor $e$ of $m$, use the root
$\zeta_e\in K_s$ which maps to $\exp(2\pi i/e)$ under the chosen
complex identification.  All these roots belong to $K_s$ by
\Cref{prop:mixed-lift}.
Represent the based puncture loop as an approach path, a positive
circle around the puncture, and the reverse approach path.
In the chart $z\mapsto z^e$, lifting the positive circle sends $z$ to
$\zeta_e z$. Let $Q$ be the point above the puncture selected by the
lifted approach path, and let $g\in J$ be the inertia element acting
this way in its local chart. Apply $g$ to the lifted approach path.
The reverse of the resulting path is the lift of the return path
starting at the endpoint of the circle lift. Thus the complete based
loop sends the chosen fiber point to its image under $g$.
Under the preceding fiber identification, its monodromy element is
therefore $g$. The inertia group is cyclic, and $g$ is its unique
generator acting on the tangent line at $Q$ by $\zeta_e$.
Choosing another point above the puncture conjugates the element in $J$.
This recovers its conjugacy class from the algebraic tangent action.
(Reversing all local-loop conventions replaces all these generators by
their inverses consistently.)

A Galois automorphism over $K_s$ sends the tangent action at $Q$ to
the tangent action at its conjugate point, and fixes $\zeta_e$.
Consequently it preserves the local $J$-conjugacy class at the transported
branch point.  It permutes the branch points, since their union is
defined over $K_s$, and hence preserves their prescribed class multiset
and the condition that all inertia orders are divisible by $p$.
This proves invariance of $\mathcal T_{\mathcal C}$.
Its finiteness follows from the finite tuple description.
Although Galois can permute the marked points, an isomorphism of the
marked covers is over the fixed geometric base curve. The cover classes
are therefore still parametrized by simultaneous inner conjugacy,
without quotienting by those permutations.
\end{proof}

For a cover with its group action specified, the obstruction to descent
is center-valued; compare the marked-cover analysis of
D\`ebes and Douai \cite[Section~3.2]{DebesDouai1997}. In the following
proposition we realize that obstruction as an explicit finite group
extension and split it by a Sylow argument. Thus a field of moduli is
not silently assumed to be a field of definition.

\begin{proposition}[Descent from a small Sylow orbit]\label{prop:small-orbit-descent}
Assume $O_p(J)=1$ and fix a multiset $\mathcal C$ as in
\Cref{lem:marked-cover-classification}.  There is a finite Galois
extension $M/K_s$ through which the action on $\mathcal T_{\mathcal C}$
factors.  Let $P_\Gamma$ be a Sylow $p$-subgroup of
$\Gamma=\operatorname{Gal}(M/K_s)$.
If a $P_\Gamma$-orbit has size less than $p^h$, one of its marked covers
descends to a finite extension $L/K_s$ satisfying
\begin{equation}\label{eq:small-orbit-degree}
                [L:K_s]_p