\section{A smooth proper curve in characteristic zero carrying the parameter}
\label{fl:sec:geometrization}

This section constructs the data of
Setup~\ref{rw:setup:auxiliary-curve} from the given parameter
$\sigma$: a smooth proper curve in characteristic zero with an
unramified local system $\rho$, related to $(X,\sigma)$ by
specialization. The parameter $\sigma$ lives on an open subset of
$X$ in characteristic $p$ and may be wildly ramified there. The
construction has four steps.
\begin{enumerate}
\item \emph{Lifting} (Theorem~\ref{fl:thm:marked-lifting}). Lift
$X$ to a curve $\mathcal X$ over a ring of Witt vectors, and lift
$\sigma$, with its whole compact image, to a representation of the
fundamental group of the complement of a divisor
$D\subset\mathcal X$. The divisor must not depend on the finite
quotient of the image under consideration, and this is where
$p\nmid|W_G|$ is used: it makes the images of wild inertia finite
abelian with a tame action on them.
\item \emph{A second component}
(Lemma~\ref{fl:lem:weillocal-reversal} and
Proposition~\ref{fl:prop:geometrization-theta}). Construct a cyclic
cover $C_2\to\mathcal X$ of degree $d$ branched along $D$, and
glue it to $\mathcal X$ along $D$. The degree is chosen so that the
pullback of the local system to $C_2$ has inertia inverse to that
of the original one, up to conjugation by elements commuting with
Frobenius.
\item \emph{Smoothing}
(Proposition~\ref{fl:prop:geometrization-theta}). The
curve $C_0=\mathcal X\cup_DC_2$, which has two components, is
Gorenstein and is nodal on the fibre of characteristic zero,
deforms to a smooth proper curve over
$R_0[[\varepsilon]]$, with a square root of its dualizing line.
\item \emph{Extending the local system}
(Proposition~\ref{fl:lem:balanced-parameter-tower} and
Proposition~\ref{fl:prop:geometrized-parameter}). Because the
inertia on the two branches at each node of the intermediate fibre,
in characteristic zero, is inverse, the local
system extends over the smoothing, and on the smooth fibre it is
unramified. The ramification of $\sigma$ has been traded for
genus; the parameter itself is still present on the component
$\mathcal X$ of the special fibre.
\end{enumerate}


Throughout this section $p\nmid|W_G|$; recall that the Weyl groups
of $G$ and $\widehat G$ have the same order. We put
$Q=\operatorname{im}(\sigma)$, a compact group, and choose a
finite extension $E$ of $\mathbb Q_\ell$ and a lattice stable
under $Q$ as in
Lemma~\xref{I}{fl:lem:arithmetic-finite-coefficients}.

\subsection{Lifting the compact representation on one open}
\label{fl:sec:marked-parameter-lifting}

\begin{theorem}[Lifting a parameter with its ramification]
\label{fl:thm:marked-lifting}
Let $X/\mathbb F_q$ be smooth, projective and geometrically connected,
let $U=X-\Sigma$ for a finite set $\Sigma$, and let
$\sigma:\pi_1(U)\to H(k)$ be continuous, where $H$ is connected
semisimple and $p\nmid|W_H|$. Fix a smooth projective lift
$\mathcal X$ of $X$ over $W(\mathbb F_q)$. After a finite totally
ramified extension $R_0/W(\mathbb F_q)$ there is a divisor
$D\subset\mathcal X_{R_0}$, finite and flat over $R_0$, whose
special fibre has support $\Sigma$, and a continuous
representation
\[
 \sigma^{\rm lift}:\pi_1(\mathcal X_{R_0}-D)\longrightarrow Q
\]
restricting to $\sigma$ on the special fibre. Its image is $Q$, and
the image of the geometric fundamental group of the generic fibre
contains the image under $\sigma$ of the geometric fundamental
group of $U$. The same divisor $D$ serves for all finite quotients
of $Q$: the lift is constructed as a compatible system of torsors
under these quotients on $\mathcal X_{R_0}-D$, with base points
compatible with those of $\sigma$ and with the action of
Frobenius.
\end{theorem}

The point of the theorem is that $D$ does not depend on the finite
quotient. If each finite quotient were lifted separately, the
branch loci could grow without bound. In the proof one first lifts
the tame quotient of the representation, with ramification only
along the lift of $\Sigma$, and then corrects the lift by a cocycle
with values in
a fixed finite group, which accounts for all further ramification.

\begin{lemma}[Pro-$p$ subgroups of the image]
\label{fl:lem:compact-finite-coefficients}
Every pro-$p$ subgroup of $Q$ is finite and abelian. If $P$ is a
normal pro-$p$ subgroup of a closed subgroup $Q_0$ of $Q$, the
image of the homomorphism $Q_0\to\operatorname{Aut}(P)$ given by
conjugation has order prime to $p$.
\end{lemma}

\begin{proof}
The group $Q$ has an open normal subgroup which is pro-$\ell$, a
congruence subgroup for the chosen lattice. A pro-$p$ subgroup
meets it trivially, so it embeds into the finite quotient and is a
finite $p$-group. A finite nilpotent subgroup of $H(k)$ normalizes
a maximal torus \cite[\S3.3, Theorem~3$''$]{Serre2007}. The image
of our $p$-group in the Weyl group of this torus is a $p$-group,
hence trivial since $p\nmid|W_H|$; so the group is contained in
the torus and is abelian.

For the second assertion let $S$ be a Sylow pro-$p$ subgroup of
$Q_0$. It contains the normal subgroup $P$, and it is abelian by
the first part, so it acts trivially on $P$. The image of $S$ in
the finite group $\operatorname{im}(Q_0\to\operatorname{Aut}(P))$
is a Sylow $p$-subgroup of this group, and it is trivial.
\end{proof}

The correction mentioned above is a cohomology class, and the
next lemma will be used to lift it from the special fibre. For a
complete discrete valuation ring $R$ with uniformizer $\pi$ and
residue field $\kappa_R$, write
\[
 A_R=R[[z]],\qquad K_R=\operatorname{Frac}(A_R),\qquad
 K_R^{\rm vert}=\operatorname{Frac}
 \widehat{(A_R)_{(\pi)}}.
\]
The field $K_R^{\rm vert}$ is the completion of $K_R$ along the
special fibre $\pi=0$ of $\operatorname{Spec}A_R$. It is a
complete discretely valued field with residue field
$\kappa_R((t))$, where $t$ is the image of $z$.

\begin{lemma}[Approximation of cohomology classes]
\label{fl:lem:local-cocycle-approximation}
Let $P$ be a finite abelian group killed by $p^n$, on which the
Galois group of $K_R$ acts through a finite quotient $\Delta$ of
order prime to $p$. Let $K'/K_R$ be the extension with group
$\Delta$. Suppose that $K'\otimes_{K_R}K_R^{\rm vert}$ is a field
and that $\mu_{p^n}\subset K_R$. Then the restriction map
\[
 H^1(K_R,P)\longrightarrow H^1(K_R^{\rm vert},P)
\]
is surjective.
\end{lemma}

\begin{proof}
Let $K'^{\rm vert}=K'\otimes_{K_R}K_R^{\rm vert}$. It is a complete
discretely valued field in which $K'$ is dense.

We first prove the surjectivity for $K'$ in place of $K_R$. Over
$K'$ the module $P$ is a sum of groups $\mathbb Z/p^h$ with
trivial action, and $\mu_{p^h}\subset K'$. By Kummer theory it is
enough to show that every $x\in(K'^{\rm vert})^\times$ is the
product of an element of $K'^\times$ and a $p^h$-th power. Choose
$y\in K'^\times$ with $v(x/y-1)>2v(p^h)$. By Hensel's lemma,
applied to the polynomial $X^{p^h}-x/y$ and its approximate root
$1$, the element $x/y$ is a $p^h$-th power.

Now let $\xi\in H^1(K_R^{\rm vert},P)$, and let
$\beta\in H^1(K',P)$ be a class whose restriction to
$K'^{\rm vert}$ is the restriction of $\xi$. Put $m=[K':K_R]$ and
choose an integer $m'$ with $mm'\equiv1\pmod{p^n}$. Since
$K'^{\rm vert}$ is a field, restriction to $K_R^{\rm vert}$
commutes with corestriction from $K'$, and the corestriction of
the restriction of $\xi$ is $m\xi$. So the class
$m'\operatorname{Cor}_{K'/K_R}\beta$ restricts to $m'm\xi=\xi$.
\end{proof}

\begin{lemma}[The local lift]
\label{fl:lem:finite-local-lift}
Let $\kappa$ be a finite field of characteristic $p$, let $Q_0$
be a closed subgroup of $Q$, and let
$\sigma_0:G_{\kappa((t))}\twoheadrightarrow Q_0$ be a continuous
surjection. Let $P$ be the image of the wild inertia group; it is
finite abelian, and the conjugation action of $Q_0$ on it factors
through a group of order prime to $p$, by
Lemma~\ref{fl:lem:compact-finite-coefficients}. Let $p^nP=0$ and
$R=W(\kappa)[\zeta_{p^n}]$. Then there are a divisor
$D^{\rm hor}\subset\operatorname{Spec}R[[z]]$, finite and flat
over $R$ and containing $z=0$, and a continuous homomorphism
\[
 \widetilde\sigma_0:
 \pi_1\bigl(\operatorname{Spec}R[[z]]-D^{\rm hor}\bigr)
 \longrightarrow Q_0
\]
whose composite with
$G_{K_R^{\rm vert}}\to
\pi_1(\operatorname{Spec}R[[z]]-D^{\rm hor})$ is the composite of
$\sigma_0$ with $G_{K_R^{\rm vert}}\to G_{\kappa((t))}$. The
equality is one of homomorphisms for compatible base points, and
not only up to conjugation. The divisor $D^{\rm hor}$ serves for
every finite quotient of $Q_0$.
\end{lemma}

\begin{proof}
In step~1 we choose a splitting over the tame quotient, and in
step~2 we lift the homomorphism so obtained to characteristic
zero, by means of the standard tame coverings. Its difference with
$\sigma_0$ is a cocycle with values in the finite group $P$, which
is lifted by Lemma~\ref{fl:lem:local-cocycle-approximation}.
Step~3 is bookkeeping: it shows that the ramification of the
result is contained in one divisor.

\step{1}{splitting off the tame part} Let
$\mathbb T=\widehat{\mathbb Z}^{(p')}(1)\rtimes G_\kappa$ be the
tame quotient of $G_{\kappa((t))}$. Since $P$ is the image of
wild inertia, $\sigma_0$ induces
$\bar\sigma_0:\mathbb T\to Q_0/P$. Consider the pullback by
$\bar\sigma_0$ of the extension
\[
 1\to P\to Q_0\to Q_0/P\to1 .
\]
It is an extension of $\mathbb T$ by the finite abelian $p$-group
$P$, and we claim that it splits. Extensions of profinite groups
with finite kernel have continuous sections, so they are
classified by continuous cohomology, and it suffices to show that
$H^2(\mathbb T,P)=0$. The tame inertia group
$\widehat{\mathbb Z}^{(p')}(1)$ has order prime to $p$, so it has
no cohomology in positive degrees with coefficients in $P$. By
the Hochschild--Serre spectral sequence $H^2(\mathbb T,P)$ is
then $H^2$ of $G_\kappa$ with coefficients in a finite module,
which is zero because $G_\kappa\simeq\widehat{\mathbb Z}$. A
splitting is a continuous homomorphism
\[
 \eta:\mathbb T\longrightarrow Q_0
\]
whose composite with $Q_0\to Q_0/P$ is $\bar\sigma_0$.

\step{2}{the correction} The unramified extensions of $R$ and the
coverings obtained by adjoining $z^{1/m}$, for $m$ prime to $p$,
define a homomorphism
\[
 \tau:\pi_1(\operatorname{Spec}R[[z]][1/z])\longrightarrow
       \mathbb T .
\]
The reductions of these coverings are the unramified and the
tamely ramified coverings of $\kappa((t))$, so the restriction of
$\tau$ to $G_{K_R^{\rm vert}}$ is the composite of
$G_{K_R^{\rm vert}}\to G_{\kappa((t))}$ with the tame quotient
map. We regard $\sigma_0$ as a homomorphism on
$G_{K_R^{\rm vert}}$ in the same way. Then $\sigma_0$ and
$\eta\tau$ have the same image in $Q_0/P$, and
\[
 \mathsf c_{\rm vert}(\gamma)=
 \sigma_0(\gamma)\,(\eta\tau(\gamma))^{-1}\in P
\]
is a continuous $1$-cocycle on $G_{K_R^{\rm vert}}$ for the
action of this group on $P$ by conjugation through $\eta\tau$.
This action is defined on $G_{K_R}$, and it factors through a
finite quotient $\Delta$ of order prime to $p$. The corresponding
extension of $K_R$ remains a field after tensoring with
$K_R^{\rm vert}$, because the restriction of $\tau$ to
$G_{K_R^{\rm vert}}$ is still surjective. So
Lemma~\ref{fl:lem:local-cocycle-approximation} applies: the class
of $\mathsf c_{\rm vert}$ is the restriction of a class in
$H^1(K_R,P)$. Let $\mathsf c$ be a cocycle in this class. Its
restriction differs from $\mathsf c_{\rm vert}$ by the coboundary
of an element of $P$, and after changing $\mathsf c$ by the same
coboundary the restriction of $\mathsf c$ is equal to
$\mathsf c_{\rm vert}$. The cocycle relation says that
\[
 \widetilde\sigma_0(\gamma)
   =\mathsf c(\gamma)\,\eta\tau(\gamma)
\]
is a homomorphism on $G_{K_R}$, and its restriction to
$G_{K_R^{\rm vert}}$ is $\sigma_0$.

\step{3}{the divisor} The homomorphism $\widetilde\sigma_0$ is
determined by $\tau$, which is unramified outside $z=0$, and by
the homomorphism
\[
 G_{K_R}\longrightarrow P\rtimes\Delta
\]
with components $\mathsf c$ and the action on $P$, which has
finite image. Let $B$ be the normalization of $R[[z]]$ in the
finite \'etale algebra over $K_R$ defined by the latter. It is a
finite $R[[z]]$-module since $R[[z]]$ is excellent, and it is
free: a normal ring of dimension two is Cohen--Macaulay, and a
finite Cohen--Macaulay module of maximal dimension over a regular
local ring is free by the formula of Auslander and Buchsbaum. Let
$\delta$ be the discriminant of a basis of $B$. After completion
along $\pi=0$ the algebra is the one defined by
$\mathsf c_{\rm vert}$ and the action, which come from
$G_{\kappa((t))}$; so $B$ is unramified at the prime $(\pi)$ and
$\delta\notin(\pi)$. By the Weierstrass preparation theorem
$z\delta$ is the product of a unit and a distinguished polynomial
in $z$. Hence
\[
 D^{\rm hor}=V(z\delta)
\]
is finite and flat over $R$, and its special fibre is supported
at $t=0$.

The algebra $B$ is \'etale outside $V(\delta)$, and the coverings
which define $\tau$ are \'etale outside $z=0$. So
$\widetilde\sigma_0$ factors through
$\pi_1(\operatorname{Spec}R[[z]]-D^{\rm hor})$. For a finite
quotient of $Q_0$ we compose $\widetilde\sigma_0$ with the
quotient map. No new discriminant and no new branch point occurs,
and this is the uniformity asserted in the lemma.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{fl:thm:marked-lifting}]
Step~1 lifts the curve. Steps~2 to~4 glue the local lifts of
Lemma~\ref{fl:lem:finite-local-lift} to the unique lift over the
part of the curve where $\sigma$ is unramified; this is an
application of formal patching, and the only point to watch is
that the divisor is constructed before a finite quotient of $Q$
is chosen. Step~5 proves the statement on the geometric
fundamental group, which is used in
Proposition~\ref{fl:prop:geometrized-parameter}.

\step{1}{the curve} The obstruction to lifting a smooth curve to
the next infinitesimal thickening lies in $H^2(X,T_X)$, and the
obstruction to lifting an ample line bundle lies in
$H^2(X,\mathcal O_X)$; both groups vanish. So there is a formal
lift with an ample line bundle, and it is algebraic by
\cite[III$_1$, Theorem~5.4.5]{EGAIII}. The locus where the lift
is not smooth is closed and proper over the base and does not
meet the special fibre, so it is empty.

\step{2}{the divisor} Let $p^n$ kill the images of the wild
inertia groups at the points of $\Sigma$, and let
$R_0=W(\mathbb F_q)[\zeta_{p^n}]$. At each point of $\Sigma$ the
completed local ring of $\mathcal X_{R_0}$ is of the form
$R[[z]]$ as in Lemma~\ref{fl:lem:finite-local-lift}, and that
lemma gives a divisor $V(z\delta)$ and a lift of the restriction
of $\sigma$. On the formal completion of $\mathcal X_{R_0}$ along
$U$, finite \'etale coverings of $U$ lift uniquely. On the
completed punctured discs around the points of $\Sigma$ the two
constructions agree, with their base points: finite \'etale
coverings of such a punctured disc correspond to those of its
special fibre $\operatorname{Spec}\kappa((t))$, and both
constructions restrict to the given covering there.

The patching theorem for formal schemes
\cite[Theorem~3.4]{Pries2000} is an equivalence between coherent
modules on $\mathcal X_{R_0}$ and compatible systems of modules
on the formal completion along $U$ and on the discs. We apply it
first to the ideals $(z\delta)$ on the discs and the unit ideal
on the formal completion along $U$. The result is an invertible
ideal of $\mathcal X_{R_0}$ whose quotient is flat over $R_0$ and
has finite support in the special fibre. The divisor $D$ which it
defines is proper and quasi-finite over $R_0$, hence finite.

\step{3}{the torsors} Let $\Delta$ be a finite quotient of $Q$.
Patching the normalizations $B$ on the discs with the \'etale
algebra on the formal completion along $U$ we obtain a finite
flat algebra $\mathcal B_\Delta$ on $\mathcal X_{R_0}$ with an
action of $\Delta$. We claim that it is a $\Delta$-torsor outside
$D$. Let $\mathcal I_D$ be the ideal of $D$. On each disc some
power of $\mathcal I_D$ annihilates the module of relative
differentials of $\mathcal B_\Delta$ and the kernel and cokernel
of the map
\[
 \mathcal B_\Delta\otimes\mathcal B_\Delta
 \longrightarrow\prod_{a\in\Delta}\mathcal B_\Delta,
 \qquad b\otimes b'\longmapsto(b\cdot a(b'))_a ,
\]
since the algebra is a torsor outside $V(z\delta)$. On the formal
completion along $U$ these modules are zero. There are finitely
many discs, so one power of $\mathcal I_D$ annihilates the three
modules on all the pieces, and since patching is faithful the
same holds on $\mathcal X_{R_0}$. So the modules vanish outside
$D$, which is the claim.

\step{4}{compatibility} The divisor $D$ was constructed before
$\Delta$ was chosen. For a surjection $\Delta'\to\Delta$ of
finite quotients of $Q$, the maps between the algebras on the
discs and on the formal completion along $U$ agree on the
punctured discs, by the uniqueness of the lift of an \'etale
covering with base point. Since patching is fully faithful they
define a map $\mathcal B_\Delta\to\mathcal B_{\Delta'}$, and the
maps for three quotients are compatible. The same argument
applies to the actions of the groups, to the base points and to
the descent data for Frobenius. The compatible system of torsors
is a continuous homomorphism
\[
 \pi_1(\mathcal X_{R_0}-D)\longrightarrow
     \varprojlim_\Delta\Delta=Q .
\]
Its restriction to the special fibre is $\sigma$, which is
surjective; so its image is $Q$. If $\Sigma=\varnothing$ there
are no discs, $D=\varnothing$, and the argument reduces to the
lifting of \'etale coverings of a proper scheme.

\step{5}{the geometric image} It suffices to prove the assertion
in each finite quotient $\Delta$. Let $\mathcal Y$ be the
$\Delta$-torsor over $\mathcal X_{R_0}-D$. It is connected: a
connected component is finite \'etale over the connected base,
hence surjective, so it meets the special fibre of $\mathcal Y$,
which is connected because $\sigma$ maps onto $\Delta$. Let
$L=\operatorname{Frac}R_0$ and let $L'$ be the algebraic closure
of $L$ in the function field of $\mathcal Y$. The Galois group of
$L'/L$ is the quotient of $\Delta$ by the image of the geometric
fundamental group of the generic fibre.

We claim that $L'/L$ is unramified. A closed point of the special
fibre of $\mathcal Y$ lifts, since $\mathcal Y$ is smooth over
$R_0$, to a point with values in a finite unramified extension
$R_1$ of $R_0$. The elements of $L'$ integral over $R_0$ are
regular functions on the normal scheme $\mathcal Y$, and
evaluation at this point embeds $L'$ into
$\operatorname{Frac}R_1$.

Let $R'$ be the ring of integers of $L'$. The corresponding
quotient cover is
$(\mathcal X_{R_0}-D)\times_{R_0}\operatorname{Spec}R'$:
the evaluation argument gives the map to $\operatorname{Spec}R'$,
and the two finite \'etale covers agree on the generic fibre.
On the special fibre this is a constant-field cover, so the
geometric fundamental group of $U$ acts trivially on it. So the
image of this group in $\Delta$ is contained in the image of the
geometric fundamental group of the generic fibre.
\end{proof}

The divisor $D$ is flat over $R_0$, so it is the closure of its
generic fibre. In the sequel we replace $D$ by the closure of the
reduced generic fibre; this does not change the complement, and
the generic fibre of $D$ is now reduced. We are also free to
enlarge $D$ by divisors, finite and flat over $R_0$, along which
$\sigma^{\rm lift}$ is unramified.

\paragraph{Fibres at tangential base points.}
Let $x$ be a point of $D(\overline L)$ and $z$ a local coordinate
of $\mathcal X_{\overline L}$ at $x$. A finite \'etale covering of
$\mathcal X_{\overline L}-D$ whose inertia at $x$ has order
dividing $n$ becomes unramified at $x$ after the substitution
$z=\zeta^n$: the normalization of its pullback is finite \'etale
over the disc with coordinate $\zeta$. Its fibre at $\zeta=0$ does
not depend on $n$, up to canonical isomorphism, and is called the
fibre at the tangential base point defined by $z$. The group
$\mu_n$ of the substitutions $\zeta\mapsto\theta\zeta$ acts on it,
and this is the action of inertia. If $z'=za$ with $a(x)=1$, the
series $a^{1/n}$ with constant term one identify the fibres for
$z$ and for $z'$, compatibly with $n$; so the fibre depends only
on the tangent vector dual to $z$. If $a(x)=c\ne1$ an
identification requires a compatible system of roots of $c$.

Suppose that the tangent vector is defined over the residue field
of $x$; this field is fixed by $\phi^r$ if the orbit of $x$ under
$\phi$ has length $r$. Then $\phi^r$ acts on the fibre at the
tangential base point, and for a torsor under $Q$ this action is
given, in a frame of the fibre, by an element $g_x\in Q$. It
depends on the tangent vector: replacing $z$ by $cz$, with $c$ in
the residue field, multiplies $g_x$ by the image under the inertia
homomorphism of the Kummer class
$(\phi^r(c^{1/n})/c^{1/n})_n$ of $c$.

\begin{corollary}[Inertia and Frobenius at the boundary]
\label{fl:cor:boundary-return-relations}
Fix an algebraic closure $\overline L$ of $L$ and a lift $\phi$
of the arithmetic Frobenius to $\overline L$. Let $x$ be a point
of $D(\overline L)$ whose orbit under $\phi$ has length $r$. The
image under $\sigma^{\rm lift}$ of the inertia group at $x$ is
quasi-unipotent: it is of the form
\[
 \rho_x(\gamma)=s_x^{a(\gamma)}\exp(t_\ell(\gamma)N_x),
\]
with $s_x$ of finite order, $N_x$ nilpotent and commuting with
$s_x$, $t_\ell$ the $\ell$-adic tame character and $a$ a
character with values in $\mathbb Z/\operatorname{ord}(s_x)$. Every
element $g_x\in Q$ by which $\phi^r$ acts on the fibre at a
tangential base point at $x$ satisfies
\[
 g_xs_x g_x^{-1}=s_x^c,\quad(c,\operatorname{ord}s_x)=1,
 \qquad \operatorname{Ad}(g_x)N_x=q^rN_x.
\]
\end{corollary}

\begin{proof}
The inertia group at a point of a curve in characteristic zero
is $\widehat{\mathbb Z}(1)$, and $\phi^r$ acts on it through the
cyclotomic character. On the $\ell$-part this character has the
value $q^r$, because the roots of unity of order a power of
$\ell$ are unramified over $L$.

An open subgroup of the inertia group is mapped into a pro-$\ell$
congruence subgroup of $Q$, hence factors through
$\mathbb Z_\ell(1)$. Let $A$ be the image of a generator. Then
$g_xAg_x^{-1}=A^{q^r}$. So the map $\lambda\mapsto\lambda^{q^r}$
permutes the eigenvalues of $A$, in a faithful representation,
and every eigenvalue satisfies $\lambda^{q^{rm}}=\lambda$ for
some $m>0$. The eigenvalues are therefore roots of unity, a power
of $A$ is unipotent, and the image of inertia is quasi-unipotent.
The relation for $N_x$ is the logarithm of
$g_xAg_x^{-1}=A^{q^r}$ for this power, and the relation for $s_x$
follows from the action of $\phi^r$ on the finite quotient of the
inertia group, where the cyclotomic character has a value $c$
prime to the order. All the coverings are defined over $R_0$, so
these relations are compatible with the descent data for
Frobenius.
\end{proof}

\subsection{Reversing inertia without changing Frobenius}

Two punctured curves with local systems can be glued at a node
only if the inertia at the two branches is inverse to each other.
The pullback to a cyclic covering of degree $d$ branched at the
point raises the inertia to the power $d$. The next lemma chooses
$d$, together with the elements which will identify the fibres at
the nodes.

\begin{lemma}[Reversing inertia]
\label{fl:lem:weillocal-reversal}
Let $\rho_x$, $s_x$ and $N_x$ be as in
Corollary~\ref{fl:cor:boundary-return-relations}, for the
finitely many points $x$ of $D(\overline L)$, and let $r_x$ be the
length of the orbit of $x$ under $\phi$. There are an odd integer
$d>1$ prime to $p$, an integer $n\ge1$ and elements
$u_x\in\mathbb Z_\ell$, which depend only on $Q$, on the orders of
the $s_x$ and on the integers $r_x$, with the following property.
For every $x$ and every element $g_x\in Q$ by which $\phi^{r_x}$
acts on the fibre at a tangential base point at $x$, the element
\[
 P_x=(g_x^n)^{u_x}\quad\text{if }N_x\ne0,\qquad
 P_x=1\quad\text{if }N_x=0,
\]
of $Q$ is defined and satisfies
\[
 P_xg_x=g_xP_x,\qquad
 P_x\rho_x(\gamma)^{-d}P_x^{-1}=\rho_x(\gamma).
\]
If the tangential base points are permuted by $\phi$ on every
orbit, so are the elements $P_x$.
\end{lemma}

The point of this formulation is that $d$ is chosen before the
tangent vectors: the curve of the next subsection depends on $d$,
and the tangent vectors at its nodes are defined by its equation.

\begin{proof}
Let $Q_1$ be an open normal subgroup of $Q$ contained in a
congruence subgroup on which the $\ell$-adic logarithm and
exponential are defined and inverse to each other (level $\ell$
for odd $\ell$, level $4$ for $\ell=2$). Choose $n\ge1$ divisible
by the index of $Q_1$ in $Q$ and by the orders of the groups
$(\mathbb Z/\operatorname{ord}(s_x))^\times$. Let $g_x$ be as in
the statement and write $r=r_x$. Then $g_x^n\in Q_1$, so
$(g_x^n)^u$ is defined for $u\in\mathbb Z_\ell$; and $g_x^n$
commutes with $s_x$, because $g_xs_xg_x^{-1}=s_x^c$ and
$c^n\equiv1$ modulo the order of $s_x$. If $N_x\ne0$, the relation
$\operatorname{Ad}(g_x^n)N_x=q^{rn}N_x$ shows that $q^{rn}$ is
congruent to one modulo the level, and $\log(q^{rn})\ne0$ because
$q$ is not a root of unity.

Choose $d>1$ odd with $d\equiv-1$ modulo the orders of all the
$s_x$, modulo $p$, and modulo $\ell^m$, where $m\ge2$ is at least the
valuation of $\log(q^{r_xn})$ for every $x$ with $N_x\ne0$. Then
$d$ is prime to $p$, and
\[
 u_x=\frac{\log(-1/d)}{\log(q^{rn})}\in\mathbb Z_\ell .
\]
These choices involve only $Q$, the orders of the $s_x$ and the
integers $r_x$. The element $P_x=(g_x^n)^{u_x}$ commutes with
$g_x$ and with $s_x$, and
\[
 \operatorname{Ad}(P_x)N_x=(q^{rn})^{u_x}N_x=-\tfrac1dN_x .
\]
Therefore
\[
 P_x\rho_x(\gamma)^{-d}P_x^{-1}
 =s_x^{-da(\gamma)}\exp\bigl(-d\,t_\ell(\gamma)
        \cdot(-\tfrac1d)N_x\bigr)=\rho_x(\gamma),
\]
since $d\equiv-1$ modulo the order of $s_x$. If $N_x=0$ the same
congruence gives $\rho_x^{-d}=\rho_x$. Finally, if the tangential
base points are permuted by $\phi$, the elements $g_x$ at the
points of an orbit are transported by $\phi$, and so are the
$P_x$. After $r$ steps one returns to $g_xP_xg_x^{-1}=P_x$, so
this is consistent.
\end{proof}

\subsection{The curve and its square root of the dualizing line}

Let $\Omega_{\mathcal X}$ be the relative dualizing sheaf of
$\mathcal X$ over $R_0$, and let $d$ be the integer of
Lemma~\ref{fl:lem:weillocal-reversal}. Choose a sufficiently
ample line bundle $\mathcal A$ on $\mathcal X$ and put
\[
 \mathcal L=\Omega_{\mathcal X}\otimes \mathcal A^2,
 \qquad \kappa_1=\Omega_{\mathcal X}^{(d+1)/2}\otimes \mathcal A^d,
\]
so that $\kappa_1^2=\Omega_{\mathcal X}\otimes\mathcal L^d$. Let
$D_{\rm ur}$ be a nonempty divisor, finite \'etale over $R_0$ and
disjoint from the divisor of Theorem~\ref{fl:thm:marked-lifting}.
Such a divisor exists: take a closed point of the special fibre
outside that divisor; over the corresponding finite unramified
extension of $R_0$ it lifts to a section, by smoothness and
Hensel's lemma; the conjugates of this section have disjoint
special fibres, hence are disjoint, and their union is defined
over $R_0$. We
choose a section $s\in H^0(\mathcal X,\mathcal L^d)$ whose
divisor contains the divisor of the theorem and $D_{\rm ur}$, has
a simple zero along $D_{\rm ur}$ in every geometric fibre, and
has reduced generic fibre. From now on $D$ denotes the divisor of
$s$; the local system $\sigma^{\rm lift}$ is unramified at the
points which have been added.

Such a section exists if $\mathcal A$ is ample enough. Sections
of $\mathcal L^d$ vanishing on the old divisor correspond to
sections of a twist of $\mathcal L^d$, and by the vanishing
theorem of Serre one can prescribe the value of the latter along
$D_{\rm ur}$ and on the first infinitesimal neighbourhood of any
point. The sections over $R_0$ with a prescribed reduction are
Zariski dense in the space of sections over $L$, and those whose
divisor on the generic fibre is not reduced form a proper closed
subset, because the linear system separates tangent vectors. So
a section with the required properties exists over $R_0$ itself.

\begin{proposition}[The curve with two components and its smoothing]
\label{fl:prop:geometrization-theta}
In the total space of $\mathcal L$, with tautological coordinate $w$, set
\begin{equation}\label{fl:eq:integral-composite}
 C_2:\ w^d=s,
 \qquad C_0:\ w(w^d-s)=0.
\end{equation}
Let $\pi_0:C_0\to\mathcal X$ be the projection.
\begin{enumerate}
\item The curve $C_0$ is the union of the zero section
$\mathcal X$ and of $C_2$, glued along $D$. It is flat, projective
and Gorenstein over $R_0$, with geometrically reduced and
connected fibres, and its dualizing sheaf is
$\pi_0^*\kappa_1^2$. So $\pi_0^*\kappa_1$ is a theta
characteristic.
\item There is a flat projective family over
$R_0[[\varepsilon]]$ with fibre $C_0$ at $\varepsilon=0$, to which
the theta characteristic extends, and whose fibre over the
fraction field is smooth. Its dualizing sheaf has positive degree
on every component of every fibre.
\end{enumerate}
\end{proposition}

Over the generic fibre of $R_0$ the divisor $D$ is reduced and
its points are ordinary double points of $C_0$. Over the closed
fibre $D$ may be nonreduced, and then that fibre of $C_0$ is not
nodal: the local equation $w(w^3-z^2)=0$, in characteristic
greater than three, defines a reduced Gorenstein curve whose
singular point is not an ordinary double point. Nothing below uses
that the closed fibre is nodal; the properties used are those
of~(1). The calculations in the proof are made for the whole
family over $R_0$.

\begin{proof}
\step{1}{the curve $C_0$} The two equations are monic in $w$, so
$C_2$ and $C_0$ are finite and free over $\mathcal X$, of degrees
$d$ and $d+1$. They are Cartier divisors in the total space of
$\mathcal L$, which is smooth over $R_0$; so they are Gorenstein.
Let $h:C_2\to\mathcal X$ be the projection. The curve $C_2$ meets
the zero section in the divisor $D_2=(w=0)$ of $C_2$, which $h$
maps isomorphically onto $D$, and $h^*D=dD_2$. The equality of
ideals $(w)\cap(w^d-s)=(w(w^d-s))$ shows that $C_0$ is the
pushout of $\mathcal X$ and $C_2$ along $D$, and this remains
true after any base change. At a point of $D_{\rm ur}$ the
section $s$ is a uniformizer in every geometric fibre, so
$w^d-s$ is an Eisenstein polynomial and the geometric fibres of
$C_2$ are irreducible; they are reduced because $d$ is
invertible. Hence the fibres of $C_0$ are geometrically reduced
and connected.

\step{2}{the dualizing sheaf} The canonical bundle of the total
space of $\mathcal L$ is the pullback of
$\Omega_{\mathcal X}\otimes\mathcal L^{-1}$, and $C_0$ is the
divisor of a section of the pullback of $\mathcal L^{d+1}$. The
adjunction formula gives
\begin{equation}\label{fl:eq:composite-theta}
 \Omega_{C_0/R_0}\simeq
 \pi_0^*(\Omega_{\mathcal X}\otimes \mathcal L^d)
 =\pi_0^*\kappa_1^2.
\end{equation}
On the component $\mathcal X$ this is
$\Omega_{\mathcal X}(D)$. The derivative of $w(w^d-s)$ with
respect to $w$ is $-s$ on the component $w=0$ and $ds$ on the
component $w^d=s$. Write $\theta_1$ and $\theta_2$ for the resulting local
trivializations of the dualizing line, corresponding to the same
local frame of $\kappa_1^2$ on the two components. Then
\begin{equation}\label{fl:eq:theta-sign}
 \theta_2=(-1/d)\,c_h(h^*\theta_1),
\end{equation}
where $c_h$ is the isomorphism of the logarithmic
Riemann--Hurwitz formula. This is the condition that the residues
of the dualizing sheaf at the two branches of a node are
opposite. If the degree of $\mathcal A$ is large,
$\pi_0^*\kappa_1^2$ has positive degree on both components, and
the generic fibre has at least three nodes.

\step{3}{the smoothing} Since $\mathcal L^d\simeq\mathcal O(D)$
and $H^1(\mathcal X,\mathcal L)=0$, the restriction
$H^0(\mathcal X,\mathcal L^{d+1})\to
H^0(D,\mathcal L^{d+1}|_D)$ is surjective. The scheme $D$ is finite
over the local ring $R_0$, hence semilocal, so $\mathcal L|_D$ has
a frame $b_D$. We choose a section $t$ of $\mathcal L^{d+1}$ with
\begin{equation}\label{rw:eq:t-normalized}
 t|_D=-b_D^{\,d+1}.
\end{equation}
In particular $t$ is invertible on $D$. The normalization
\eqref{rw:eq:t-normalized} is used in
Section~\ref{rw:sec:extending-parameter}: it makes the tangent
vectors of the two branches at a node correspond to each other.
For
an integer $e>0$ let $\mathcal C$ be the curve over
$R_0[[\varepsilon]]$ with equation
\begin{equation}\label{fl:eq:explicit-smoothing}
 w(w^d-s)=\varepsilon^e t
\end{equation}
in the total space of $\mathcal L$, and $\pi:\mathcal C\to
\mathcal X$ the projection. As before it is finite and free over
$\mathcal X$, projective and Gorenstein, with
$\pi_*\mathcal O_{\mathcal C}=\bigoplus_{i=0}^d\mathcal L^{-i}$
and dualizing sheaf $(\pi^*\kappa_1)^2$. The first formula shows
that the fibres are connected, and the second that the dualizing
sheaf is positive on every component and that $\pi^*\kappa_1$
extends the theta characteristic. The locus of the base over
which the fibre is geometrically reduced is open
\cite[Tag~0C0E]{StacksProject}; it contains the closed point of
the local base, so it is the whole base.

It remains to see that the fibre over the fraction field of
$R_0[[\varepsilon]]$ is smooth. Let $Z\subset\mathcal C$ be the
closed subset where $\mathcal C$ is not smooth over
$R_0[[\varepsilon]]$. Consider the fibre of $\mathcal C$ over the
generic point of the divisor $\varepsilon=0$ of the base, which
is $C_0\otimes L$. Outside the nodes it is smooth. At a node, $w$
and $w^d-s$ are part of a system of local coordinates of the
total space of $\mathcal L$, because $s$ has a simple zero there,
and $t$ is a unit; so, in a local frame of $\mathcal L$, with
$x=w$ and $y=(w^d-s)/t$ the curve
$\mathcal C$ is given near the node by $xy=\varepsilon^e$, and
$Z$ is contained in $\varepsilon=0$ in a neighbourhood of the
node. Now suppose that $Z$ contains a point $z$ of the fibre over
the fraction field. The closure of $z$ is proper over the base
and maps onto it, so $z$ specializes to a point of $Z$ in
$C_0\otimes L$, necessarily a node. Every neighbourhood of this
node then contains the point $z$ of $Z$, at which
$\varepsilon\ne0$; this is a contradiction.
\end{proof}

\begin{remark}
The line bundle $\mathcal A$ is taken very positive for three
reasons: there must be a section $s$ with the prescribed divisor,
there must be a direction of smoothing, and the dualizing line of
$C_0$ must be positive on both components. The degree of
$\mathcal A$ plays no role afterwards.
\end{remark}

\subsection{Extending the parameter across the smoothing}
\label{rw:sec:extending-parameter}

On the component $\mathcal X$ of $C_0$ we have the local system
$\sigma^{\rm lift}$, and on $C_2-D_2$ we take its pullback by
$h$. We work over $\overline L$ and use, near each point of $D$, a
local frame $b$ of $\mathcal L$ which extends the frame $b_D$ of
\eqref{rw:eq:t-normalized}. In this frame $w$, $s$ and $t$ are
functions, $t=-1$ on $D$, and
\[
 x=w,\qquad y=(w^d-s)/t
\]
are the coordinates in which $\mathcal C$ is $xy=\varepsilon^e$
near the node. On the branch $\mathcal X$, where $w=0$, the
function $y=-s/t$ is a local coordinate; it is $s$ times a
function with value one at the node. On the branch $C_2$ the
function $x=w$ is a local coordinate, and
\begin{equation}\label{rw:eq:branch-coordinates}
 h^*(y|_{\mathcal X})=-w^d/t=x^d\,a(x),\qquad a(0)=1 .
\end{equation}
Another frame extending $b_D$ changes $x$ and $y$ by functions
with value one at the node. So the tangent vectors defined by $y$
on $\mathcal X$ and by $x$ on $C_2$ are well defined, and they are
permuted by $\phi$, since $b_D$ is defined over $R_0$. By
\eqref{rw:eq:branch-coordinates} the fibre of
$h^*\sigma^{\rm lift}$ at the tangential base point $x$ of $C_2$
is identified with the fibre of $\sigma^{\rm lift}$ at the
tangential base point $y$ of $\mathcal X$: one substitutes
$y^{1/n}=x^{d/n}a^{1/n}$, with the root of $a$ whose constant term
is one. This identification commutes with the action of
$\phi^{r}$, which is therefore given on both fibres by the same
element $g_x$ of $Q$; and it transforms the action of
$\theta\in\mu_n$ on the fibre for $C_2$ into that of $\theta^d$ on
the fibre for $\mathcal X$. So the inertia on the branch in $C_2$
is $\rho_x^d$. With a section $t$ whose value on $D$ is not
normalized, $a(0)$ would be a unit $c$ of the residue field, and
the two actions of $\phi^r$ would differ by the image under
$\rho_x$ of the Kummer class of $c$; this is the reason for
\eqref{rw:eq:t-normalized}.

From now on $g_x$ denotes the action of $\phi^r$ at these
tangential base points, and $P_x$ the element of
Lemma~\ref{fl:lem:weillocal-reversal} for this $g_x$. At a node
over a point $x$ of the old divisor we identify the fibres on the
two branches by $P_x$, and at a node over an added point by the
identity. The two branches of a node have opposite orientations,
so the condition for gluing is that $\rho_x$ on the first branch
corresponds to the inverse of $\rho_x^d$ on the second, and this
is the relation $P_x\rho_x^{-d}P_x^{-1}=\rho_x$. The relation
$P_xg_x=g_xP_x$ says that the identifications are compatible with
Frobenius.

Fix a compatible system of roots $\varepsilon^{1/m}$ of
$\varepsilon$, put
\[
 R_{\rm P}=\bigcup_m\overline L[[\varepsilon^{1/m}]],
\]
and extend $\phi$ to $R_{\rm P}$ by its action on the
coefficients, so that it fixes all the $\varepsilon^{1/m}$. The
fraction field of $R_{\rm P}$ is algebraically closed.

\begin{proposition}[Extension of the coverings to the smoothing]
\label{fl:lem:balanced-parameter-tower}
Let $\mathcal C$ be the family \eqref{fl:eq:explicit-smoothing},
considered over $\overline L[[\varepsilon]]$, and let $Q$ be a
profinite group with a countable basis of neighbourhoods of the
identity. Suppose given
\begin{itemize}
\item a continuous homomorphism to $Q$ of the fundamental group
of $\mathcal X_{\overline L}-D$, and on $C_2-D_2$ its pullback by
$h$;
\item for each node an element of $Q$ identifying the fibres of
the two branches at the tangential base points defined by $y$ and
by $x$, such that the inertia homomorphisms of the two branches
become inverse to each other;
\item an action of $\phi$ on these data, such that the
identification at each node is carried by $\phi$ to the
identification at its image.
\end{itemize}
Then these data define a continuous homomorphism to $Q$ of the
fundamental group of the smooth locus of
$\mathcal C\otimes R_{\rm P}$ over $R_{\rm P}$, and in particular
of its geometric generic fibre, which is smooth and proper. Its
image is contained in the closed subgroup generated by the image
of the given homomorphism and by the elements attached to the
nodes, and it contains the image of the given homomorphism. The
automorphism $\phi$ acts on the result, compatibly with all finite
quotients of $Q$ and with composition.
\end{proposition}

\begin{proof}
The construction is made for one finite quotient $\Delta$ of $Q$
at a time, and the coverings are then shown to be compatible. The
essential point is in step~2: the condition that the inertia on
the two branches of a node is inverse is what is needed for the
two coverings to glue to a finite \'etale covering of a stack
whose coarse space is the nodal curve. Choose a decreasing
sequence of open normal subgroups of $Q$ which is a basis of
neighbourhoods of the identity. (The group $Q$ to which the proposition
is applied has such a sequence because it is a compact subgroup of
$\operatorname{GL}_n(E)$.)

\step{1}{a stack of roots} Let $n$ be an integer divisible by $e$
and by the orders of the images in $\Delta$ of the inertia groups
at the points of $D$. Put
\[
 \tau_n=\varepsilon^{e/n},\qquad A_n=\overline L[[\tau_n]],
\]
so that $\tau_n^n=\varepsilon^e$. On the open subset $t\ne0$ of
$\mathcal C_{A_n}=\mathcal C\otimes A_n$, which contains the
nodes, $w$ is a section of the pullback of $\mathcal L$, the
quotient $y=(w^d-s)/t$ is a section of the pullback of
$\mathcal L^{-1}$, and $wy=\tau_n^n$. Let $\mathscr C_n$ be the
stack over this open subset whose objects are
\[
 \bigl(\mathcal M,\ \mathcal M^{\otimes n}\simeq\pi^*\mathcal L,\
   u\in\Gamma(\mathcal M),\ v\in\Gamma(\mathcal M^{-1})\bigr),
 \qquad u^n=w,\quad v^n=y,\quad uv=\tau_n,
\]
with $\mathcal M$ a line bundle. Where $w$ is invertible, $u$
trivializes $\mathcal M$, and the object is unique and has no
automorphisms; the same holds where $y$ is invertible. So
$\mathscr C_n$ is isomorphic to $\mathcal C_{A_n}$ outside the
nodes of the closed fibre, and we glue it with the complement of
these nodes in $\mathcal C_{A_n}$ to obtain a stack over
$\mathcal C_{A_n}$, still denoted $\mathscr C_n$. In a local frame
of $\mathcal L$ near a node it is the quotient
\begin{equation}\label{rw:eq:node-chart}
 [\{uv=\tau_n\}/\mu_n],\qquad x=u^n,\quad y=v^n,
 \quad \theta(u,v)=(\theta u,\theta^{-1}v).
\end{equation}
Thus $\mathscr C_n$ is a Deligne--Mumford stack, proper over
$A_n$, with coarse space $\mathcal C_{A_n}$. It is regular, since
the chart $uv=\tau_n$ is and $\mathcal C_{A_n}$ is smooth over
$A_n$ outside the nodes. It coincides with $\mathcal C_{A_n}$ over
the locus where $\mathcal C_{A_n}$ is smooth over $A_n$, in
particular over the generic fibre. The closed
fibre $\mathscr C_{n,0}$ has the chart $[\{uv=0\}/\mu_n]$ at each
node. Its normalization is the union of the stack of $n$-th roots
of $\mathcal X_{\overline L}$ along $D$, with $v^n=y$, and of that
of $C_{2,\overline L}$ along $D_2$, with $u^n=x$; and
$\mathscr C_{n,0}$ is obtained by gluing them along the residual
gerbes $B\mu_n$ of the nodes.

If $n'=mn$, then $\tau_n=\tau_{n'}^m$, and
$(\mathcal M',u',v')\mapsto(\mathcal M'^{\otimes m},u'^m,v'^m)$
defines a morphism
$\mathscr C_{n'}\to\mathscr C_n\otimes_{A_n}A_{n'}$, which is the
identity over the smooth locus. The automorphism $\phi$ acts on
all these stacks and morphisms, because it fixes the $\tau_n$ and
$\mathcal L$, $s$ and $t$ are defined over $R_0$.

\step{2}{a covering of the closed fibre} The given torsor under
$\Delta$ on $\mathcal X_{\overline L}-D$ extends uniquely to a
finite \'etale torsor on the stack of $n$-th roots of
$\mathcal X_{\overline L}$ along $D$: in the chart with coordinate
$v$ the normalization of its pullback is finite \'etale, because
$n$ is divisible by the order of inertia, and the group $\mu_n$
acts on it. Its restriction to the residual gerbe at a node is the
fibre at the tangential base point defined by $y$, with the action
of $\mu_n$; by \eqref{rw:eq:node-chart} an element $\theta$ acts
by $\rho_x(\theta)^{-1}$. In the same way the pullback by $h$
extends to the stack of roots of $C_{2,\overline L}$, and its
restriction to the gerbe is the fibre at the tangential base point
defined by $x$, which has been identified with the previous one;
on it $\theta$ acts by $\rho_x(\theta)^d$. The element of $Q$
attached to the node, taken in $\Delta$, is an isomorphism between
these two fibres, and the hypothesis that it makes the inertia
homomorphisms inverse to each other says that it commutes with
the two actions of $\mu_n$. So it is an isomorphism between the
restrictions of the two torsors to the gerbe $B\mu_n$.

Locally for the \'etale topology, a finite \'etale covering of the
chart $[\{uv=0\}/\mu_n]$ is the same as finite \'etale coverings
of the two branches, with actions of $\mu_n$, and an equivariant
isomorphism between their fibres at the origin. Indeed the ring of
$uv=0$ is the fibre product of the rings of the two branches over
the residue field of the origin; finite projective modules over a
fibre product of rings along a surjection are obtained by patching
(\cite[\S2]{Milnor1971}, \cite[Tag~0D2J]{StacksProject}); and a
finite locally free algebra over
this ring is \'etale if it is \'etale over both branches. Hence
the two torsors and the isomorphisms at the nodes define a finite
\'etale torsor $\mathcal Y_{\Delta,0}$ under $\Delta$ on
$\mathscr C_{n,0}$, whose restrictions to the two components are
the given ones. The element attached to the node is part of the
construction, so the torsor comes with its identification of
fibres, and not only up to isomorphism.

\step{3}{extension to the family} The stack $\mathscr C_n$ is
proper over the complete local ring $A_n$. Therefore restriction
to the closed fibre is an equivalence between the categories of
finite \'etale coverings of $\mathscr C_n$ and of
$\mathscr C_{n,0}$. Indeed, a finite \'etale covering extends
uniquely over each infinitesimal neighbourhood of the closed
fibre, with its morphisms. The resulting compatible system of
coherent sheaves is algebraic by the existence theorem of
Grothendieck for proper stacks \cite[Theorem~1.4]{Olsson2005}. The
theorem gives an equivalence of categories of coherent sheaves
which is compatible with tensor products, so the multiplication of
the algebras, the action of the group and the morphisms between
coverings are algebraic as well. The algebra obtained is \'etale
over a neighbourhood of the closed fibre, and the closed subset
where it is not \'etale has a closed image in
$\operatorname{Spec}A_n$ which does not contain the closed point;
so it is empty. Let $\mathcal Y_\Delta$ be the torsor on
$\mathscr C_n$ which extends $\mathcal Y_{\Delta,0}$. Over the
smooth locus of $\mathcal C_{A_n}$ it is a finite \'etale torsor
on a scheme.

\step{4}{the maps between the coverings} Let $\Delta'\to\Delta$ be
two quotients of the sequence, with integers $n'=mn$. On the
closed fibre of $\mathscr C_{n'}$ the torsor under $\Delta$
induced from $\mathcal Y_{\Delta',0}$ and the pullback of
$\mathcal Y_{\Delta,0}$ are the same: they agree on the two
components, the fibres at the tangential base points formed with
$n$ and with $n'$ are identified canonically, and at each node
both torsors are glued by the image of the same element of $Q$.
Since restriction to the closed fibre is fully faithful, this
isomorphism extends uniquely to $\mathscr C_{n'}$. For three
quotients, the composite of two of these isomorphisms and the
third have the same restriction to the closed fibre, so they are
equal. On the smooth locus the morphisms
$\mathscr C_{n'}\to\mathscr C_n$ are the identity. So the torsors
$\mathcal Y_\Delta$, pulled back to $R_{\rm P}$, form an inverse
system of finite \'etale torsors on the smooth locus of
$\mathcal C\otimes R_{\rm P}$. This is the continuous homomorphism
to $Q$.

\step{5}{the action of $\phi$} On the closed fibre the action of
$\phi$ on the data is a system of isomorphisms
$\Phi_\Delta:\phi^*\mathcal Y_{\Delta,0}\simeq
\mathcal Y_{\Delta,0}$ compatible with the maps of step~4. On the
two components they are the given ones, and they are compatible
with the gluing because the identification at each node is carried
by $\phi$ to the identification at its image. By the equivalence
of step~3, $\Phi_\Delta$ and its inverse extend to $\mathscr C_n$,
the extensions are inverse to each other and compatible with the
maps of step~4, and the extensions of the powers of $\phi$ are the
powers of the extension. The action of $\phi$ itself is
extended, and not only that of the power which fixes a given node.
This is possible because $\phi$ fixes $\tau_n$.

\step{6}{the image} All the data have values in the closed
subgroup of $Q$ generated by the image of the given homomorphism
and the elements attached to the nodes, so the homomorphism
constructed has values in it.

For the lower bound, fix $\Delta$ and let $\mathcal Y'$ be a
connected component of $\mathcal Y_\Delta$. Since $\mathcal Y'$ is
proper over $A_n$, its closed fibre is connected: the idempotents
of the closed fibre lift. The stack $\mathcal Y'$ is regular and
connected, hence integral. Its closed fibre is a finite \'etale
covering of the connected stack $\mathscr C_{n,0}$, so it has a
point over the smooth locus of $\mathcal C_{A_n}$; as
$\mathcal Y'$ is smooth over $A_n$ at this point, the point lifts
to a section of $\mathcal Y'$ over $A_n$. The ring of global
functions of $\mathcal Y'$ is a domain, finite over $A_n$, and the
section defines a homomorphism from it to $A_n$ which is the
identity on $A_n$. The kernel of this homomorphism is a prime
ideal which meets $A_n$ in zero, hence is zero. So the ring of
global functions of $\mathcal Y'$ is $A_n$, and the generic fibre
of $\mathcal Y'$ is geometrically connected. The same holds after
replacing $n$ by a multiple. So the connected components of the
torsor on the geometric generic fibre correspond to those of
$\mathcal Y_{\Delta,0}$.

Now the restriction of $\mathcal Y_{\Delta,0}$ to
$\mathcal X_{\overline L}-D$ is the torsor of the given
homomorphism, and a connected component of this restriction lies
in one connected component of $\mathcal Y_{\Delta,0}$. So the
image of the fundamental group of the geometric generic fibre in
$\Delta$ contains the image of the given homomorphism. The image
in $Q$ is compact, hence closed, and a closed subgroup of $Q$
which has this property in every quotient of the chosen sequence
contains the image in $Q$ of the given homomorphism.

The elements attached to the nodes themselves need not lie in the
image. Only the products of these elements around the cycles of
the dual graph, conjugated along paths, contribute to it.
\end{proof}

\begin{proposition}[The unramified parameter on the smooth curve]
\label{fl:prop:geometrized-parameter}
Let $\mathsf{Mon}_{\sigma}^{\rm ar}$ and
$\mathsf{Mon}_{\sigma}^{\rm geom}$ be the arithmetic and the
geometric monodromy group of the given elliptic parameter
$\sigma$, and $g=\sigma(\operatorname{Frob})$ the image of the
chosen Frobenius element. The geometric generic fibre of the
smoothing of Proposition~\ref{fl:prop:geometrization-theta}
carries an unramified $\widehat G$-local system $\rho$ whose
geometric monodromy group $\mathsf{Mon}_{\rho}$ satisfies
\[
 \mathsf{Mon}_{\sigma}^{\rm geom}\subset \mathsf{Mon}_{\rho}\subset \mathsf{Mon}_{\sigma}^{\rm ar},\qquad
 \overline{\langle \mathsf{Mon}_{\rho},g\rangle}=\mathsf{Mon}_{\sigma}^{\rm ar}.
\]
The group $\mathsf{Mon}_\rho$ is reductive and normalized by $g$.
The local system $\rho$ extends to the smooth locus of the family
over $R_{\rm P}$, and the restriction of the extension to the closed
fibre is $\sigma^{\rm lift}$ on $\mathcal X_{\overline L}-D$ and the
pullback of $\sigma^{\rm lift}$ by $h$ on $C_{2,\overline L}-D_2$,
compatibly with the action of Frobenius.
\end{proposition}

\begin{proof}
Apply Proposition~\ref{fl:lem:balanced-parameter-tower} to the local
systems on the two components and to the elements $P_x$. All
these data have values in $Q$, so the image of $\rho$ is
contained in $Q$ and $\mathsf{Mon}_\rho\subset
\mathsf{Mon}_\sigma^{\rm ar}$. Take the base point on the
component which carries $\sigma^{\rm lift}$. By the proposition the image
contains the image of the geometric fundamental group of the
generic fibre of $\mathcal X_{R_0}-D$ under $\sigma^{\rm lift}$,
which
contains the image of the geometric fundamental group of $U$
under $\sigma$ by Theorem~\ref{fl:thm:marked-lifting}. This gives
the first inclusion.

The quotient
$\mathsf{Mon}_{\sigma}^{\rm ar}/\mathsf{Mon}_{\sigma}^{\rm geom}$
is finite and cyclic by
Lemma~\xref{I}{fl:lem:arithmetic-monodromy-quotient}. So every group
between the two is normal in $\mathsf{Mon}_\sigma^{\rm ar}$, in
particular normalized by $g$, and has the same identity
component, which is reductive. Finally
$\mathsf{Mon}_\sigma^{\rm geom}$ and $g$ generate a dense
subgroup of $\mathsf{Mon}_\sigma^{\rm ar}$, which gives the last
equality.
\end{proof}