\section{Specializing the whole Hecke family}
\label{fl:sec:hecke-extension}

Section~\ref{fl:sec:punctured-spectral-fibre} constructed a family
of automorphic sheaves $P\mapsto\mathcal F_P$ on the smooth curve in
characteristic zero, and Section~\ref{fl:sec:geometrization} showed
that this curve degenerates, first to the nodal curve
$\mathcal X\cup_DC_2$ in characteristic zero and then to
characteristic $p$. The goal of this section is
Theorem~\ref{fl:thm:compact-automorphic-family}: the family extends
over a base $S$ that contains both degenerations, as a family of
complexes which are universally locally acyclic (ULA) over $S$,
together with its Hecke identities and its compatibility with
Frobenius. The whole family is extended, with all maps between its
members, because the argument of
Lemma~\ref{fl:lem:scalar-coefficient-covariance} needs naturality
in $P$.

The tool is the theorem of Hansen and Scholze
(Input~\ref{rw:in:hs}): over an absolutely integrally closed
valuation ring, ULA complexes are the same as constructible
complexes on the generic fibre. So objects extend automatically and
uniquely, and maps between ULA complexes extend uniquely. What has
to be proved is that everything appearing in a Hecke identity is
ULA. For the sheaf this holds by construction, for the local
systems on the curve it requires that they extend over $S$
(Lemma~\ref{fl:lem:composed-parameter-gluing}), and for the Satake
kernels it is a theorem of Gaitsgory and Raskin over a Witt ring,
which we transfer to $S$
(Lemma~\ref{fl:lem:relative-satake-kernels}).

\subsection{The base and the total local system}
\label{fl:sec:valued-nodal-family}

Let $\overline L$ be the algebraic closure of
$\operatorname{Frac}R_0$ chosen above, and $R_{\rm m}$ its valuation
ring. Introduce
\[
 R_{\rm P}=\bigcup_{n\ge1}\overline L[[\varepsilon^{1/n}]],
 \qquad R=R_{\rm P}\times_{\overline L}R_{\rm m},
 \qquad S=\operatorname{Spec}R,
\]
so that $R$ consists of the series in $R_{\rm P}$ whose constant
term lies in $R_{\rm m}$. It is a valuation ring of rank two. Its
spectrum $S$ has three points $\eta$, $\xi$, $s$, with residue
fields the field of Puiseux series over $\overline L$, the field
$\overline L$, and $\overline{\mathbb F}_q$. Over $\xi$ the curve
$\mathcal C_S$ is the nodal curve $C_0$ in characteristic zero,
and over $s$ it is its reduction. The semilinear automorphism
$\phi$ acts on the coefficients of the series and fixes the chosen
roots of $\varepsilon$.

\begin{lemma}[The valuation ring]
\label{fl:lem:puiseux-algebraic-closure}
The field $\bigcup_n\overline L((\varepsilon^{1/n}))$ is
algebraically closed. The ring $R$ is a valuation ring of this
field, and it is absolutely integrally closed.
\end{lemma}

\begin{proof}
A polynomial over the union has its coefficients in a field
$\overline L((u))$ with $u=\varepsilon^{1/n}$. A finite extension
of this complete discretely valued field has residue field
$\overline L$, and it is of the form $\overline L((v))$ for a
uniformizer $v$. Write $u=v^ea(v)$ with $a$ a unit. Since the
residue characteristic is zero, $a$ has an $e$-th root, and after
replacing $v$ by $va(v)^{1/e}$ we have $u=v^e$. So the extension
is contained in the union.

Let $f$ be a nonzero element of the field. If its leading
exponent is positive, it lies in $R_{\rm P}$ with constant term
zero, hence in $R$; if the exponent is negative, $f^{-1}$ lies in
$R$. If the exponent is zero, $f$ is a unit of $R_{\rm P}$ whose
constant term is a nonzero element of $\overline L$, and either
this constant term or its inverse lies in $R_{\rm m}$. So $R$ is
a valuation ring. A valuation ring with algebraically closed
fraction field is absolutely integrally closed.
\end{proof}

Let $\mathcal C_S$ be the base change to $S$ of the family of
Proposition~\ref{fl:prop:geometrization-theta}, and
$U=\mathcal C_S^{\rm sm}$ its smooth locus over $S$. Over $R_{\rm m}$
the family is the Gorenstein curve $C_0$ with its two components,
nodal on its generic fibre, whose smooth locus carries
$\sigma^{\rm lift}$ and its pullback by $h$; over $R_{\rm P}$ it
carries the homomorphism of
Proposition~\ref{fl:lem:balanced-parameter-tower}.

\begin{lemma}[The local system over the base]
\label{fl:lem:composed-parameter-gluing}
These homomorphisms glue to a continuous homomorphism from the
fundamental group of $U$ to $Q$, and so define, for every
representation of $\widehat G$, a local system on $U$, compatibly
with tensor products. We denote this system by $\mathcal E$. The
gluing is compatible with base points, with the finite quotients
of $Q$ and with the action of $\phi$.
\end{lemma}

\begin{proof}
By construction the two homomorphisms agree on the fibre over
$\overline L$, which is the smooth locus of $C_0\otimes\overline L$.
The equality $R=R_{\rm P}\times_{\overline L}R_{\rm m}$ says that
$S$ is the pushout of $\operatorname{Spec}R_{\rm P}$ and
$\operatorname{Spec}R_{\rm m}$ along
$\operatorname{Spec}\overline L$, which is closed in the first.
Such pushouts have the expected universal property for maps to
algebraic stacks with suitable diagonal, and they are compatible
with flat base change
\cite[Theorem~4.2 and Lemma~4.3(2)]{AHHLR2024}; so $U$ is the
pushout of its restrictions to the two rings.

A torsor under a finite quotient $Q/K$ of $Q$ is a map to the
classifying stack $B(Q/K)$, whose diagonal is finite. By the
universal property the torsors on the two pieces, with their
identification over $\overline L$, define a torsor on $U$, and
maps of torsors glue in the same way. Since a map out of the
pushout is determined by its restrictions, the glued maps satisfy
the same identities; this applies to the transition maps, to the
action of the groups and to the action of $\phi$. The inverse
limit over $K$ is the required continuous homomorphism.
\end{proof}

\begin{lemma}[The auxiliary puncture]
\label{fl:lem:closed-return-puncture}
Let $y$ be a closed point, of degree $d_{\rm aux}$, of the special
fibre of $C_2$ over $\mathbb F_q$ which does not lie on $D_2$.
There is a divisor $D_{\rm aux}$ of the family $\mathcal C$,
finite \'etale over $R_0[[\varepsilon]]$, whose closed fibre is
$y$. On the generic fibre $\phi$ permutes its points cyclically,
and $\phi^{d_{\rm aux}}$ acts on the fibre of $\mathcal E$ at such
a point as the Frobenius element of $y$ acts on the fibre of
$h^*\sigma$. In particular its adjoint action is pure of weight
zero.
\end{lemma}

\begin{proof}
The family $\mathcal C$ is smooth over $R_0[[\varepsilon]]$ at
$y$. Let $R_1/R_0$ be the unramified extension of degree
$d_{\rm aux}$. By Hensel's lemma $y$ lifts to a section over
$R_1[[\varepsilon]]$. Its $d_{\rm aux}$ conjugates are pairwise
disjoint: the locus where two of them coincide is a closed
subscheme of the local scheme
$\operatorname{Spec}R_1[[\varepsilon]]$ which does not contain the
closed point. Their union descends to a divisor $D_{\rm aux}$
finite \'etale over $R_0[[\varepsilon]]$.

After base change to $S$ each of the sections is a section over
$S$. An absolutely integrally closed valuation ring is strictly
henselian, so the restriction of $\mathcal E$ to a section is
constant, and its fibres over $\eta$ and over $s$ are identified.
The identification commutes with $\phi^{d_{\rm aux}}$, which maps
the section to itself. Over $s$ this automorphism is the Frobenius
element of $y$. The last assertion follows from
Theorem~\xref{I}{fl:thm:arithmetic-strong-purity}, applied to the
image of $y$ in $X$.
\end{proof}

Put $U^\circ=U-D_{\rm aux}$; this is where the legs of Hecke
modifications are allowed. It is dense in every component of every
geometric fibre. Since the puncture $D_{\rm aux}$ lies on the
auxiliary component, $U^\circ$ contains all of
$X-\operatorname{supp}D$ on the original component.

\subsection{ULA extension and relative Satake}

For a map $Y\to S$, a complex is universally locally acyclic
(ULA) if every local specialization map is an isomorphism after
every base change. By Input~\ref{rw:in:hs}, for $Y/S$ separated and
of finite presentation, restriction to the generic fibre is an
equivalence
\begin{equation}\label{rw:eq:ula-restriction}
 D^{\rm ULA}(Y/S,k)\xrightarrow{\sim}D_c(Y_\eta,k),
 \qquad\text{with inverse }Rj_*.
\end{equation}
This includes rational and algebraic $\ell$-adic coefficients
\cite[Theorem~4.1]{HS}. It commutes with external products
\cite[Corollary~4.2(iv)]{HS}, and with smooth pullback, because
smooth pullback preserves universal local acyclicity. We use it
in the following form: \emph{a map between the generic fibres of
two ULA complexes extends uniquely}, and an identity between such
maps holds over $S$ if it holds over $\eta$.

On a stack with a smooth atlas, a complex is called
\emph{locally ULA} if its pullbacks to the schemes of a smooth
atlas are ULA over $S$.

\begin{lemma}[ULA extension on the stack of bundles]
\label{fl:lem:ula-stack-descent}
Let $\mathcal B=\operatorname{Bun}_G(\mathcal C_S/S)$. Restriction
to the generic fibre is an equivalence between the locally ULA
complexes on $\mathcal B$ which are bounded and constructible on
every quasi-compact open substack, and the complexes on
$\mathcal B_\eta$ which are bounded and constructible on every
quasi-compact open substack.
\end{lemma}

\begin{proof}
The stack $\mathcal B$ is locally of finite presentation over $S$
with affine diagonal. Choose a smooth atlas by affine schemes of
finite presentation over $S$; the fibre products of its members
over $\mathcal B$ are then separated and of finite presentation,
and \eqref{rw:eq:ula-restriction} applies to all of them. A
complex on $\mathcal B$, or on $\mathcal B_\eta$, is a compatible
system of complexes on the schemes $Y_n$ of the \v Cech nerve of
the atlas $Y_0$. Let a complex on $\mathcal B_\eta$ be given. Let
$E_0$ be the ULA extension to $Y_0$ of its restriction to the
generic fibre of the atlas, and let $E_n$ be the pullback of $E_0$
to $Y_n$ by the first projection. Smooth pullback preserves ULA
complexes and commutes with restriction to the generic fibre, so
$E_n$ is the ULA extension of the $n$-th member of the given
system. Let $f:Y_n\to Y_m$ be a map of the nerve. The composite of
$f$ with the first projection of $Y_m$ is one of the projections
of $Y_n$ to $Y_0$, which are smooth. So $f^*E_m$ is ULA; this
includes the degeneracy maps, which are not smooth. By the full
faithfulness in \eqref{rw:eq:ula-restriction} the isomorphisms
$f^*E_m\simeq E_n$ of the given system extend, together with the
identities and the higher compatibilities between them. The
$E_n$ therefore form a complex on $\mathcal B$ which extends the
given one, and the same argument applies to maps between
complexes. On a quasi-compact open substack one can take a finite
atlas, which gives the boundedness; constructibility on each
member of the atlas is part of the theorem of Hansen and Scholze.
\end{proof}

To extend a Hecke identity, both of its sides must be ULA. For the
side that is a Hecke transform this requires a Satake kernel over
$S$ that is ULA over the space of legs, including the loci where
legs collide. The results available in the literature are over
other bases. Zhu's fusion theorem is over a field
\cite[\S5.4, Proposition~5.4.2, Remark~5.4.12,
Theorem~5.5.17 and Corollary~5.5.19]{Zhu2017}. Gaitsgory and
Raskin construct the kernel in mixed characteristic over a ring
of Witt vectors and prove that it is ULA for both projections
\cite[Theorem~4.2.2, \S4.2.3 and (4.14)]{GR2025}. The following
lemma transfers these kernels to $S$.

\begin{lemma}[Satake kernels over the base $S$]
\label{fl:lem:relative-satake-kernels}
Let $V$ be a representation of $\widehat G^I$. On the Hecke
correspondence over $S$ with legs in $U^\circ$, bounded by the
weights of $V$, there is a complex
$\mathcal S^{\rm val}_{I,V}$ with the following properties.
\begin{enumerate}
\item Its restrictions to the fibres over $\eta$, $\xi$ and $s$ are
the Satake kernels of $V$ over those fields.
\item It is ULA over the product of the stack of bundles and the
space of legs, for the projection to either end.
\item The isomorphisms expressing compatibility of Satake kernels
with maps of representations, with convolution, with changes of
local coordinate and with fusion along the diagonals extend from
the generic fibre.
\end{enumerate}
\end{lemma}

\begin{proof}
Steps~1 and~2 define the kernel, by extending the Satake sheaf of
the generic fibre. The content of the lemma is in step~3: over a larger base, which contains a
ring of Witt vectors, the kernel so defined coincides with the one
constructed by Gaitsgory and Raskin, whose properties are known.
Steps~4 and~5 descend these properties to $S$ and extend the
compatibilities.

\step{1}{local models} The question is local on the space of
legs. Fix a geometric point of $(U^\circ)^I$, and divide the legs
into groups: two legs are in the same group if they coincide at
this point. For each group choose an \'etale coordinate on the
smooth curve $U^\circ$ near the common image of its legs. We work
on the locus where legs of different groups do not meet. By
completion along the graphs of the legs, the space of
modifications at the legs of one group is identified with the
Beilinson--Drinfeld Grassmannian of the affine line, pulled back
by the coordinates; for several groups one obtains the product.
On the part bounded by the weights of $V$, the action of the
group of arcs and the changes of coordinates factor through jets
of finite order. Adding a trivialization of the jets of that
order is a smooth covering, because the group of jets is smooth.
In this way the construction is reduced to the bounded Schubert
varieties in the Grassmannians, which are separated and of finite
presentation over $S$, with the diagonals included.

\step{2}{definition of the kernel} On each of these Schubert
varieties we apply \eqref{rw:eq:ula-restriction} to the Satake
sheaf of the generic fibre, and call the extension
$\mathcal S^{\rm val}_{I,V}$. It is ULA over $S$ by construction.
Properties (1) and (2) do not follow from this, and are proved in
the next two steps by comparison with the kernel of Gaitsgory and
Raskin.

\step{3}{comparison over a larger base} Let $C_{\rm mix}$ be the
completion of $\overline L$ and $\mathcal O_{\rm mix}$ its
valuation ring, which contains $W(\overline{\mathbb F}_q)$. The
field $C_{\rm mix}$ is algebraically closed
\cite[\S3.4.1, Proposition~3]{BGR1984}. Define
\[
 R^{\rm W}=
 \left(\bigcup_n C_{\rm mix}[[\varepsilon^{1/n}]]\right)
 \times_{C_{\rm mix}}\mathcal O_{\rm mix},
 \qquad S^{\rm W}=\operatorname{Spec}R^{\rm W}.
\]
By the proof of Lemma~\ref{fl:lem:puiseux-algebraic-closure} this
is again an absolutely integrally closed valuation ring. The
inclusion $R\to R^{\rm W}$ is local and injective, so
$R^{\rm W}$ is torsion free over the valuation ring $R$, hence
flat, and faithfully flat
\cite[Tags~0539 and 00HQ]{StacksProject}.

We take the kernels from the projective line. By
\cite[Theorem~4.2.2 and \S4.2.3]{GR2025}, applied to
$\mathbb P^1$ over $W(\overline{\mathbb F}_q)$, with the legs
restricted to $\mathbb A^1$ and after pullback along the section
of either end given by the trivial bundle, there is a kernel on
the Beilinson--Drinfeld Grassmannian of the affine line which is
ULA over the space of legs and restricts to the Satake kernels
over the fields. Pull it back to $R^{\rm W}$. With the
trivializations of jets of step~1, the kernel for either end is
the pullback of this one; it does not depend on the
trivializations, because the kernel on the model is equivariant
for the changes of trivialization. The
ULA property is preserved by base change, so the pullback is ULA
over the same product for our local model. It is therefore also
ULA over $S^{\rm W}$: the space of legs is smooth over
$S^{\rm W}$, and so are the charts of the stack of bundles,
because the obstruction to deforming a bundle on a curve lies in
the $H^2$ of the adjoint bundle, which is zero.

We now have two complexes on the local model over $S^{\rm W}$:
the pullback of $\mathcal S^{\rm val}_{I,V}$ and the pullback of
the kernel of Gaitsgory and Raskin. Both are ULA over
$S^{\rm W}$, and on the generic fibre both are the Satake sheaf
of $V$, with the same shifts and twists. By
\eqref{rw:eq:ula-restriction} over $S^{\rm W}$ they are
isomorphic, by a unique isomorphism extending the identification
on the generic fibre. Hence the pullback of
$\mathcal S^{\rm val}_{I,V}$ has properties (1) and (2).

\step{4}{descent} The map $S^{\rm W}\to S$ is faithfully flat and
quasi-compact, hence a covering for the topology in which the
ULA property descends
\cite[Tag~0ETC]{StacksProject}, \cite[Proposition~3.7]{HS}; the
same holds after base change to the charts of the product of the
stack of bundles and the space of legs. So
$\mathcal S^{\rm val}_{I,V}$ has property (2) over $S$. Property
(1) over $S$ follows from step~3: each of the three algebraically
closed fields over which $S$ has a fibre embeds in the
corresponding field for $S^{\rm W}$, and groups of morphisms
between constructible complexes do not change under an extension
of algebraically closed fields.
The isomorphisms which express
the invariance of the kernel under changes of coordinates and
under the action of jets are maps between complexes which are ULA
over $S$, and are given on the generic fibre; so they extend
uniquely, and the extensions agree on overlaps of charts. This
defines $\mathcal S^{\rm val}_{I,V}$ on the Hecke correspondence
of the curve $\mathcal C_S$.

\step{5}{compatibilities} A map of representations induces a map
of kernels on the generic fibre, which extends uniquely. For
convolution the complex to be pushed forward is ULA, by the
argument with local products of
Lemma~\ref{fl:lem:hecke-legwise-ula}, and the proper direct image
from the bounded convolution space preserves this property. For a
diagonal, the restriction of the kernel is again ULA over $S$ by
(2); so the fusion isomorphism of the generic fibre extends
uniquely, and by the comparison of step~3 the extension is the
fusion isomorphism on the special fibre. Associativity and the
other identities between these isomorphisms hold because they
hold on the generic fibre. Finally, the construction involves
only the formal neighbourhoods of the legs, which lie in the
smooth locus; the nodes of the fibres play no role.
\end{proof}

\begin{lemma}[Hecke transforms are ULA over the legs]
\label{fl:lem:hecke-legwise-ula}
Let $\mathcal K$ be a locally ULA complex on $\mathcal B$, $I$ a
finite set and $V$ a representation of $\widehat G^I$. The Hecke
transform of $\mathcal K$ with kernel $\mathcal S^{\rm val}_{I,V}$
is locally ULA over $(U^\circ)^I_S$, the diagonals included. The
same holds for a composite of such transforms.
\end{lemma}

\begin{proof}
Let $a$ be large enough for the bound defined by $V$. We
trivialize the bundles along the divisor $a\sum_i\Gamma_i$, where
$\Gamma_i$ is the graph of the $i$-th leg. In an \'etale
coordinate $t$ this divisor has the equation
$\prod_{i\in I}(t-t_i)^a=0$, which is monic of degree $a|I|$; so
the divisor is finite and locally free over the space of legs,
also where legs coincide. By the infinitesimal lifting criterion, the Weil restriction of
$G$ along it is smooth. Let $T=(U^\circ)^I_S$. When a
trivialization at the first end is added to the data, we get
a smooth covering $B'\to\mathcal B\times_ST$, over which the
bounded Hecke correspondence becomes the product
$B'\times_T\operatorname{Gr}_{I,\le V}$ with the local model
$\operatorname{Gr}_{I,\le V}$ of the Grassmannian over $T$. On
this product the complex to be pushed forward is the exterior
product over $T$ of the pullback of $\mathcal K$ to $B'$ and of
the Satake complex of the local model. The first factor is ULA
over $T$, by base change and smooth pullback; the second is, by
Lemma~\ref{fl:lem:relative-satake-kernels}(2); and an exterior
product over the common base of two ULA complexes is ULA
\cite[Corollary~4.2(iv) and Corollary~3.10]{HS}. The property
descends along the smooth covering and is preserved by the proper
direct image. For a composite of several transforms, give the
legs of the successive transforms different labels, apply the
statement for one transform repeatedly, each time over the space
of the legs already introduced, and finally pull back along the
map which identifies the labels that are to coincide. On a
diagonal,
the fusion isomorphism of
Lemma~\ref{fl:lem:relative-satake-kernels}(3) and proper base
change identify the restriction of the transform with the
transform for the representation obtained by fusion.
\end{proof}

The next statement concerns Hecke families in the sense of
Definition~\ref{rw:def:hecke-family}, on $\mathcal B_\eta$ and
with legs in $U^\circ_\eta$. It applies in the same way when the
family is indexed by a small stable category, as the family
$P\mapsto\mathcal F_P$ of
Lemma~\ref{fl:lem:coarse-covariance} is, and not only by the
representations of $Z$: the proof uses only that the Hecke
identities are isomorphisms between complexes which are ULA.

\begin{proposition}[Extension of a Hecke family]
\label{fl:prop:hecke-equivariant-extension}
Consider a Hecke family on $\mathcal B_\eta$ whose members are
bounded and constructible on every quasi-compact open substack,
and suppose that the local systems in its Hecke identities extend
to local systems on powers of $U^\circ$. Then the family has a
unique locally ULA extension to $\mathcal B$, and the Hecke
identities extend to it. If the family carries an action of a
semilinear automorphism, or of a smooth group, compatible with
the Hecke identities, the action extends as well.
\end{proposition}

\begin{proof}
The members of the family extend by
Lemma~\ref{fl:lem:ula-stack-descent}. In a Hecke identity the
source is a Hecke transform of an extended member, which is ULA
by Lemma~\ref{fl:lem:hecke-legwise-ula}, and the target is the
exterior product of an extended member with a local system, which
is ULA. Both restrict on the generic fibre to the two sides of
the given identity, by
Lemma~\ref{fl:lem:relative-satake-kernels}(1). So the isomorphism
and its inverse extend uniquely, and every identity between such
isomorphisms holds for the extensions.

The same reasoning applies to all the data at once. A family
with its Hecke identities and their compatibilities is an object
of a limit of categories, indexed by the finite sets of legs, the
maps of representations and the diagrams of convolution and
fusion; an action of a group, or of an automorphism, adds the
terms of its nerve to the index. Restriction to the generic fibre
is an equivalence on each term of the limit, hence on the limit.
Each finite part of these data is defined over a finite extension
of $\mathbb Q_\ell$, so the passage to coefficients in $k$ causes
no difficulty. The extension is functorial in the member of the
family, so it applies to an exact functor on a small stable
category.
\end{proof}

\begin{theorem}[The family over the valuation base]
\label{fl:thm:compact-automorphic-family}
The family $P\mapsto\mathcal F_P$ of
Section~\ref{fl:sec:punctured-spectral-fibre} extends uniquely to
a family of locally ULA complexes on
$\operatorname{Bun}_G(\mathcal C_S)$, bounded and constructible on
every quasi-compact open substack, together with the maps induced
by maps in $\operatorname{Perf}(\mathfrak F_\rho)$, the Hecke
identities for all finite sets of legs in $U^\circ$, and the
action of Frobenius.
\end{theorem}

\begin{proof}
We apply Proposition~\ref{fl:prop:hecke-equivariant-extension} to
the functor $P\mapsto\mathcal F_P$ on
$\operatorname{Perf}(\mathfrak F_\rho)$. Its Hecke identities
\eqref{fl:eq:puncture-family-hecke} involve the local systems
$L_a=\operatorname{Hom}_{Z_\rho}(W_a,\mathcal E(V)_\eta)$, and we
have to show that they extend to local systems on $U^\circ$. The
local system $\mathcal E(V)$ is defined on $U$ by
Lemma~\ref{fl:lem:composed-parameter-gluing}. The action of
$Z_\rho$ on $\mathcal E(V)_\eta$ has its matrix coefficients in a
subcoalgebra $C_V\subset\mathcal O(Z_\rho)$ of finite dimension,
and is given by a coaction
\[
 \mathcal E(V)_\eta\longrightarrow
 \mathcal E(V)_\eta\otimes C_V.
\]
Both sides are restrictions of local systems on $U$, which are
ULA. So the coaction extends, with the identities which express
that it is counital and coassociative and its compatibility with
maps of representations. Since $Z_\rho$ is reductive, the
extension of $L_a$ is then defined as a direct summand of a local
system, and is a local system. The action of Frobenius on the
family (Lemma~\ref{fl:lem:coarse-covariance}(2)) extends by the
proposition.
\end{proof}

\begin{remark}
The theorem gives a family on the special fibre, but it gives no
control over its trace function. Restriction from $S$ to the closed
fibre is not conservative, and nothing proved so far excludes that
the trace function of $\mathcal F_{P,s}$ vanishes identically.
Sections~\ref{fl:sec:noncharacteristic}
and~\ref{fl:sec:conservation} address this.
\end{remark}