\section{Geometric setup}
\label{rw:sec:geometric-route}

This section fixes the geometric objects which are used from
Section~\ref{fl:sec:punctured-spectral-fibre} on: the curves, the
base over which they vary, the Hecke correspondences, and the
notion of a Hecke family. It contains no proofs.

\subsection{Curves and specializations}

The curve of the closed fibre is the union \(C_0=X\cup_D C_2\) of
the original curve and of an \emph{auxiliary component} \(C_2\).
It is Gorenstein and geometrically reduced, but it is not nodal
when \(D\) is not reduced. Its construction
is expressed by the pushout square
\begin{equation}\label{rw:eq:curve-pushout-guide}
\begin{tikzcd}[column sep=large,row sep=large]
 D \arrow[r,hook] \arrow[d,hook]
   & C_2 \arrow[d,hook] \\
 X \arrow[r,hook] & C_0 .
\end{tikzcd}
\end{equation}
The original parameter and the Hecke legs remain on \(X\).
The component \(C_2\) carries the extra puncture that produces the
exterior algebra of
Section~\ref{fl:sec:punctured-spectral-fibre}. On the intermediate fibre, in
characteristic zero, the two local systems have opposite inertia
along \(D\); this is the condition under which the glued local
system extends to the smoothing. The divisor \(D\) may have multiplicities; its scheme
structure records the level retained at the end of the construction.

Two specializations are involved: the smooth curve degenerates to
a nodal one, and the nodal curve in characteristic zero reduces to
characteristic \(p\). Section~\ref{fl:sec:valued-nodal-family} constructs
a rank-two valuation base \(S\) with geometric points
\(\eta,\xi,s\) in this order. The fibres have the following roles:
\begin{equation}\label{rw:eq:valuation-flag-guide}
\begin{tikzcd}[column sep=8.5em,row sep=large]
 \mathcal C_\eta
   \arrow[r,rightsquigarrow,"\text{the nodes form}"]
   \arrow[d]
 & \mathcal C_\xi
   \arrow[r,rightsquigarrow,"\text{reduction mod }p"]
   \arrow[d]
 & C_0\otimes\overline{\mathbb F}_q \arrow[d] \\
 \eta \arrow[r,rightsquigarrow] & \xi \arrow[r,rightsquigarrow] & s .
\end{tikzcd}
\end{equation}
The first fibre is smooth and proper in characteristic zero; the
middle fibre has the two components of
\eqref{rw:eq:curve-pushout-guide} in characteristic zero; the last
is the curve \(C_0\) in characteristic \(p\), which is
geometrically reduced and Gorenstein. In general only the middle
fibre is nodal.
The wavy arrows indicate specialization. All comparisons are made along this ordered
flag.

For a finite set of legs \(I\) in the smooth open \(U\), the
Hecke functor is defined by the correspondence
\begin{equation}\label{rw:eq:hecke-correspondence-guide}
\begin{tikzcd}[column sep=small,row sep=large]
 \Bun_G(\mathcal C)\times U^I \arrow[dr]
 & \operatorname{Hk}_{G,I}\arrow[l,"h_0"']
       \arrow[r,"h_1"]\arrow[d]
 & \Bun_G(\mathcal C)\times U^I\arrow[dl] \\
 & U^I . &
\end{tikzcd}
\end{equation}
After bounding the modifications by a Satake representation, the
appropriate projection is proper.

\subsection{The object which is transported: Hecke families}

One kind of object is constructed in characteristic zero
(Section~\ref{fl:sec:punctured-spectral-fibre}), extended over the
base $S$ (Section~\ref{fl:sec:hecke-extension}), restricted to the
curve $X$ (Section~\ref{fl:sec:exceptional-assembly}), and finally
used to produce a function
(Section~\ref{fl:sec:arithmetic-trace}). We define it here once,
in the generality needed for all four steps.

Let $\mathcal Y$ be a stack of $G$-bundles on a curve, with or
without level structure, over a field or over $S$, and let $U'$ be
an open subset of the smooth locus of the curve, disjoint from the
level. For a finite set $I$ and a representation $V$ of
$\widehat G^I$, modifications of bundles at $I$ points of $U'$,
with the Satake kernel of $V$, define the Hecke functor
$\mathsf H_{I,V}$ from complexes on $\mathcal Y$ to complexes on
$\mathcal Y\times U'^I$; when $I$ has one element we write
$\mathsf H_V$. Let $\rho$ be a $\widehat G$-local system
on $U'$, and $Z$ a reductive group over $k$ which acts on $\rho$ by
automorphisms. For $V$ as above let $V_\rho$ be the local system
on $U'^I$ attached to $\rho$ and $V$; the group $Z$ acts on it.
For a local system $\mathcal W$ with an action of $Z$ on a scheme
$B$, and a functor $W\mapsto\mathcal F_W$ on representations of
$Z$, put
\[
 \mathcal F(\mathcal W)=\bigoplus_W\mathcal F_W\boxtimes
 \underline{\operatorname{Hom}}_Z(W,\mathcal W)
 \qquad\text{on }\mathcal Y\times B,
\]
the sum being over the irreducible representations of $Z$.

\begin{definition}[Hecke families]
\label{rw:def:hecke-family}
\begin{enumerate}
\item A \emph{Hecke family} on $\mathcal Y$, with legs in $U'$ and
eigenvalue $\rho$, is a $k$-linear functor $W\mapsto\mathcal F_W$
from the finite-dimensional representations of $Z$ to complexes
on $\mathcal Y$, together with isomorphisms
\[
 \mathsf H_{I,V}(\mathcal F_W)\simeq\mathcal F(W\otimes V_\rho)
 \qquad\text{on }\mathcal Y\times U'^I
\]
for all $I$, $V$ and $W$. They are required to be natural in $W$
and in $V$, to be the identity for the trivial representation
$V$, and to be compatible with the restriction to the diagonals
of $U'^I$ and with convolution, that is, with the composition of
Hecke functors.
\item Let $\phi$ be an automorphism of the curve which preserves
$U'$ and the level structure, so that it acts on $\mathcal Y$,
let $\alpha$ be an automorphism of $Z$, and let
$\theta:\alpha^*\phi^*V_\rho\simeq V_\rho$ be isomorphisms
commuting with $Z$ and compatible with the operations on $V$. A
\emph{structure of compatibility with $\phi$} on a Hecke family
is a system of isomorphisms
\[
 c_W:\phi^*\mathcal F_W\simeq\mathcal F_{\alpha^*W},
\]
natural in $W$, such that for all $I$, $V$ and $W$ the two
isomorphisms from $\phi^*\mathsf H_{I,V}(\mathcal F_W)$ to
$\mathcal F(\alpha^*W\otimes V_\rho)$ are equal: the Hecke
isomorphism for $W$ followed by $c$ and $\theta$, and
$\mathsf H_{I,V}(c_W)$ followed by the Hecke isomorphism for
$\alpha^*W$.
\item When $\mathcal Y$ is defined over $\mathbb F_q$, the
pullbacks by the Frobenius endomorphisms of $\mathcal Y$ and of
$U'^I$ are equivalences of the \'etale categories. A \emph{Hecke
family with Frobenius} is a Hecke family together with a
comparison of the Hecke functors under these pullbacks, given by
the Frobenius structure of the Satake kernels as in (F3) of
Setup~\ref{rw:setup:trace}.
\end{enumerate}
\end{definition}

In this paper the word \emph{coefficient}, used alone, refers to
the argument of such a family: a representation $W$ of $Z$, a
local system $\mathcal W$ with an action of $Z$, or more generally
a perfect complex $P$ on the stack $\mathfrak F_\rho$ of
Section~\ref{fl:sec:punctured-spectral-fibre}. Whittaker
coefficients, Fourier coefficients and fields of coefficients are
always named in full.

The conditions (1) and (2), for
$\mathcal Y=\Bun_{G,D}(X)$ and the Frobenius endomorphism, are
written out with all normalizations as (F1)--(F3) in
Setup~\ref{rw:setup:trace}. In these terms the steps of the
construction are the following.
\begin{itemize}
\item Lemma~\ref{fl:lem:coarse-covariance}: the functor
$W\mapsto\mathcal F_{P_W}$ is a Hecke family on the stack of
bundles of the smooth curve in characteristic zero, with legs
outside $D_{\rm aux}$, and it has a structure of compatibility
with the semilinear Frobenius automorphism. (The lemma is stated
for the larger family $P\mapsto\mathcal F_P$ indexed by perfect
complexes; this is needed in
Section~\ref{fl:sec:conservation}.)
\item Theorem~\ref{fl:thm:compact-automorphic-family}: the family,
with this structure, extends over $S$.
\item Proposition~\ref{fl:prop:passive-bundle-restriction}: the
restriction of the fibre over $s$ to $\Bun_{G,D}(X)$ is a Hecke
family with Frobenius.
\item Theorem~\ref{fl:thm:arithmetic-family-trace}: the trace
function of a Hecke family with Frobenius satisfies the excursion
equations of its eigenvalue.
\end{itemize}