\section{Returning to the original curve}
\label{fl:sec:exceptional-assembly}

At this point we have a family of Weil complexes on
$\Bun_G(C_0)$, for the curve $C_0=X\cup_DC_2$ over $\mathbb F_q$,
and a $G$-bundle $\mathcal G_0$ on $C_0$, defined over
$\mathbb F_q$, at which the member with coefficient $P_{\rm reg}$
has nonzero trace
(Proposition~\ref{fl:prop:composite-trace-nonzero}). We want a
function on bundles on $X$. A bundle on $C_0$ is a pair of bundles
on $X$ and on $C_2$ with an identification along $D$, so we fix the
bundle on $C_2$ and let the one on $X$ vary. The bundle on $C_2$
must be the restriction of $\mathcal G_0$: then $\mathcal G_0$
itself lies in the resulting family, and the function obtained is
not zero.
Theorem~\ref{fl:thm:weyl-prime-realization} is then assembled from
the results of Sections~\ref{fl:sec:punctured-spectral-fibre}--\ref{fl:sec:arithmetic-trace}.

\subsection{Fixing the bundle on the auxiliary component}

Recall from Definition~\ref{rw:def:hecke-family} the notion of a
Hecke family with Frobenius; for $\mathcal Y=\Bun_{G,D}(X)$ its
conditions are (F1)--(F3) of Setup~\ref{rw:setup:trace}.

\begin{proposition}[Restriction with a trivialization on the boundary]
\label{fl:prop:passive-bundle-restriction}
Let $C_0=X\cup_D C_2$ be the special fibre of the curve of
Proposition~\ref{fl:prop:geometrization-theta}; the divisor $D$,
identified with $D_2\subset C_2$, may be nonreduced. Let
$\Bun_{G,D}(X)$ be the stack of $G$-bundles on $X$ with a
trivialization along $D$. Suppose given a Hecke family with
Frobenius on $\Bun_G(C_0)$ whose legs lie in an open subset
containing $X-\operatorname{supp}D$, and a point
$\mathcal G_0\in\Bun_G(C_0)(\mathbb F_q)$ at which one member of
the family has nonzero trace. Then there is a morphism
\[
 \mathsf{gl}:\Bun_{G,D}(X)\longrightarrow\Bun_G(C_0)
\]
over $\mathbb F_q$ such that the pullback of the family by
$\mathsf{gl}$ is a family with the structure (F1)--(F3) for legs
in $X-\operatorname{supp}D$, and the pullback of that member has
nonzero trace at a point of $\Bun_{G,D}(X)(\mathbb F_q)$.
\end{proposition}

\begin{proof}
\step{1}{the morphism} Let $\mathcal G_2$ be the restriction of
$\mathcal G_0$ to $C_2$. Its restriction to $D_2$ is trivial:
$D_2$ is a finite scheme over $\mathbb F_q$, a $G$-torsor over a
finite field is trivial by the theorem of Lang, and a section
over the reduced subscheme lifts to $D_2$ because the torsor is
smooth. Choose a trivialization of $\mathcal G_2$ along $D_2$. A
bundle on $X$ with a trivialization along $D$ can then be glued
to $\mathcal G_2$ along $D\simeq D_2$, and this defines
$\mathsf{gl}$.

Since $D$ may be nonreduced, and its scheme structure determines
the level of the function we are going to construct, we give the
details of the gluing. Locally the ring of $C_0$ is the fibre
product of the rings of $X$ and of $C_2$ over the ring of $D$, by
Proposition~\ref{fl:prop:geometrization-theta}, and this remains
true after base change to any test scheme. Finite projective
modules over such a fibre product are equivalent to pairs of
finite projective modules over the two rings with an isomorphism
over the ring of $D$, compatibly with tensor products
\cite[Tag~0D2J]{StacksProject}. To pass to $G$-bundles choose a
faithful representation $G\to\operatorname{GL}_n$; the quotient
$\operatorname{GL}_n/G$ is affine, because $G$ is geometrically
reductive. A $G$-bundle is a vector
bundle of rank $n$ with a section of the associated bundle with
fibre $\operatorname{GL}_n/G$. The vector bundles glue by the
equivalence just quoted, and a section of a scheme affine over
$C_0$ is the same as a pair of compatible sections over the two
components. The same applies to isomorphisms. Hence, if
$\mathcal B_1$ and $\mathcal B_2$ are the stacks of bundles on
$X$ and on $C_2$ with a trivialization along $D$,
\[
 \Bun_G(C_0)\simeq
 [\mathcal B_1\times\mathcal B_2/G_D],\qquad
 G_D=\operatorname{Res}_{D/\mathbb F_q}(G\times D),
\]
and the choice of $\mathcal G_2$ with its trivialization gives
\begin{equation}\label{rw:eq:fixed-auxiliary-fibre}
 \Bun_{G,D}(X)\simeq
 \Bun_G(C_0)\times_{\Bun_G(C_2)}\{\mathcal G_2\}.
\end{equation}

\step{2}{the trace} The bundle $\mathcal G_0$ defines a point
$\mathcal G_1^{\rm fr}$ of the fibre
\eqref{rw:eq:fixed-auxiliary-fibre} over $\mathbb F_q$, which
$\mathsf{gl}$ maps to $\mathcal G_0$. For the member $\mathcal K$
of the family with nonzero trace at $\mathcal G_0$, the stalk of
$\mathsf{gl}^*\mathcal K$ at $\mathcal G_1^{\rm fr}$ is the stalk
of $\mathcal K$ at $\mathcal G_0$, with the same action of
Frobenius. So
\[
 \operatorname{Str}\bigl(\operatorname{Frob}_q,
       (\mathsf{gl}^*\mathcal K)_{\mathcal G_1^{\rm fr}}\bigr)
 =\operatorname{Str}\bigl(\operatorname{Frob}_q,
       \mathcal K_{\mathcal G_0}\bigr)
 \ne0.
\]

\step{3}{the Hecke structure} A modification of a bundle on
$C_0$ at points of $X-\operatorname{supp}D$ does not change its
restriction to $C_2$. So the Hecke correspondence of
$\Bun_{G,D}(X)$ with legs in $X-\operatorname{supp}D$ is the
base change by $\mathsf{gl}$ of the Hecke correspondence of
$\Bun_G(C_0)$, for the projection to the source and for the
projection to the target. The formal neighbourhoods of the legs
are the same for $X$ and for $C_0$, so the two correspondences
have the same Satake kernels; this includes the loci where legs
coincide, where one uses the sum of the graphs of the legs. Let
$h_0$ and $h_1$ be the projections of a bounded part of the
correspondence of $C_0$ and $A_H$ the kernel, and let
$h_0^{X}$, $h_1^{X}$ and $A_H^{X}$ be the corresponding objects
for $X$. Since $h_1$ is proper on a bounded part, proper base
change gives
\[
 (\mathsf{gl}\times\mathrm{id}_{U^I})^*h_{1,!}
     (h_0^*\mathcal K\otimes A_H)
 \simeq h_{1,!}^{X}
 \bigl((h_0^{X})^*\mathsf{gl}^*\mathcal K\otimes A_H^{X}\bigr).
\]
Base change isomorphisms are compatible with maps of
representations, with convolution and with restriction to
diagonals. So the isomorphisms (F1) of the family on
$\Bun_G(C_0)$ give isomorphisms (F1) for its pullback, with the
same compatibilities. Everything is defined over $\mathbb F_q$,
so the isomorphisms (F2) pull back as well and (F3) is preserved.
\end{proof}

\subsection{Geometric realization when $p$ does not divide the order of the Weyl group}

\begin{theorem}[Geometric realization]
\label{fl:thm:weyl-prime-realization}
Let $G/F$ be split, simply connected and semisimple, and suppose
that $p\nmid|W_G|$. For every semisimple elliptic parameter
$\sigma:\Gamma_F\to\widehat G(k)$ there are an open compact
subgroup $K\subset G(\mathbb A_F)$ and a nonzero cusp form
$\varphi_\sigma$ on $G(F)\backslash G(\mathbb A_F)/K$ such that
\[
 S^{\rm fun}_{I,f,\boldsymbol\gamma}\varphi_\sigma
 =f\bigl((\sigma(\gamma_i))_{i\in I}\bigr)\varphi_\sigma
\]
for every excursion datum $(I,f,\boldsymbol\gamma)$ with
$\boldsymbol\gamma\in\Gamma_F^I$. In particular the excursion
summand of $\sigma$ in the space of cusp forms of level $K$ is
not zero.
\end{theorem}

The proof occupies the rest of this subsection. It assembles the
results of the preceding sections in three stages, which produce a
family of sheaves on the special fibre and a rational point at
which one of them has nonzero trace. The step which is new here is
Proposition~\ref{rw:prop:restricted-family}: the family restricted
to bundles on $X$ has the structure required by
Theorem~\ref{fl:thm:arithmetic-family-trace}.

If $G$ is trivial the constant function has the required
properties. Otherwise the Weyl group contains a reflection, so
$p\ne2$. We apply the construction to the parameter
$\sigma^\dagger=\vartheta_G\sigma$, which is again continuous,
finitely ramified, semisimple and elliptic. The reason is that
the function obtained at the end has the parameter
$\vartheta_G\sigma^\dagger$
(Corollary~\ref{fl:cor:arithmetic-family-eigenfunction}), which
is conjugate to $\sigma$.

\medskip\noindent\emph{The smooth curve and its local system.}\ By
Lemmas~\xref{I}{fl:lem:arithmetic-finite-coefficients}
and~\xref{I}{fl:lem:arithmetic-monodromy-quotient} the parameter
$\sigma^\dagger$ has values in a finite extension of
$\mathbb Q_\ell$, its arithmetic and geometric monodromy groups
$\mathsf{Mon}_{\sigma^\dagger}^{\rm ar}$ and
$\mathsf{Mon}_{\sigma^\dagger}^{\rm geom}$ are reductive, and the
quotient of the first by the second is finite and cyclic.
Theorem~\ref{fl:thm:marked-lifting} lifts $\sigma^\dagger$ to
characteristic zero.
Proposition~\ref{fl:prop:geometrization-theta} gives a proper
family $\mathcal C$ of curves with a theta characteristic, whose
special fibre is $C_0=X\cup_DC_2$ and whose generic fibre is
smooth of characteristic zero, and
Proposition~\ref{fl:prop:geometrized-parameter} gives a local
system $\rho$ on the geometric generic fibre such that
\[
 \mathsf{Mon}_{\sigma^\dagger}^{\rm geom}\subset \mathsf{Mon}_{\rho}\subset \mathsf{Mon}_{\sigma^\dagger}^{\rm ar},\qquad
 \overline{\langle \mathsf{Mon}_{\rho},g\rangle}
     =\mathsf{Mon}_{\sigma^\dagger}^{\rm ar},
\]
where $g$ is the image under $\sigma^\dagger$ of the chosen
arithmetic Frobenius element. These data satisfy the conditions
of Setup~\ref{rw:setup:auxiliary-curve}: $\mathsf{Mon}_\rho$ is
reductive, the quotient in (2) is finite, and the centralizer in
(3) is the centralizer of
$\mathsf{Mon}_{\sigma^\dagger}^{\rm ar}$, which is finite because
$\sigma^\dagger$ is elliptic and $G$ is semisimple. By
Lemma~\ref{fl:lem:geometric-centralizer-torus} the group
\[
 Z_\rho=C_{\widehat G}(\mathsf{Mon}_{\rho}),\qquad
 \alpha=\operatorname{Ad}(g)|_{Z_\rho},
\]
is reductive, its identity component is a torus, and $1-\alpha$
is an isogeny of this torus. Since
$\mathsf{Mon}_{\sigma^\dagger}^{\rm geom}\subset\mathsf{Mon}_\rho$,
the group $Z_\rho$ centralizes the image under $\sigma^\dagger$
of the geometric fundamental group. It will be the group $Z$ of
Setup~\ref{rw:setup:trace}.

\medskip\noindent\emph{The family and the trace of its
coefficient.}\ Choose the
divisor $D_{\rm aux}$ on the second component as in
Lemma~\ref{fl:lem:closed-return-puncture}. For a finite-dimensional
representation $W$ of $Z_\rho$ let
\[
 P_W=\mathcal O_{\mathfrak F_\rho}\otimes W,
 \qquad P_{\rm reg}=P_{k[\pi_0(Z_\rho)]} .
\]
Section~\ref{fl:sec:punctured-spectral-fibre} attaches to these
objects the sheaves $\mathcal F_{P_W}$ on the stack of bundles of
the generic fibre, with their Hecke identities and their
compatibility with Frobenius
(Lemma~\ref{fl:lem:coarse-covariance}). By
Proposition~\ref{fl:prop:puncture-exact-trace}, whose hypothesis
on weights holds by Lemma~\ref{fl:lem:closed-return-puncture},
Frobenius and its inverse have nonzero supertrace on
$\Gamma(\mathfrak F_\rho,P_{\rm reg})$.

\medskip\noindent\emph{A rational bundle with nonzero trace.}\ By
Theorem~\ref{fl:thm:compact-automorphic-family} the whole family
extends over the base $S$. The functor
$P\mapsto\mathcal F_P$ is extended, and not only the object
$\mathcal F_{P_{\rm reg}}$; so the comparison with a microstalk is
natural in $P$, and by
Lemma~\ref{fl:lem:scalar-coefficient-covariance} the two actions
of Frobenius differ by one scalar. These arguments are carried
out in Sections~\ref{fl:sec:noncharacteristic}
and~\ref{fl:sec:conservation}, and their conclusion is
Proposition~\ref{fl:prop:composite-trace-nonzero}: there is a
bundle $\mathcal G_0\in\Bun_G(C_0)(\mathbb F_q)$ at which
$\mathcal F_{P_{\rm reg},s}$ has nonzero trace of geometric
Frobenius. (Geometric Frobenius is the inverse of the Frobenius
automorphism used in Lemma~\ref{fl:lem:coarse-covariance}, see
Remark~\ref{rw:rem:trace-conventions}; this is why both Frobenius
and its inverse were considered in the preceding paragraph.)

\begin{proposition}[The family on bundles on $X$]
\label{rw:prop:restricted-family}
With the data just constructed, let
\[
 \mathcal F_W^{X}=
 \mathsf{gl}^*\mathcal F_{P_W,s}
 \qquad\text{on }\Bun_{G,D}(X),
\]
with $\mathsf{gl}$ as in
Proposition~\ref{fl:prop:passive-bundle-restriction} for the
bundle $\mathcal G_0$. Then $W\mapsto\mathcal F_W^X$ is a family
with the
structure (F1)--(F3) of Setup~\ref{rw:setup:trace}, for the group
$Z=Z_\rho$ with its automorphism $\alpha$ and for the local system
on $X-\operatorname{supp}D$ which
is the restriction of the local system $\mathcal E$ on the base
$S$, namely $\sigma^\dagger$
(Proposition~\ref{fl:prop:geometrized-parameter}). The Frobenius
structure on the Satake kernels which enters (F3) is that of
\cite[(3.5) and Propositions~4.10--4.14]{VLa18}.
\end{proposition}
\begin{proof}
By
Proposition~\ref{fl:prop:passive-bundle-restriction} it suffices
to see that the functor $W\mapsto\mathcal F_{P_W,s}$ on
$\Bun_G(C_0)$ has this structure, with legs in $U^\circ_s$, and
this is obtained from Lemma~\ref{fl:lem:coarse-covariance} as
follows.

\step{1}{the Hecke identities \textup{(F1)}} For $P=P_W$ and a decomposition
$V_\rho=\bigoplus_aW_a\boxtimes L_a$ into isotypic parts, with
$W_a$ irreducible and
$L_a=\underline{\operatorname{Hom}}_{Z_\rho}(W_a,V_\rho)$, the
Hecke identity \eqref{fl:eq:puncture-family-hecke} reads
\[
 \mathsf H_V(\mathcal F_{P_W})
  \simeq\bigoplus_a\mathcal F_{P_{W\otimes W_a}}\boxtimes L_a .
\]
The functor $W\mapsto\mathcal F_{P_W}$ is additive and $Z_\rho$
is reductive, so
$\mathcal F_{P_{W\otimes W_a}}=\bigoplus_{W'}\mathcal F_{P_{W'}}
\otimes\operatorname{Hom}_{Z_\rho}(W',W\otimes W_a)$, the sum
being over the irreducible representations. Hence the right-hand
side is
\[
 \bigoplus_{W'}\mathcal F_{P_{W'}}\boxtimes
   \underline{\operatorname{Hom}}_{Z_\rho}(W',W\otimes V_\rho)
 =\mathcal F(W\otimes V_\rho)
\]
in the notation of Setup~\ref{rw:setup:trace}. The same
computation applies to every finite set of legs. The
isomorphisms are natural in $W$ because the Hecke identities are
natural in $P$, and their compatibility with maps of
representations $V$, with restriction to diagonals and with
convolution is the compatibility asserted in
Lemma~\ref{fl:lem:coarse-covariance}(1). By
Theorem~\ref{fl:thm:compact-automorphic-family} these
isomorphisms, with their compatibilities, extend over $S$, for
the Hecke functors defined by the kernels
$\mathcal S^{\rm val}_{I,V}$. The restriction of these Hecke
functors to the fibre over $s$ is the Hecke functor of that
fibre, by proper base change and
Lemma~\ref{fl:lem:relative-satake-kernels}(1) and~(3). So the
restriction to $s$ has the isomorphisms (F1).

\step{2}{the Frobenius covariance \textup{(F2)}} Let $u_C$ be the arithmetic Frobenius of the curve
and $c:u_C^*\rho\simeq\rho$ the descent isomorphism, with matrix
$g$ in the marked fibre. Pullback of local systems by $u_C$ acts
on $BZ_\rho$ through
$z\mapsto c\,u_C^*(z)\,c^{-1}=\alpha(z)$. The spectral map of
Lemma~\ref{fl:lem:coarse-covariance} is the transport, the inverse
of this pullback, so
\[
 u_{\mathfrak F}^*P_W=P_{(\alpha^{-1})^*W},
 \qquad
 u_{\rm aut}^*\mathcal F_{P_W}
   \simeq\mathcal F_{P_{(\alpha^{-1})^*W}}.
\]
This covariance extends over $S$. On the special fibre
$u_{\rm aut}^*$ is inverse to $\operatorname{Fr}^*$, and the
inverse of the covariance is
$\operatorname{Fr}^*\mathcal F_W\simeq\mathcal F_{\alpha^*W}$.

\step{3}{the compatibility \textup{(F3)}} The inverse covariance acts on the legs by
$u_C^{-1}$, and the corresponding isomorphism of local systems is
$\theta=(u_C^{-1})^*(c^{-1})$, with matrix $g^{-1}$ in the marked
fibre. It is an equivariant map
$\alpha^*\operatorname{Fr}^*V_\rho\to V_\rho$ under $Z_\rho$. The
compatibility of the covariance with the Hecke identities, in
Lemma~\ref{fl:lem:coarse-covariance}(2), read in the inverse
direction, is (F3). It is preserved by the extension over $S$ and
by the restriction to $s$, like all the maps which enter it.

\step{4}{the Frobenius structure of the Satake kernels} The Frobenius
structure obtained in this way must be the one used in (F3). The
semilinear pullback of the Satake kernels on the generic fibre,
and the comparison maps, are compatible with convolution and with
fusion, and so are their extensions, by
Lemma~\ref{fl:lem:relative-satake-kernels}(3) and the uniqueness
of ULA extensions. So on the special fibre we have two Frobenius
structures on the Satake kernels compatible with the tensor
structure: this one, and that of
\cite[(3.5) and Propositions~4.10--4.14]{VLa18}. The kernels are
taken with their Tate twists. On the open stratum of the Schubert
variety of a coweight $\lambda$ the kernel is the constant sheaf
with the shift and twist $[d_\lambda](d_\lambda/2)$, where
$d_\lambda=\langle2\rho_G,\lambda\rangle$ is even because $G$ is
simply connected. Both structures restrict there to the canonical
structure of the constant sheaf and of the Tate twist, for which
the semilinear comparison was the canonical one. An isomorphism
between intersection complexes is determined by its restriction to
the open stratum, so the two structures agree on the kernel of
every irreducible representation, for one leg, and by naturality
on all the kernels. The tensor structure gives the same
conclusion: for one leg the ratio of the two structures is an
automorphism of the Satake functor compatible with tensor
products. By the Satake equivalence it is given by an element of
$\widehat G$ which commutes with $\widehat G$, that is, by an
element of the centre of $\widehat G$, and this centre is trivial
because $G$ is semisimple and simply connected. For several legs
the two structures agree by factorization and fusion.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{fl:thm:weyl-prime-realization}]
\step{1}{the parameter} Let $\varphi_\sigma$ be the trace function
of the member of the family of
Proposition~\ref{rw:prop:restricted-family} for the regular
representation of $\pi_0(Z_\rho)$. The sheaf
$\mathcal F_{P_{\rm reg},s}$ has nonzero trace at $\mathcal G_0$, so
$\varphi_\sigma$ is not zero
(Proposition~\ref{fl:prop:passive-bundle-restriction}). On this
representation the arithmetic closing map sends $e_\gamma$ to
$e_{\alpha(\gamma)}$, and its inverse is the map $\lambda_R$ of
Setup~\ref{rw:setup:trace}. By
Theorem~\ref{fl:thm:arithmetic-family-trace} the linear form
defined by $\varphi_\sigma$ on functions with compact support is
an eigenvector of the excursion operators for tuples in the Weil
group, with the eigenvalues of $\sigma^\dagger$. By
Corollary~\ref{fl:cor:arithmetic-family-eigenfunction},
$\varphi_\sigma$ is a cusp form with compact support and an
eigenfunction of the excursion operators for all Galois tuples,
with parameter $\vartheta_G\sigma^\dagger$, that is, $\sigma$ up
to conjugation. (See Remark~\ref{rw:rem:convention-robust} for
the dependence of this step on conventions.)

\step{2}{the level} Write $D=\sum_vn_vv$ and
\[
 K_D=\prod_{v\notin\operatorname{supp}D}G(\mathcal O_v)
 \times\prod_{v\in\operatorname{supp}D}
 \ker\!\left(G(\mathcal O_v)\longrightarrow
             G(\mathcal O_v/\mathfrak m_v^{n_v})\right).
\]
Since $G$ is smooth, $G(\mathcal O_v)$ maps onto
$G(\mathcal O_v/\mathfrak m_v^{n_v})$, so a trivialization along
$D$ is the same as a level structure for $K_D$, also when $D$ is
not reduced. For split simply connected $G$ the points of
$\Bun_{G,D}(X)$ over $\mathbb F_q$ are
$G(F)\backslash G(\mathbb A_F)/K_D$
\cite[Remark~8.21]{VLa18}, and $\varphi_\sigma$ is a function on
this set. The bundle $\mathcal G_0$, its restriction to $C_2$
and the trivialization are defined over $\mathbb F_q$, and the
values of $\varphi_\sigma$ lie in a finite extension of
$\mathbb Q_\ell$. So the conclusion holds for the given field $F$
and the given prime $\ell$.
\end{proof}

\begin{remark}[Independence of conventions]
\label{rw:rem:convention-robust}
Which of $\sigma$ and $\vartheta_G\sigma$ is the parameter of
$\varphi_\sigma$ depends on conventions of several of the works
we use: whether the Hecke functor attached to $V$ in
\cite{AGKRRV2022,GR2025} corresponds to the functor
$\mathsf H_{I,V}$ of \cite{VLa18} or to the one attached to the
dual representation, and the transposition in
Proposition~\xref{I}{fl:prop:cuspidal-excursion-character}. The two
normalizations of the Satake equivalence which occur, and the
effect of passing from one to the other, are described in
\cite[Section~\xnum{I}{rw:sec:satake-convention}]{HuI}. The
existence statement of
Theorem~\ref{fl:thm:weyl-prime-realization} does not depend on
them: by Lemma~\xref{I}{rw:lem:chevalley-dual-summand} the cuspidal
excursion summand of $\sigma$ at a given level is not zero if and
only if that of $\vartheta_G\sigma$ is not zero. For the same
reason it does not matter whether $\vartheta_G$ is an involution
or only has inner square.
\end{remark}

\begin{remark}
The condition $p\nmid|W_G|$ is used in two places. In
Theorem~\ref{fl:thm:marked-lifting} it makes the images of wild
inertia lie in tori, so that the parameter lifts to characteristic
zero with a divisor of ramification independent of the finite
quotient. In Lemma~\ref{fl:lem:fixed-maximal-detectors} it makes
the invariant form nondegenerate on all Levi subalgebras. There is
no hypothesis on the ramification of $\sigma$ or on the component
group of the centralizer of its geometric monodromy; these affect
the curves $C_2$ and $C$ and the level of $\varphi_\sigma$ only.
\end{remark}

\subsection{The example of Section~\ref{fl:sec:low-rank-example}, concluded}

For the parameter $\sigma:\Gamma_F\to S_3\subset
\operatorname{PGL}_2(k)$ over $F=\mathbb F_5(t)$ of
Sections~\ref{fl:sec:low-rank-example}
and~\ref{rw:sec:example-curve}, the
theorem gives the following. The group is
$G=\operatorname{SL}_2$, and $p=5$ does not divide $|W_G|=2$. The
genus of the smooth curve is $841$, and its special fibre is
$C_0=\mathbb P^1\cup_DC_2$, with $D$ a divisor of degree $58$ on
$\mathbb P^1$ which contains the points where $\sigma$ is
ramified. The group $Z_\rho$ has two elements, the perfect complex
$P_{\rm reg}$ is $\mathcal O\otimes k[\mu_2]$. On its global sections
the arithmetic supertrace is $(1-5)^3=-64$, and the geometric one
is $(1-5^{-1})^3=64/125$. Since these numbers are not zero,
Proposition~\ref{fl:prop:composite-trace-nonzero} provides a
bundle on $C_0$ over $\mathbb F_5$ at which the trace function of
$\mathcal F_{P_{\rm reg},s}$ does not vanish, and the theorem
provides a nonzero cusp form on
$\operatorname{SL}_2(F)\backslash\operatorname{SL}_2(\mathbb A_F)/K_D$
with parameter $\sigma$. What the construction does not give is a
formula: neither the bundle nor any value of the cusp form is
determined explicitly, and the level $K_D$ is far from minimal.
(For this group the result also follows from
\cite{HuIII}; the example is meant to show the
size of the objects.)