\section{A perfect complex whose trace cannot cancel}
\label{fl:sec:punctured-spectral-fibre}

As explained in Section~\ref{fl:sec:strategy}, the first step of
the geometric construction is to find a perfect complex whose Frobenius
trace cannot vanish by cancellation. It is constructed in this
section, in characteristic zero and before any sheaf is
specialized. Its cohomology is an exterior algebra, so the
alternating trace of Frobenius on it is a determinant, and purity
shows that no factor of this determinant is zero.

The section is organized as follows. After fixing the data
(Setup~\ref{rw:setup:auxiliary-curve}), we describe the stack on which the perfect
complexes live, a derived fibre
(Lemma~\ref{fl:lem:puncture-fibre-model}), define the corresponding
family of automorphic sheaves and compute its Whittaker coefficient
\eqref{fl:eq:puncture-vacuum}, show that the family is compatible
with Hecke functors and with Frobenius
(Lemma~\ref{fl:lem:coarse-covariance}), and finally compute the
supertrace (Proposition~\ref{fl:prop:puncture-exact-trace}).

\begin{setup}\label{rw:setup:auxiliary-curve}
Let $C$ be a smooth proper curve over an algebraically closed field
$K_{\rm geom}$ of characteristic zero, and let $\rho$ be a
semisimple $\widehat G$-local system on $C$ with coefficients in
$k=\overline{\mathbb Q}_\ell$. The curve and the local system carry
a semilinear automorphism lifting the arithmetic $q$-Frobenius;
let $g\in\widehat G(k)$ be its action on a marked fibre of $\rho$.
Let $\mathsf{Mon}_{\rho}$ be the Zariski closure of the geometric
monodromy. We assume that
\begin{enumerate}
\item $\mathsf{Mon}_{\rho}$ is reductive;
\item $\overline{\langle\mathsf{Mon}_{\rho},g\rangle}/\mathsf{Mon}_{\rho}$
is finite;
\item $C_{\widehat G}(\mathsf{Mon}_{\rho},g)$ is finite.
\end{enumerate}
\end{setup}

Such data are constructed from the given elliptic parameter in
Section~\ref{fl:sec:geometrization}. The curve lives over
$K_{\rm geom}$, and $k$ is the field of coefficients of the
sheaves.

Put $Z_\rho=C_{\widehat G}(\mathsf{Mon}_{\rho})$,
$\alpha=\operatorname{Ad}(g)|_{Z_\rho}$ and
$\Pi_\rho=\pi_0(Z_\rho)$. Thus $Z_\rho$ is the centralizer of the
geometric monodromy. Ellipticity makes the centralizer of the
arithmetic monodromy finite, while $Z_\rho$ can contain a torus.

\begin{lemma}[The centralizer of the geometric monodromy]
\label{fl:lem:geometric-centralizer-torus}
The group $Z_\rho$ is reductive, $Z_\rho^\circ$ is a torus, and
$1-\alpha:Z_\rho^\circ\to Z_\rho^\circ$ is an isogeny.
\end{lemma}

\begin{proof}
The centralizer of a reductive subgroup of a reductive group in
characteristic zero is reductive. By assumption (2) some power
$g^n$ lies in $\mathsf{Mon}_\rho$; it commutes with $Z_\rho$, so
$\alpha^n=1$. The group of fixed points of $\alpha$ is
$C_{\widehat G}(\mathsf{Mon}_\rho,g)$, which is finite by
assumption (3).

Suppose that the derived group of $Z_\rho^\circ$ is not trivial.
An automorphism of finite order of a connected reductive group
stabilizes a Borel subgroup and a maximal torus of it
\cite[\S3.3, Proposition~3(b)--(c)]{Serre2007}. Then $\alpha$
permutes the corresponding simple coroots, and the sum of the
coroots in one orbit is a nonzero cocharacter fixed by $\alpha$.
Its image is a torus of dimension one fixed by $\alpha$, which
contradicts the finiteness of the group of fixed points. So
$Z_\rho^\circ$ is a torus. The kernel of $1-\alpha$ on it is the
group of fixed points, which is finite; so $1-\alpha$ is an
isogeny.
\end{proof}

\subsection{Filling punctures as a derived intersection}

Choose a nonempty divisor $D_{\rm aux}$ on $C$, stable under the
Frobenius automorphism and consisting of distinct points, and
write $C^\circ=C-D_{\rm aux}$. For the stacks of local systems
with restricted variation of \cite{AGKRRV2022,GR2025}, consider
\[
 Z_C=\operatorname{LS}_{\widehat G}^{r}(C),\qquad
 Z_{C^\circ}=\operatorname{LS}_{\widehat G}^{r}(C^\circ),\qquad
 \mathfrak F_\rho=BZ_\rho\mathbin{\times^R}_{Z_{C^\circ}}Z_C.
\]
Here $BZ_\rho\to Z_{C^\circ}$ is the residual gerbe of the point
$\rho|_{C^\circ}$, and we write $j:\mathfrak F_\rho\to Z_C$ for
the projection. The underlying classical stack of
$\mathfrak F_\rho$ is $BZ_\rho$. The derived structure records the
equations which express that the monodromy at the punctures is
trivial; they are satisfied at $\rho$, but not transversally.

The stacks $Z_C$ and $Z_{C^\circ}$ are defined by means of lisse
$\ell$-adic sheaves. To compute with them we compare them with
stacks of representations of topological fundamental groups. The
comparison has three parts: a change of the base field, the
passage from $\ell$-adic sheaves to sheaves for the classical
topology, and the passage from local systems with restricted
variation to all local systems. We make them explicit, because the
second and the third are used for the curve $C^\circ$, which is
not proper.

\emph{The base field.} The curve $C$ and the divisor
$D_{\rm aux}$ are defined over an algebraically closed subfield
$K_0\subset K_{\rm geom}$ of finite transcendence degree over
$\mathbb Q$, and such a field can be embedded in $\mathbb C$. For
an extension of algebraically closed fields of characteristic zero
and a scheme $Y$ of finite type over the smaller field, pullback is
an equivalence between the categories $\operatorname{QLisse}$ of
$Y$ and of its base change
\cite[\S9.5.2, with its footnote, and Remark~9.5.3]{AGKRRV2022}. It
is compatible with tensor products and with restriction to open
subschemes. The stack of local systems with restricted variation is
defined in terms of the symmetric monoidal category
$\operatorname{QLisse}$ with its $t$-structure. Therefore $Z_C$,
$Z_{C^\circ}$, the map between them and the point $\rho$ do not
change when the pair $(C,D_{\rm aux})$ is replaced by its model
over $K_0$, and then by the base change of this model to
$\mathbb C$. Lemmas~\ref{fl:lem:puncture-fibre-model}
and~\ref{fl:lem:puncture-compactness} concern only these objects;
the semilinear automorphism of Setup~\ref{rw:setup:auxiliary-curve}
does not occur in them. In their proofs we may therefore assume
that $K_{\rm geom}=\mathbb C$.

\begin{lemma}[Comparison with the Betti stacks]
\label{rw:lem:betti-comparison}
Suppose that $K_{\rm geom}=\mathbb C$. For $Y=C$ or $Y=C^\circ$ let
$\operatorname{LS}^{\rm B}(Y)$ be the stack of $\widehat G$-local
systems with coefficients in $k$ on the topological space
$Y(\mathbb C)$, that is, the stack of maps from its homotopy type
to $B\widehat G$ \cite[\S4.5.1]{AGKRRV2022}.
\begin{enumerate}
\item There are monomorphisms of prestacks
$\iota_Y:Z_Y\to\operatorname{LS}^{\rm B}(Y)$, compatible with
restriction from $C$ to $C^\circ$.
\item Put
\[
 T=BZ_\rho\mathbin{\times^R}_{\operatorname{LS}^{\rm B}(C^\circ)}
   \operatorname{LS}^{\rm B}(C),
\]
the fibre product being formed with the residual gerbe of
$\rho|_{C^\circ}$ and the restriction map. The projection
$T\to\operatorname{LS}^{\rm B}(C)$ factors through $\iota_C$, and
the resulting map $T\to\mathfrak F_\rho$ is an isomorphism, under
which the projection becomes $\iota_C\circ j$.
\item Let $\widehat X_\rho$ be the formal completion of
$\operatorname{LS}^{\rm B}(C)$ along the fibre through $\rho$ of
the map to its coarse moduli space \cite[\S4.6.2]{AGKRRV2022}. The
intersection $Z_{C,\rho}$ of $Z_C$ and $\widehat X_\rho$ in
$\operatorname{LS}^{\rm B}(C)$ is open and closed in each of them,
and $j$ factors through $Z_{C,\rho}$.
\end{enumerate}
\end{lemma}

\begin{proof}
Write $\operatorname{LS}^{{\rm B},r}(Y)$ for the stack of local
systems with restricted variation in the classical topology
\cite[\S4.7]{AGKRRV2022}; the results quoted from there apply to
$C$ and to $C^\circ$, which are smooth connected varieties over
$\mathbb C$.

\step{1}{the monomorphisms} The comparison functor from
$\ell$-adic sheaves to ind-constructible sheaves of $k$-vector
spaces for the classical topology is fully faithful and commutes
with inverse images \cite[\S2.2.1]{GR2025}; on lisse sheaves it is
symmetric monoidal \cite[\S2.2.4]{GR2025}. It preserves the usual
$t$-structure, filtered colimits and lisse sheaves, so it maps the
full subcategory $\operatorname{QLisse}(Y)$ of the first kind
\cite[Definition~1.2.6]{AGKRRV2022} to that of the second kind, and
the functor so obtained is fully faithful and symmetric monoidal,
being the restriction of a fully faithful functor to full
subcategories.
The functor from the second category $\operatorname{QLisse}$ to
all locally constant sheaves is fully faithful as well
\cite[Proposition~4.7.2]{AGKRRV2022}. A fully faithful symmetric
monoidal functor remains fully faithful after tensoring with
$\operatorname{QCoh}(S)$, which is dualizable, and therefore
induces a monomorphism between the prestacks of symmetric monoidal
functors from $\operatorname{Rep}(\widehat G)$; this is the
argument of \cite[Lemma~4.1.4]{AGKRRV2022}, used in
\cite[\S4.7.4]{AGKRRV2022} for the second functor. So we have
monomorphisms
\[
 Z_Y\longrightarrow\operatorname{LS}^{{\rm B},r}(Y)
 \longrightarrow\operatorname{LS}^{\rm B}(Y),
\]
and $\iota_Y$ is their composite. Both functors commute with
restriction to $C^\circ$. This proves (1).

\step{2}{the projection factors through local systems with
restricted variation} The monomorphism
$\operatorname{LS}^{{\rm B},r}(C)\to\operatorname{LS}^{\rm B}(C)$ is
a formal isomorphism \cite[Proposition~4.7.8]{AGKRRV2022}. By the
definition \cite[\S4.1.6]{AGKRRV2022} this means that a point of
$\operatorname{LS}^{\rm B}(C)$ with values in an affine scheme $S$
lies in $\operatorname{LS}^{{\rm B},r}(C)$ as soon as its
restriction to the reduced classical scheme underlying $S$ does.
So let $S$ be reduced and classical. On classical schemes the
Betti stack classifies families of homomorphisms from the
fundamental group to $\widehat G$, up to conjugation
\cite[\S4.5]{AGKRRV2022}. A point of $T$ with values in $S$ is
therefore given by a $Z_\rho$-torsor on $S$, a family of local
systems on $C$ parametrized by $S$, and an isomorphism between the
restriction of this family to $C^\circ$ and the twist by the torsor
of the constant family with fibre $\rho|_{C^\circ}$. Since
$\pi_1(C^\circ)\to\pi_1(C)$ is surjective, the family on $C$ is the
twist by the torsor of the constant family with fibre $\rho$.
Locally on $S$ for the \'etale topology the torsor is trivial, and
the family is the image of the point $\rho$ of
$\operatorname{LS}^{{\rm B},r}(C)$ under $S\to\mathrm{pt}$. The
prestack $\operatorname{LS}^{{\rm B},r}(C)$ is a stack for the
\'etale topology \cite[Main Theorem~1.4.5]{AGKRRV2022}, so the
point lies in it.

\step{3}{the projection factors through $Z_C$} The map
$Z_C\to\operatorname{LS}^{{\rm B},r}(C)$ is an isomorphism onto a
union of connected components \cite[Lemma~2.2.5]{GR2025}, that is,
onto an open and closed substack. Its complement is open and
closed as well. By step~2 every point of $T$ with values in a field
maps to the point $\rho$, after an extension of the field, and
$\rho$ is a point of $Z_C$. So the inverse image in $T$ of the
complement has no points and is empty, and the projection factors
through $Z_C$.

\step{4}{the isomorphism} If a map $A\to Y$ factors through a
monomorphism $Y'\to Y$, then $A\times_YY'\simeq A$; and if two maps
$A\to Y$ and $B\to Y$ factor through $Y'$, then
$A\times_{Y'}B\simeq A\times_YB$. Both follow from
$Y'\times_YY'\simeq Y'$. Applying the second to the monomorphism
$\iota_{C^\circ}$ and the first to $\iota_C$ and the projection of
$T$, which factors through it by step~3, we get
\[
 \mathfrak F_\rho
 =BZ_\rho\times_{Z_{C^\circ}}Z_C
 \simeq BZ_\rho\times_{\operatorname{LS}^{\rm B}(C^\circ)}Z_C
 \simeq T\times_{\operatorname{LS}^{\rm B}(C)}Z_C
 \simeq T ,
\]
all fibre products being derived. The composite is inverse to the
map of (2), and it is compatible with the projections to
$\operatorname{LS}^{\rm B}(C)$. This proves (2).

\step{5}{the formal completion} The substack
$\operatorname{LS}^{{\rm B},r}(C)$ of $\operatorname{LS}^{\rm B}(C)$
is the disjoint union of the formal completions of the fibres of
the map to the coarse moduli space
\cite[Corollary~4.8.5]{AGKRRV2022}. So $\widehat X_\rho$ is open
and closed in $\operatorname{LS}^{{\rm B},r}(C)$, and so is $Z_C$
by step~3; hence their intersection is open and closed in each.
The map $j$ factors through $Z_C$ by construction, and through
$\widehat X_\rho$ because every point of $\mathfrak F_\rho$ with
values in a reduced classical scheme maps into the fibre through
$\rho$, by step~2. This proves (3).
\end{proof}

\begin{lemma}[The derived fibre]
\label{fl:lem:puncture-fibre-model}
There is an isomorphism
\begin{equation}\label{fl:eq:puncture-fibre}
 \mathfrak F_\rho\simeq
 [\operatorname{Spec}\Lambda(\mathsf B_\rho[1])/Z_\rho],\qquad
 \mathsf B_\rho=\bigoplus_{x\in D_{\rm aux}}\widehat{\mathfrak g}_x^*,
\end{equation}
where $\Lambda(\mathsf B_\rho[1])$ is the exterior algebra of
$\mathsf B_\rho$, placed in degrees $\le0$ with zero differential,
and $Z_\rho$ acts through the coadjoint representation at each
point. In particular $H^0\Gamma(\mathfrak F_\rho,\mathcal O)=k$.
The objects $\mathcal O\otimes W$, for $W$ a representation of
finite dimension of $Z_\rho$, generate
$\operatorname{Perf}(\mathfrak F_\rho)$ under finite cones and
retracts.
\end{lemma}

\begin{proof}
We may assume that $K_{\rm geom}=\mathbb C$. By
Lemma~\ref{rw:lem:betti-comparison}(2) the stack $\mathfrak F_\rho$
is then the fibre $T$ of the restriction map between the Betti
stacks, which we now compute.

Let $h$ be the genus of $C$ and $n=|D_{\rm aux}|$. The fundamental
group of the punctured surface has generators
$a_1,b_1,\dots,a_h,b_h,c_1,\dots,c_n$ and the relation
$\prod_i[a_i,b_i]\prod_jc_j=1$. We eliminate $c_n$: the
punctured surface retracts onto a graph with $2h+n-1$ loops, so
its Betti stack is $[\widehat G^{2h+n-1}/\widehat G]$. The closed
surface is obtained from it by attaching a disc along each
boundary loop, and this is a homotopy pushout. The stack of maps
to $B\widehat G$ turns it into a derived fibre product: the Betti
stack of the closed surface is the fibre product of that of the
punctured surface and of $B\widehat G$ over
$[\widehat G^n/\widehat G]$, for the map which records the images
of $c_1,\dots,c_n$ and the inclusion of the identity. Since
$\rho$ has trivial monodromy at the punctures, the fibre over the
residual gerbe of $\rho|_{C^\circ}$ is the quotient by $Z_\rho$
of the spectrum of
\[
 k\otimes^L_{\mathcal O(\widehat G^n)}k ,
\]
both copies of $k$ being the residue field at the identity. The
identity is a smooth point with cotangent space
$\mathsf B_\rho$, and the Koszul resolution shows that the
cohomology of this ring is the exterior algebra of
$\mathsf B_\rho$, with $\mathsf B_\rho$ in degree $-1$.

To obtain an isomorphism of algebras with an action of $Z_\rho$,
and not only of their cohomology, we use formal coordinates at the
identity which are compatible with the action. They exist: since
$Z_\rho$ is reductive, the surjections
$\mathfrak m/\mathfrak m^{a+1}\to\mathfrak m/\mathfrak m^2$ of
representations of $Z_\rho$ are split, compatibly in $a$, and the
image of a splitting is a space of coordinates. In such
coordinates the Koszul complex is a resolution of $k$ by modules
with an action of $Z_\rho$, and after tensoring with $k$ its
differential is zero. This gives \eqref{fl:eq:puncture-fibre} for
$T$, hence for $\mathfrak F_\rho$.

The identification of the cohomology in degree $-1$ with the
cotangent space, and of the graded algebra of cohomology with its
exterior algebra, is canonical. So an automorphism which preserves
the fibre acts on the graded algebra of cohomology by the exterior
powers of its action in degree $-1$. The splitting which was used
to produce a model with zero differential is a choice, and it need
not be compatible with Frobenius; it is not used for the action.

The functions of degree zero on $\mathfrak F_\rho$ are the
invariants of $Z_\rho$ in the part of degree zero of the exterior
algebra, which is $k$. For a complex $Q$ of modules over the
exterior algebra with a compatible action of $Z_\rho$, the
complex of maps from $\mathcal O\otimes W$ to $Q$ is
$(W^*\otimes Q)^{Z_\rho}$. This functor of $Q$ commutes with
filtered colimits, because taking invariants under a reductive
group does. If $Q$ has a nonzero cohomology class, the class lies
in a finite-dimensional subrepresentation $W$ of a cohomology
group, and gives a nonzero map from a shift of
$\mathcal O\otimes W$. So the objects $\mathcal O\otimes W$ are
compact generators, and the perfect complexes, which are the
compact objects, are obtained from them by finite cones and
retracts.
\end{proof}

\begin{lemma}[Compactness]
\label{fl:lem:puncture-compactness}
For every $P\in\operatorname{Perf}(\mathfrak F_\rho)$, the object
$j_*P$ is compact in $\operatorname{QCoh}(Z_C)$.
\end{lemma}

\begin{proof}
By Lemma~\ref{fl:lem:puncture-fibre-model} it is enough to
consider $P=\mathcal O\otimes W$. We may assume that
$K_{\rm geom}=\mathbb C$, and we use the notation of
Lemma~\ref{rw:lem:betti-comparison} and of the preceding proof.
Write $X=\operatorname{LS}^{\rm B}(C)$ and
$j^{\rm B}=\iota_C\circ j:\mathfrak F_\rho\to X$.

\step{1}{the direct image on the Betti stack is perfect} The
variety $\widehat G^{2h+n-1}$ is smooth, and the orbit of the
semisimple representation $\rho|_{C^\circ}$ in it is closed and
smooth; so the inclusion of this orbit is a regular immersion. The
sheaf on
$[\widehat G^{2h+n-1}/\widehat G]=\operatorname{LS}^{\rm B}(C^\circ)$
supported on the orbit and defined by the representation $W$ of
the stabilizer $Z_\rho$ is therefore perfect. It is the direct
image of $W$ from the residual gerbe. The inclusion of the residual
gerbe is a closed immersion, because the orbit is closed; so its
base change $j^{\rm B}$ is an affine morphism, and direct images
along these two maps commute with base change. By base change in
the square which defines $T$, and by
Lemma~\ref{rw:lem:betti-comparison}(2), the pullback of this sheaf
to $X$ is $j^{\rm B}_*(\mathcal O\otimes W)$. Hence
$j^{\rm B}_*(\mathcal O\otimes W)$ is perfect. The stack $X$ is the
quotient of an affine derived scheme $\operatorname{Spec}R$ by
$\widehat G$ \cite[\S\S4.5--4.6.1]{AGKRRV2022}. Since $\widehat G$
is reductive, the functor of global sections on $X$ commutes with
colimits, and perfect complexes on $X$ are compact in
$\operatorname{QCoh}(X)$.

\step{2}{sheaves on the formal completion} Let
$\hat\iota:\widehat X_\rho\to X$ be the inclusion of the formal
completion of Lemma~\ref{rw:lem:betti-comparison}(3). The coarse
moduli space of $X$ is the spectrum of the invariants
$R^{\widehat G}$, and its underlying classical scheme is of finite
type \cite[\S\S4.6.1--4.6.2]{AGKRRV2022}; let
$f_1,\ldots,f_m\in H^0(R)^{\widehat G}$ generate the maximal ideal
of the image of $\rho$ in it. Then $\widehat X_\rho$ is the quotient by
$\widehat G$ of the formal completion of $\operatorname{Spec}R$
along the locus $f_1=\cdots=f_m=0$, which is a formal affine
scheme in the sense of \cite[\S7.1]{AGKRRV2022}. Let
$\operatorname{QCoh}(X)_\rho\subset\operatorname{QCoh}(X)$ be the
full subcategory of the objects whose restriction to the
complement of the fibre is zero; it is closed under colimits. We
claim that $\hat\iota^*$ restricts to an equivalence
\[
 \operatorname{QCoh}(X)_\rho\xrightarrow{\ \sim\ }
 \operatorname{QCoh}(\widehat X_\rho).
\]
For the affine scheme $\operatorname{Spec}R$ in place of $X$ this
is \cite[Proposition~7.1.5]{AGKRRV2022}, and the same holds for
the schemes $\widehat G^a\times\operatorname{Spec}R$ with the
pullbacks of the functions $f_i$. These schemes form the \v Cech
nerve of $\operatorname{Spec}R\to X$, and their formal completions
form the \v Cech nerve of the corresponding cover of
$\widehat X_\rho$. The equivalences are given by pullback, so they
are compatible with the maps of the two nerves, and the condition
on the support is compatible with these maps, because pullback
commutes with restriction to the open complement of the fibre. The
claim follows by descent for quasi-coherent sheaves.

\step{3}{conclusion} The complex
$j^{\rm B}_*(\mathcal O\otimes W)$ is supported at the point
$\rho$, so it lies in $\operatorname{QCoh}(X)_\rho$. It is compact
in $\operatorname{QCoh}(X)$ by step~1. Hence it is compact in the
full subcategory $\operatorname{QCoh}(X)_\rho$: the inclusion
commutes with colimits, so maps from the complex into a filtered
colimit formed in the subcategory are computed in
$\operatorname{QCoh}(X)$. By step~2 the complex
$\hat\iota^*j^{\rm B}_*(\mathcal O\otimes W)$ is compact in
$\operatorname{QCoh}(\widehat X_\rho)$. The map $j^{\rm B}$ factors
through the monomorphism $\hat\iota$, so
$\mathfrak F_\rho\times_X\widehat X_\rho=\mathfrak F_\rho$, and by
base change $\hat\iota^*j^{\rm B}_*(\mathcal O\otimes W)$ is the
direct image of $\mathcal O\otimes W$ under
$\mathfrak F_\rho\to\widehat X_\rho$.

By Lemma~\ref{rw:lem:betti-comparison}(3) this map factors through
$Z_{C,\rho}$, which is open and closed in $\widehat X_\rho$ and in
$Z_C$. So $\operatorname{QCoh}(Z_{C,\rho})$ is a direct factor of
$\operatorname{QCoh}(\widehat X_\rho)$ and of
$\operatorname{QCoh}(Z_C)$, and the direct images of
$\mathcal O\otimes W$ to $\widehat X_\rho$ and to $Z_C$ both come
from its direct image to $Z_{C,\rho}$. An object of a direct factor
which is compact in the whole category is compact in the factor,
and an object which is compact in a direct factor is compact in the
whole category. Therefore $j_*(\mathcal O\otimes W)$ is compact in
$\operatorname{QCoh}(Z_C)$.
\end{proof}

\subsection{The automorphic family and its Whittaker coefficient}

We use the equivalence of categories of the geometric Langlands
correspondence with restricted variation in characteristic zero
\cite[Theorem~1.3.9(ii)]{GR2025}, which we denote by
$\mathbb L_G^r$. Let $\mathsf e$ be the fully faithful embedding
of the category of quasi-coherent complexes into the spectral
category with nilpotent singular support, normalized as in
\cite[\S\S1.3.1--1.3.4]{GR2025}; the normalization includes the
graded line of the dualizing complex. Define
\[
 \mathcal F_P=(\mathbb L_G^r)^{-1}\mathsf e(j_*P),
 \qquad P\in\operatorname{Perf}(\mathfrak F_\rho).
\]
The functor $P\mapsto\mathcal F_P$ is exact and $k$-linear. Since
$j_*P$ is compact, \cite[Theorem~1.1.7 and \S1.3.4]{GR2025} shows
that $\mathcal F_P$ is compact in the category of all automorphic
sheaves; its restriction to every smooth chart of finite type is
a bounded constructible complex.

\subsubsection*{The Whittaker coefficient}
Let $\operatorname{Poinc}^{\rm Vac}_!$ be the Poincar\'e object
defined by $!$-pushforward from the Whittaker stack, with the
regular representation of the finite centre as coefficient (it is
recalled in the proof of
Lemma~\ref{fl:lem:coarse-covariance}). The Whittaker coefficient
is the functor $\operatorname{Wh}$ which it corepresents. The
projection of $\operatorname{Poinc}^{\rm Vac}_!$ to the category
of sheaves with nilpotent singular support corepresents the
functor to $\operatorname{QCoh}(Z_C)$ followed by global sections.
The two objects are different, but on our family the two functors
agree:
\begin{equation}\label{fl:eq:puncture-vacuum}
 \operatorname{Wh}(\mathcal F_P)
 \simeq\Gamma_!(Z_C,j_*P)
 \simeq\Gamma(\mathfrak F_\rho,P).
\end{equation}
The first isomorphism is \cite[Proposition~1.2.5]{GR2025}. For the
second, $\Gamma_!$ agrees with $\Gamma$ on compact objects
\cite[Remark~7.6.2]{AGKRRV2022}, and
$\Gamma(Z_C,j_*P)=\Gamma(\mathfrak F_\rho,P)$. In terms of the
projected object, the adjunction for the functor
$\mathbb L^r_{G,\rm coarse}$ to $\operatorname{QCoh}(Z_C)$ gives
\begin{equation}\label{fl:eq:projected-vacuum-coefficient}
 \operatorname{RHom}
 (\operatorname{Poinc}^{\rm Vac}_{!,\operatorname{Nilp}},\mathcal F_P)
 \simeq\Gamma(Z_C,j_*P),
\end{equation}
since $\mathbb L^r_{G,\rm coarse}\mathcal F_P=j_*P$: the right
adjoint of the fully faithful functor $\mathsf e$ maps
$\mathsf e(j_*P)$ to $j_*P$
\cite[Corollary~1.3.6(ii)]{GR2025}.

\subsubsection*{Theta pairs}
The curves in this paper carry a \emph{theta pair}: a line bundle
$L$ with an isomorphism $L^2\simeq\Omega_C$. Let
$\lambda_{\rm pr}$ be the sum of the positive coroots of $G$, a
cocharacter of the maximal torus $T$, and
$\mathcal P_T=\lambda_{\rm pr}(L)$ the $T$-bundle induced from
$L$. The theta pair enters the definition of the Whittaker stack
through $\mathcal P_T$, and that of the canonical bundle in
Section~\ref{fl:sec:boundary-whittaker-geometry-section}.

\paragraph{Pullback and transport.}
A semilinear automorphism $u_C$ of the curve acts on the moduli
stacks in two ways which are inverse to each other: an object can
be pulled back by $u_C$, or transported by it. The direction of
Frobenius in the rest of the paper depends on keeping the two
apart, so we give them different names:
\[
 u_B(\mathcal E)=u_C^*\mathcal E,
 \qquad u_L(\tau)=u_C^*\tau,
 \qquad
 u_{\rm aut}=u_B^{-1},\quad u_{\rm spec}=u_L^{-1}.
\]
Thus $u_{\rm aut}$ and $u_{\rm spec}$ are transport by $u_C$, that
is, pullback of bundles and of local systems by $u_C^{-1}$. When
$u_C$ is the arithmetic Frobenius acting on the coefficients,
$u_{\rm aut}$ is the arithmetic Frobenius of the stack of bundles.
The same convention must be used on the two stacks. Indeed the
pullback of a modification at $x$ is a modification at
$u_C^{-1}x$, and for the universal local system $V_{\rm univ}$
evaluation gives a canonical isomorphism
\[
 (u_L\times u_C^{-1})^*V_{\rm univ}\simeq V_{\rm univ}.
\]
The corresponding statements for the inverse maps $u_{\rm aut}$
and $u_{\rm spec}$ follow. The directions of all the Frobenius
maps which occur in the paper are collected in the table of
Remark~\ref{rw:rem:trace-conventions}.

Let $c:u_C^*\rho\xrightarrow{\sim}\rho$ be the chosen descent
isomorphism. The point $\rho$ is fixed by $u_L$, and on the
residual gerbe $BZ_\rho$ of this point $u_L$ acts by the
automorphism
\[
 \delta:Z_\rho\longrightarrow Z_\rho,
 \qquad z\longmapsto c\,u_C^*(z)\,c^{-1}.
\]
So $u_{\rm spec}=u_L^{-1}$, and its lift $u_{\mathfrak F}$ to the
derived fibre, act on the gerbe through $\delta^{-1}$. The
pullback of a vector bundle on $BZ_\rho$ along the map induced by
an automorphism $\gamma$ of $Z_\rho$ is the restriction of the
representation along $\gamma$. Hence
\begin{equation}\label{rw:eq:arithmetic-coefficient-pullback}
 u_{\mathfrak F}^*P_W\simeq P_{(\delta^{-1})^*W}.
\end{equation}

We compute $\delta$ for the arithmetic Frobenius of
Setup~\ref{rw:setup:auxiliary-curve}. The
semilinear map $u_C$ is the one with $u_C^\#(a)=a^q$ on the
residue field. Let $a$ be the chosen arithmetic lift of Frobenius
in the fundamental group, used to base it, and $g$ its image. Then
$u_C$ acts on the geometric fundamental group by
$\gamma\mapsto a^{-1}\gamma a$, so that
\[
 (u_C^*\rho)(\gamma)=g^{-1}\rho(\gamma)g,
 \qquad c=g,
\]
in the marked fibre, and $\delta=\operatorname{Ad}(g)=\alpha$. The
orientation can be seen on a finite \'etale torsor: pulling back a
section with coordinates $b$ along $u_C$ replaces them by $b^q$,
so the canonical descent map on the fibre is the arithmetic
Frobenius; the local system is a limit of such torsors. Therefore
\eqref{rw:eq:arithmetic-coefficient-pullback} reads
\[
 u_{\mathfrak F}^*P_W=P_{(\alpha^{-1})^*W}.
\]

\begin{lemma}[Hecke action and semilinear covariance]
\label{fl:lem:coarse-covariance}
\begin{enumerate}
\item Let $V$ be a representation of $\widehat G$, and let
$V_\rho|_{C^\circ}=\bigoplus_aW_a\boxtimes L_a$ be a decomposition
of the associated local system, with $W_a$ a representation of
$Z_\rho$ and $L_a$ a local system on $C^\circ$. Then
\begin{equation}\label{fl:eq:puncture-family-hecke}
 \mathsf H_{V}(\mathcal F_P)
 \simeq\bigoplus_a\mathcal F_{P\otimes W_a}\boxtimes L_a
\end{equation}
over $C^\circ$. The isomorphism does not depend on the
decomposition. The same holds for every finite set $I$ of legs
and every representation of $\widehat G^I$, over $(C^\circ)^I$,
and these isomorphisms are compatible with tensor products,
composition, permutations of the legs and fusion. We refer to
them as the \emph{Hecke identities} of the family; for $P=P_W=\mathcal O_{\mathfrak F_\rho}\otimes W$
they make $W\mapsto\mathcal F_{P_W}$ a Hecke family in the sense
of Definition~\ref{rw:def:hecke-family}(1).
\item Let $u_C=u$ be a semilinear isomorphism of the curve $C$ which
preserves the pinning, the theta pair and the divisor
$D_{\rm aux}$, together with an isomorphism $u^*\rho\simeq\rho$.
It induces an autoequivalence $u_{\mathfrak F}^*$ of
$\operatorname{Perf}(\mathfrak F_\rho)$ and a natural isomorphism
\[
 u_{\rm aut}^*\mathcal F_P
 \simeq\mathcal F_{u_{\mathfrak F}^*P},
\]
compatible with the Hecke identities. For a composite of two
semilinear isomorphisms the isomorphism is the composite of the
two. In particular the chosen lift of Frobenius and all its
powers act on the family compatibly.
\end{enumerate}
\end{lemma}

\begin{proof}
The Hecke identities concern the spectral action. The covariance
concerns the object to which the spectral action is applied, and
it is this part which determines the Frobenius structure of the
family.

\step{1}{the Hecke identities} On
$\mathfrak F_\rho\times C^\circ$ the universal local system is
pulled back from $BZ_\rho\times C^\circ$, where the local system
attached to $V$ is $\bigoplus_aW_a\boxtimes L_a$ with $W_a$
regarded as a vector bundle on $BZ_\rho$. The Hecke functor
$\mathsf H_V$ corresponds under $\mathbb L^r_G$ to the tensor
product with the local system attached to $V$, and by the
projection formula for $j$
\[
 j_*P\otimes V_{\rm univ}|_{Z_C\times C^\circ}
   \simeq\bigoplus_a j_*(P\otimes W_a)\boxtimes L_a .
\]
The left-hand side does not involve the decomposition. The
compatibilities for all finite sets of legs are part of the
equivalence between the spectral action and the action of Hecke
functors on sheaves with nilpotent singular support
\cite[Theorems~8.1.4, 14.3.2 and \S14.3.3]{AGKRRV2022}.

\step{2}{a formula for the family} Let $\mathcal P_C$ be the
projection of $\operatorname{Poinc}^{\rm Vac}_!$ to the category
of sheaves with nilpotent singular support. For
$A\in\operatorname{QCoh}(Z_C)$ and an automorphic sheaf
$\mathcal K$,
\[
 \operatorname{RHom}((\mathbb L_G^r)^{-1}\mathsf e A,\mathcal K)
 \simeq\operatorname{RHom}
 (A,\mathsf e^R\mathbb L_G^r\mathcal K)
 \simeq\operatorname{RHom}(A,\mathbb L_{G,\rm coarse}^r\mathcal K),
\]
where $\mathsf e^R$ is the right adjoint of $\mathsf e$. So
$(\mathbb L_G^r)^{-1}\mathsf e$ is the left adjoint of
$\mathbb L^r_{G,\rm coarse}$. By the construction of the latter
\cite[\S\S1.1.3--1.1.8]{GR2025} this left adjoint is
$A\mapsto A\star\mathcal P_C$, where $\star$ denotes the spectral
action. Hence
\begin{equation}\label{rw:eq:family-by-spectral-action}
 \mathcal F_P\simeq(j_*P)\star\mathcal P_C.
\end{equation}
This formula does not involve the choice of an inverse of
$\mathbb L_G^r$, and it fixes the normalization of $\mathsf e$.

\step{3}{covariance of the Poincar\'e object} Let
$\mathscr W_C=\operatorname{Bun}_{N,\mathcal P_T}/T$ be the
Whittaker stack and $r$ the rank of $G$. The Poincar\'e object is
defined by the diagram
\[
 \operatorname{Bun}_G(C)\xleftarrow{\ p_C\ }
 \mathscr W_C\xrightarrow{\ \chi_C\ }
 [\mathbb A^r/T]\xleftarrow{\ j_{\rm reg}\ }BZ(G)
\]
as $p_{C,!}\chi_C^*$ of the sheaf
$j_{{\rm reg},*}\operatorname{Reg}_{Z(G)}[-r]$, where
$\operatorname{Reg}_{Z(G)}=(\operatorname{pt}\to BZ(G))_!k$, with
the shifts of \cite[\S\S3.3.3--3.3.7]{GLCI2024}. The component of
$\chi_C$ attached to a simple root sends an $N$-torsor to the
class of the extension it defines, an element of
$H^1(C,\Omega_C)$, followed by the trace
$H^1(C,\Omega_C)\to K_{\rm geom}$ of Serre duality.

With the convention of transport fixed above, $u$ acts on the
$N$-torsors, on their quotients
attached to the simple roots and on the theta pair, and the trace
of Serre duality commutes with semilinear pullback. So $u$
carries the whole diagram to the diagram of the curve obtained by
base change. Here $\chi_C$ is a morphism of stacks over
$K_{\rm geom}$, along which the sheaves, with coefficients in $k$,
are pulled back. Pullback by an isomorphism
commutes with $p_!$, $\chi^*$ and $j_*$ and preserves
$\operatorname{Reg}_{Z(G)}$. These base change isomorphisms give
$u_{\rm aut}^*\operatorname{Poinc}^{\rm Vac}_!
\simeq\operatorname{Poinc}^{\rm Vac}_!$.

\step{4}{covariance of the projection} The projection to sheaves
with nilpotent singular support is given by the universal Hecke
projector of \cite[\S13.4.1, Corollary~13.4.6 and
\S15.4.7]{AGKRRV2022}. It is a colimit of Hecke functors, indexed
by diagrams formed from finite sets of points of the curve, their
collisions, representations of $\widehat G$ and the fibres of the
universal local system. The pullbacks by $u_{\rm aut}$, on the
automorphic side, and by $u_{\rm spec}$, on the spectral side, send
each of these diagrams to the corresponding diagram for the curve
obtained by base change, and they commute with tensor products,
with colimits and with $\Gamma_!$ on the spectral side. So they
commute with the projector. The comparison is canonical. By
\cite[Theorem~1.1.7]{GR2025} and
\cite[Proposition~17.2.3]{AGKRRV2022} the projector is the right
adjoint of the inclusion of the sheaves with nilpotent singular
support; the equivalence $u_{\rm aut}^*$ preserves this
subcategory, and an equivalence which commutes with a fully
faithful functor commutes with its right adjoint, compatibly with
the counits. Applied to the Poincar\'e
object this gives $u_{\rm aut}^*\mathcal P_C\simeq\mathcal P_C$.

\step{5}{covariance of the family} For the same reason
\[
 u_{\rm aut}^*(A\star\mathcal K)
 \simeq(u_{\rm spec}^*A)\star(u_{\rm aut}^*\mathcal K).
\]
The Cartesian square which defines $\mathfrak F_\rho$ is
preserved by $u$, so
$u_{\rm spec}^*j_*P\simeq j_*u_{\mathfrak F}^*P$. Substituting in
\eqref{rw:eq:family-by-spectral-action} we obtain the asserted
isomorphism, that is, a natural isomorphism which makes the
square
\[
\begin{tikzcd}[column sep=large]
 \operatorname{Perf}(\mathfrak F_\rho)
   \arrow[r,"P\mapsto\mathcal F_P"]\arrow[d,"u_{\mathfrak F}^*"'] &
 \operatorname{Shv}_{\rm Nilp}(\operatorname{Bun}_G(C))
   \arrow[d,"u_{\rm aut}^*"]\\
 \operatorname{Perf}(\mathfrak F_\rho)
   \arrow[r,"P\mapsto\mathcal F_P"'] &
 \operatorname{Shv}_{\rm Nilp}(\operatorname{Bun}_G(C))
\end{tikzcd}
\]
commute. For two semilinear isomorphisms $u$ and $v$, the
isomorphism attached to $uv$ and the composite of those attached
to $u$ and $v$ are both base change maps for the same diagrams,
and base change maps are compatible with composition. The same
argument applies to the maps of creation, fusion and annihilation
in the Hecke action, so the isomorphisms are compatible with the
Hecke identities.
\end{proof}

\subsection{Why the whole family is needed}

The comparison with a microstalk in
Section~\ref{fl:sec:boundary-whittaker-geometry-section} is at
first a comparison over an algebraically closed field, without
Frobenius. For a single complex this would give no control of the
trace of Frobenius. The control comes from the following property
of the functor of global sections on the whole category
$\operatorname{Perf}(\mathfrak F_\rho)$.

\begin{lemma}[Rigidity of the functor of global sections]
\label{fl:lem:scalar-coefficient-covariance}
Let $\Gamma_\rho(P)=\Gamma(\mathfrak F_\rho,P)$ on
$\operatorname{Perf}(\mathfrak F_\rho)$. Every natural
endomorphism of degree zero of the functor $\Gamma_\rho$ acts on
the cohomology groups $H^i\Gamma_\rho(P)$ by one scalar, the same
for all $P$ and all $i$. For this it suffices that the
endomorphism be natural for the morphisms of the homotopy category
and commute with the shift. Consequently, if a functor $\mu$ is
isomorphic to $\Gamma_\rho\otimes L$ for a graded line $L$, and
both functors are compatible with Frobenius, the two actions of
Frobenius on cohomology differ, under the isomorphism, by a nonzero
scalar which does not depend on $P$ or on the degree.
\end{lemma}

\begin{proof}
In the homotopy category of $\operatorname{Perf}(\mathfrak F_\rho)$
the functor $P\mapsto H^0\Gamma_\rho(P)$ is
$\operatorname{Hom}(\mathcal O,P)$: it is corepresented by the
structure sheaf, which is a perfect complex. By the Yoneda lemma
for this ordinary category, its endomorphisms are the compositions
with the elements of
\[
 \operatorname{End}(\mathcal O_{\mathfrak F_\rho})
 =H^0\Gamma(\mathfrak F_\rho,\mathcal O)=k,
\]
the last equality by Lemma~\ref{fl:lem:puncture-fibre-model}.
Concretely, a natural endomorphism is determined by its value on
the identity of $\mathcal O$, an endomorphism of $\mathcal O$, and
its value on a section $v:\mathcal O\to P$ is the composite of $v$
with this endomorphism. So the given endomorphism acts on
$H^0\Gamma_\rho(P)$ by a scalar $c$ which does not depend on $P$.
Since $H^i\Gamma_\rho(P)=H^0\Gamma_\rho(P[i])$ and the endomorphism
commutes with the shift, it acts by $c$ on $H^i\Gamma_\rho(P)$.

For the second assertion, transport the action of Frobenius on
$\mu$ to $\Gamma_\rho\otimes L$ by the isomorphism, and compose it
with the inverse of the action of Frobenius on
$\Gamma_\rho\otimes L$. The result is a natural automorphism of
the functor $\Gamma_\rho\otimes L$, of degree zero, and the first
part applies to it after the choice of a generator of $L$; the
scalar is not zero because the automorphism is invertible.
\end{proof}

\begin{remark}[Which naturality is used]
\label{rw:rem:homotopy-naturality}
Only the Yoneda lemma of an ordinary category enters. So the
isomorphisms of functors to which the lemma is applied below, in
particular the comparison with the microstalk of
Proposition~\ref{fl:lem:vacuum-microstalk-comparison}, are needed only
as isomorphisms of functors on homotopy categories which commute
with the shift. This is enough, because the functors take values
in complexes of vector spaces over a field and all conclusions
concern supertraces.
\end{remark}

\begin{remark}
The lemma would be of no use for the single object
$P_{\rm reg}$. Its cohomology
$\Gamma(\mathfrak F_\rho,P_{\rm reg})$ lives in several degrees, and
an automorphism acting by different scalars in different degrees
can turn a nonzero alternating trace into zero. Naturality on all
of $\operatorname{Perf}(\mathfrak F_\rho)$ forces the same scalar in
every degree.
\end{remark}

\begin{proposition}[The supertrace is a nonzero determinant]
\label{fl:prop:puncture-exact-trace}
Suppose that every orbit of Frobenius in $D_{\rm aux}$ consists of
the points which lift a closed point of degree $d_{\rm aux}$ of a
fibre in characteristic $p$, and that the adjoint action of the
Frobenius element $g_D\in\widehat G(k)$ of that point is pure of
weight zero. Let
\[
 P_{\rm reg}=\mathcal O_{\mathfrak F_\rho}\otimes k[\Pi_\rho]
\]
with the action of Frobenius induced by $\alpha$, which permutes
the basis by $e_\gamma\mapsto e_{\alpha(\gamma)}$. Then Frobenius
and its inverse have nonzero supertrace on
$\Gamma(\mathfrak F_\rho,P_{\rm reg})$.
\end{proposition}

\begin{proof}
Write $\phi$ for the action of Frobenius. We compute in the
graded algebra of cohomology
$\Lambda=\Lambda(\mathsf B_\rho[1])$, on which Frobenius acts by
the exterior powers of its action in degree $-1$.
The group $Z_\rho$ is linearly reductive, so
$H^*\Gamma(\mathfrak F_\rho,P_{\rm reg})
=(k[\Pi_\rho]\otimes\Lambda)^{Z_\rho}$.

\step{1}{removing the group of components} For a complex
$\mathcal K$ of representations of $Z_\rho$ the map
\[
 \mathcal K^{Z_\rho^\circ}\longrightarrow(k[\Pi_\rho]\otimes \mathcal K)^{Z_\rho},
 \qquad v\longmapsto\sum_{\gamma\in\Pi_\rho}
 e_\gamma\otimes\widetilde\gamma v ,
\]
where $\widetilde\gamma\in Z_\rho$ is a representative of
$\gamma$, is an isomorphism: it does not depend on the
representatives because $v$ is invariant under $Z_\rho^\circ$,
and its inverse is the coefficient of $e_1$. For
$\mathcal K=\Lambda$ it commutes with $\phi$, since
$\phi(\widetilde\gamma v)=\alpha(\widetilde\gamma)\phi(v)$. So
the supertrace to be computed is that of $\phi$ on
$\Lambda^{Z_\rho^\circ}$.

\step{2}{removing the torus} Decompose $\Lambda$ according to the
characters of the torus $Z_\rho^\circ$. The map $\phi$ sends the
part of weight $\chi$ to the part of weight
$\chi\circ\alpha^{-1}$. If $\chi\circ\alpha^{-1}=\chi$, then
$\chi$ is trivial on the image of $1-\alpha$, which is
$Z_\rho^\circ$ by
Lemma~\ref{fl:lem:geometric-centralizer-torus}; so $\chi=0$. The
parts of nonzero weight are therefore permuted without fixed
points and do not contribute to the trace, in any degree. Hence
$\phi$ has the same supertrace on $\Lambda^{Z_\rho^\circ}$ and on
$\Lambda$, and since $\Lambda^i\mathsf B_\rho$ is in degree $-i$,
\begin{equation}\label{fl:eq:puncture-exact-trace}
 \operatorname{Str}\bigl(\phi;
 \Gamma(\mathfrak F_\rho,P_{\rm reg})\bigr)
 =\sum_i(-1)^i\operatorname{tr}(\phi;\Lambda^i\mathsf B_\rho)
 =\det(1-\phi;\mathsf B_\rho).
\end{equation}

\step{3}{the determinant} Take an orbit of punctures; its length
is $d=d_{\rm aux}$. At one of its points choose a local
arithmetic lift $a_D$ of Frobenius and a generator $\tau$ of the
tame inertia. On the $\ell$-primary tame quotient
\[
 a_D\tau a_D^{-1}=\tau^{q^d}.
\]
Write $g_D=\rho(a_D)$, and let
$\operatorname{Ad}(g_D)^\vee(\xi)
=\xi\circ\operatorname{Ad}(g_D^{-1})$ be the coadjoint action on
$\widehat{\mathfrak g}^{\,*}$. The arithmetic map on the spectral
side is the transport $u_{\rm spec}$. Let $M$ be the monodromy at
the puncture of a local system near $\rho$, a formal variable near
the identity. After $d$ steps, and after identifying the base
local system by $g_D^{-1}$, the transport is
\[
 M\longmapsto g_D^{-1}M^{q^d}g_D ,
\]
whose linearization on the Lie algebra is
$q^d\operatorname{Ad}(g_D^{-1})$. So the pullback of the equation
$M=1$ acts on its conormal space by
$q^d\operatorname{Ad}(g_D)^\vee$. This is the action on $H^{-1}$
of the Koszul complex of the derived intersection, and its
exterior powers give the action on all the cohomology groups; the
complex itself need not be given an action of Frobenius. Hence the
arithmetic operator $\phi$ on $\mathsf B_\rho$ permutes the
summands of the orbit cyclically, and
\[
 \phi^d|_{\widehat{\mathfrak g}_x^*}
   =q^d\operatorname{Ad}(g_D)^\vee.
\]
The determinant of $1-\phi$ on the cyclic sum is
$\det(1-\phi^d;\widehat{\mathfrak g}_x^*)$. Its factors are
$1-q^d\lambda$, with $\lambda$ an eigenvalue of
$\operatorname{Ad}(g_D)^\vee$, and $|\lambda|=1$ for every complex
embedding by purity. So no factor vanishes. The geometric operator
is the inverse of the arithmetic one, on the conormal space and on
the representation of the group of components together; its
factors are $1-q^{-d}\lambda^{-1}$, which do not vanish either.
What is essential is purity: compactness of the image of the
parameter gives only $\ell$-adic information, and by itself would
not exclude an eigenvalue of complex absolute value $q^{\pm d}$.
\end{proof}

\begin{remark}
The hypothesis on weights cannot be dropped: the determinant
$\det(1-\phi;\mathsf B_\rho)$ vanishes as soon as $\phi$ has the
eigenvalue $1$. In the application the puncture is chosen over a
closed point of the original curve
(Lemma~\ref{fl:lem:closed-return-puncture}), so that
Theorem~\xref{I}{fl:thm:arithmetic-strong-purity} applies to $g_D$.
\end{remark}