\section{Overview of the construction}
\label{fl:sec:strategy}

This section describes the proof of
Theorem~\ref{rw:thm:II-main} without giving it. Its main
difficulty is not visible from the statement, and the first
subsection explains it. The last subsection lists the statements in
which the steps are carried out and shows how they depend on one
another.

\subsection{The difficulty: a nonzero sheaf can have zero trace}

Geometric Langlands produces sheaves, and we need a function. The
function attached to a complex of sheaves \(\mathcal K\) with an
action of Frobenius is the alternating sum of the traces of
Frobenius on the cohomology of its stalks. This function can vanish
identically for a nonzero \(\mathcal K\): on a point, the complex
\(k\oplus k[-1]\) with the identity as Frobenius already has trace
zero. So a theorem that produces a nonzero Hecke eigensheaf does
not, by itself, produce a nonzero automorphic function. To exclude
this cancellation, over the given finite field and without
enlarging it, is the central problem of the geometric construction.

The way out is to arrange that the alternating trace in question
is a determinant. If \(\phi\) is an endomorphism of a
finite-dimensional vector space \(V\), then the alternating trace
of \(\phi\) on the exterior algebra of \(V\), with \(V\) placed in
degree \(-1\), is
\[
 \sum_i(-1)^i\operatorname{tr}(\phi;\Lambda^iV)=\det(1-\phi;V).
\]
A determinant is nonzero as soon as no eigenvalue of \(\phi\) is
equal to one, and this is a condition that purity can verify.

\subsection{Where the exterior algebra comes from}

The parameter \(\sigma\) lives on an open part of the curve \(X\)
in characteristic \(p\), and may be ramified. The first step of
the construction (Section~\ref{fl:sec:geometrization}) replaces
this situation by an unramified one in characteristic zero: there
is a smooth projective curve \(C\) over an algebraically closed
field of characteristic zero, with an everywhere unramified local
system \(\rho\). This curve degenerates, through a nodal curve in
characteristic zero, to a curve with two components in
characteristic \(p\), which is geometrically reduced and
Gorenstein. One component is \(X\), with the parameter
\(\sigma\); the other is auxiliary. On \(C\) the
geometric Langlands equivalence of Gaitsgory and Raskin is
available. It attaches to a coherent sheaf \(P\) on the space of
local systems an automorphic sheaf \(\mathcal F_P\) on the stack
of \(G\)-bundles on \(C\), and it identifies a certain functional
of \(\mathcal F_P\), its Whittaker coefficient, with the space of
global sections of \(P\).

It remains to choose \(P\). The obvious choice is the structure
sheaf of the residual gerbe of \(\rho\), whose space of global
sections is \(k\). But its direct image on the stack of local
systems need not be compact, and compactness is used twice below:
for constructibility, and for the formula which expresses the
Whittaker coefficient by ordinary global sections. Instead
we remove a few points from the auxiliary component, fix the
restriction of \(\rho\) to the punctured curve, and consider all
ways of extending it across the punctures. As a set this space of
extensions is small. As a derived space it also remembers the
equations which say that the monodromy around each puncture is
trivial, one equation for each direction in the Lie algebra, and
these equations contribute exterior generators: the space is
\[
 \mathfrak F_\rho
   \simeq[\operatorname{Spec}\Lambda(\mathsf B_\rho[1])/Z_\rho],
\]
where \(\mathsf B_\rho\) is the sum of one copy
\(\widehat{\mathfrak g}_x^*\) of the dual of the Lie algebra of
\(\widehat G\) for each point \(x\) which is removed, and
\(Z_\rho\) is the centralizer of the monodromy of \(\rho\)
(Section~\ref{fl:sec:punctured-spectral-fibre}). For an orbit
of length \(d\) of Frobenius on the points which are removed, and
a point \(x\) of the orbit, the \(d\)-th power of Frobenius acts on
the summand \(\widehat{\mathfrak g}_x^*\) by \(q^d\) times
the coadjoint action of the image \(g_D\) of Frobenius under the
parameter. The trace we need is therefore, up to the corrections
discussed in the next paragraph,
\[
\det\bigl(1-q^d\operatorname{Ad}(g_D)^\vee;\widehat{\mathfrak g}_x^*\bigr),
\]
and it is not zero: by the purity theorem of
Section~\xref{I}{fl:sec:arithmetic-purity} the eigenvalues of
\(\operatorname{Ad}(g_D)\) have absolute value one, so none of
them is \(q^{-d}\). In the example of
Section~\ref{fl:sec:low-rank-example}, where \(q=5\), \(d=1\),
\(g_D=1\) and the Lie algebra has dimension three, this determinant
is \((1-5)^3=-64\).

Two corrections are needed because of the quotient by \(Z_\rho\).
Global sections on the quotient are the invariants of \(Z_\rho\),
and taking invariants could reintroduce cancellation. The group of
components \(\Pi_\rho\) of \(Z_\rho\) is finite; we tensor the
structure sheaf with its regular representation,
\[
 P_{\mathrm{reg}}
   =\mathcal O_{\mathfrak F_\rho}\otimes k[\Pi_\rho],
\]
because the invariants of a finite group in a representation
tensored with the regular representation are the representation
itself. The identity component of \(Z_\rho\) is a torus, since
\(\sigma\) is elliptic, and Frobenius fixes no nontrivial character
of it; so the parts of nonzero weight, which the invariants
discard, would not have contributed to the trace in any case. The
result is Proposition~\ref{fl:prop:puncture-exact-trace}.

\subsection{Why the comparison is made for all coefficients}

We have computed a trace on the side of local systems, and we need
it on the side of automorphic sheaves, in characteristic \(p\).
The equivalence of Gaitsgory and Raskin is an equivalence over an
algebraically closed field. It identifies the two vector spaces,
but it says nothing about the action of Frobenius on them, and an
isomorphism of vector spaces that does not respect Frobenius is of
no use for traces. If we transported only the object
\(P_{\mathrm{reg}}\), the two actions of Frobenius could differ by
an arbitrary automorphism.

For this reason we transport the functor and not one object. On
the side of local systems the functor is
\[
 \Gamma_\rho:
 \operatorname{Perf}(\mathfrak F_\rho)\longrightarrow
 \operatorname{Vect}_k,\qquad
 P\longmapsto\Gamma(\mathfrak F_\rho,P),
\]
and on the automorphic side it is \(P\mapsto\) the Whittaker
coefficient of \(\mathcal F_P\). The two Frobenius actions then
differ by an automorphism of the functor \(\Gamma_\rho\), and such
automorphisms are very few: by the Yoneda lemma they are given by
the invertible functions of degree zero on \(\mathfrak F_\rho\),
which are the nonzero constants. So the two actions differ by one
scalar \(c\in k^\times\), the same for all \(P\) and in all
degrees (Lemma~\ref{fl:lem:scalar-coefficient-covariance}). A
scalar multiplies the alternating trace of \(P_{\mathrm{reg}}\) by
a nonzero number and cannot make it vanish.

\subsection{From characteristic zero to characteristic \texorpdfstring{$p$}{p}}

The price is that the whole family \(P\mapsto\mathcal F_P\), with
all the maps between its members and all its Hecke identities, has
to be carried from the smooth curve \(C\) in characteristic zero to
the curve with two components in characteristic \(p\). There are
two degenerations, the formation of the nodes and the reduction
modulo \(p\), and we treat them together over a valuation ring of
rank two (Section~\ref{fl:sec:hecke-extension}).

The family is extended by relative nearby cycles. To compare its
Whittaker coefficient before and after specialization, we replace
this coefficient by a local
invariant, the vanishing cycles of \(\mathcal F_P\) for a suitable
function \(f\) on the stack of bundles at a suitable point, and
we must show that this invariant is the same before and after
specialization. It could change only at points where the
differential of \(f\) lies in the singular support of the
sheaves, in a suitable relative sense. For Hecke eigensheaves over
a field the singular support is contained in the nilpotent cone;
Theorem~\ref{fl:thm:noncharacteristic-hecke-comparison} is a form
of this statement over a valuation ring: the restriction of the
family to the hypersurface \(f=0\) still specializes well at
every point where the differential of \(f\) is not nilpotent.
The proof moves the point of a Hecke modification until the
direction in question becomes visible there. The Hecke identity
controls only the direct image of the sheaf along a proper map from
the Hecke correspondence, and the theorem is a statement about one
stalk before the direct image is taken. So one argument isolates
the stalk inside a fibre of the correspondence
(Lemma~\ref{rw:lem:isolated-hecke-contribution}), and another
removes the auxiliary variables introduced on the way
(Lemmas~\ref{rw:lem:exact-hyperbolic-coordinates}
and~\ref{fl:lem:quadric-detection}). One nilpotent
direction remains, the regular one. There \(f\) is chosen
homogeneous of weight two for an action of the multiplicative
group, and the part of the vanishing cycles on which \(-1\) acts
by \(-1\) is shown to be the same at the three points of the base
(Section~\ref{fl:sec:conservation}).

\subsection{Returning to the original curve and field}

At this stage we know that a vanishing cycle of the sheaf in
characteristic \(p\) has nonzero Frobenius trace. This is still
not a value of the function. Because \(f\) is homogeneous, the
trace on its vanishing cycles can be written as a sum of traces
on stalks of the sheaf, at points defined over \(\mathbb F_q\),
some of them twisted by the action of \(-1\). The twisted terms
are ordinary traces at other rational points: if \(x\) satisfies
\(-\operatorname{Fr}_q(x)=x\) and \(a^{q-1}=-1\), then \(ax\) is
fixed by \(\operatorname{Fr}_q\), and the equivariance of the sheaf
identifies the twisted trace at \(x\) with the ordinary trace at
\(ax\). So if the function vanished at every rational point, the
trace on the vanishing cycles would vanish too. Hence the function
is not zero at some bundle defined over \(\mathbb F_q\); no
extension of the finite field is made.

That bundle lives on the curve with two components. We restrict
it to the auxiliary component, keep this restriction fixed, and
let the bundle on \(X\) vary. The result is a nonzero function on
bundles on \(X\) with a level structure along the points where the
components meet (Section~\ref{fl:sec:exceptional-assembly}). The
order of the two steps matters: the auxiliary bundle is the one
found by the trace argument, and an arbitrary choice made
beforehand would not be known to give a nonzero function.

\subsection{The parameter}

It remains to see that this function has parameter \(\sigma\), in
the sense of \eqref{fl:eq:main-full-label}. Its Hecke eigenvalues
at the unramified places are those of \(\sigma\), because it comes
from a Hecke family attached to \(\sigma\). But for a general group
the Hecke
eigenvalues do not determine the parameter, and
\eqref{fl:eq:main-full-label} asks for more: the eigenvalues of
all excursion operators. An excursion operator is built from the
cohomology of spaces of shtukas with several legs, and its three
steps, creation, action of Galois elements, and annihilation, have
counterparts for the family of sheaves: Hecke functors with
several legs and their compatibilities when legs collide. Taking
traces of Frobenius of these compatibilities gives the excursion
equations (Theorem~\ref{fl:thm:arithmetic-family-trace}). They
come out for the Chevalley dual of the parameter of the sheaf, so
the construction is applied to the Chevalley dual of $\sigma$.
This is
why the Hecke identities of the family, with all legs and all
collisions, were carried through every step above.

Two further points complete the argument. The trace argument
gives the excursion equations for tuples of elements of the Weil
group. The function is cuspidal. Otherwise one of its constant
terms along a proper parabolic subgroup would have a parameter
with values in a proper Levi subgroup of the dual group, and an
elliptic parameter is not of this form
(Section~\xref{I}{fl:sec:arithmetic-cuspidality}). On the space of
cusp forms the equations extend to tuples in the Galois group by
continuity
(Corollary~\xref{I}{fl:cor:weil-to-galois-nonzero}).

\subsection{The statements and their dependencies}
\label{rw:sec:dependencies}

\begin{figure}[tbp]
\centering
\begin{tikzpicture}[
 box/.style={draw,rounded corners=2pt,align=center,text width=5.9cm,
             inner sep=6pt,font=\small},
 >=Stealth]
 \node[box] (coeff) {Coefficient and Frobenius trace\\
   Proposition~\ref{fl:prop:puncture-exact-trace}};
 \node[box,right=11mm of coeff] (curve) {Lift and smooth curve\\
   Theorem~\ref{fl:thm:marked-lifting};
   Proposition~\ref{fl:prop:geometrized-parameter}};
 \node[box,below=12mm of curve] (family) {The entire Hecke family over \(S\)\\
   Theorem~\ref{fl:thm:compact-automorphic-family}};
 \node[box,below=12mm of coeff] (phase) {Transverse test function\\ and microstalk\\
   Theorem~\ref{fl:thm:noncharacteristic-hecke-comparison};
   Proposition~\ref{fl:lem:vacuum-microstalk-comparison}};
 \node[box,below=12mm of phase] (trace) {Conserved trace and rational bundle\\
   Theorems~\ref{fl:thm:scaling-character-conservation},
   \ref{fl:thm:homogeneous-detector}};
 \node[box,below=12mm of family] (excursion) {Excursion equations\\
   Theorem~\ref{fl:thm:arithmetic-family-trace}};
 \node[box,below=14mm of trace,xshift=3.55cm] (original)
   {Restriction to \(X\) and cuspidality\\
   Theorem~\ref{fl:thm:weyl-prime-realization}};
 \draw[->] (curve)--(family);
 \draw[->] (coeff)--(phase);
 \draw[->] (family)--(phase);
 \draw[->] (phase)--(trace);
 \draw[->] (family)--(excursion);
 \draw[->] (trace)--(original);
 \draw[->] (excursion)--(original);
\end{tikzpicture}
\caption{The two inputs to geometric realization: a nonzero rational
trace and the excursion equations. The final restriction uses
both.}
\label{rw:fig:geometry-dependencies}
\end{figure}

The nonzero number of the first subsections is followed through
three statements.
\begin{enumerate}
\item Proposition~\ref{fl:prop:puncture-exact-trace}: on the side of
local systems, the Frobenius supertrace on the global sections of
a certain perfect complex is a determinant, and it is nonzero by
purity.
\item Proposition~\ref{fl:prop:composite-trace-nonzero}: the
corresponding automorphic sheaf, specialized to characteristic
\(p\), has nonzero trace at some bundle defined over
\(\mathbb F_q\) on the curve with two components.
\item Proposition~\ref{fl:prop:passive-bundle-restriction}: the
trace remains nonzero after the bundle on the auxiliary component
is fixed, which gives a function on bundles on \(X\).
\end{enumerate}
Each passage must be compatible with Frobenius, and most of the
work lies in the passage from (1) to (2).
Figure~\ref{rw:fig:geometry-dependencies} shows how these and the
other main results depend on one another; an arrow means that the
result at its tail is used in the result at its head. The
calculation of the trace on the side of local systems and the
construction of the curve can be read separately, up to the
comparison under specialization. The proof of
Theorem~\ref{fl:thm:weyl-prime-realization}, in
Section~\ref{fl:sec:exceptional-assembly}, assembles these results.