\section{Introduction}
\label{fl:sec:introduction}

Let \(F\) be the function field of a curve over a finite field and
let \(G\) be a split reductive group over \(F\). Vincent Lafforgue
decomposed the space of cusp forms on \(G\) according to semisimple
Langlands parameters
\(\Gamma_F\to\widehat G(\overline{\mathbb Q}_\ell)\). His theorem
attaches a parameter to each summand of the decomposition. It does
not say which parameters occur.

This paper and its two sequels \cite{HuII,HuIII} address the
converse. We show that every semisimple
elliptic parameter is the parameter of a cusp form on \(G\), for
classical groups in every positive characteristic and for the
exceptional groups once the characteristic avoids a short list of
primes. Three features of the result should be
stated at once. The cusp form lives over \(F\) itself: neither the
field nor its field of constants is extended. The parameter may be
ramified at finitely many places and may have disconnected, even
finite, monodromy. And the parameter is recovered in the strongest
available sense, through every excursion operator, so the form
determines \(\sigma\) up to conjugacy and not only the conjugacy
classes of its Frobenius elements.

\subsection{Elliptic parameters}

There are three theorems.
Theorem~\ref{fl:thm:semisimple-elliptic} concerns elliptic
parameters of semisimple groups, which are realized by cusp forms.
Theorem~\ref{fl:thm:main} concerns all semisimple parameters of
reductive groups. A parameter which is not elliptic is realized by
an automorphic form which is not cuspidal, and the definitions
which precede the
theorem describe the space in which this form lives.
Theorem~\ref{fl:thm:central-period-realization} says which lattices
in the centre are compatible with a given parameter. The first
theorem is stated in this subsection, and
Section~\ref{rw:sec:coverage} says how it is proved; a reader
interested only in the elliptic case can stop there.

Let \(X/\mathbb F_q\) be a smooth projective geometrically connected
curve, let \(F=\mathbb F_q(X)\), and put \(p=\operatorname{char}F\).
Fix a prime \(\ell\ne p\), put \(k=\overline{\mathbb Q}_\ell\), and
write \(\Gamma_F=\operatorname{Gal}(F^{\mathrm{sep}}/F)\). For a
split connected reductive group \(G/F\), the dual group
\(\widehat G\) is taken over \(k\).

\begin{definition}\label{fl:def:intro-parameter}
A \emph{parameter} is a continuous homomorphism
\(\sigma:\Gamma_F\to\widehat G(k)\) unramified at all but finitely
many places. It is \emph{semisimple} if the identity component of
the Zariski closure of its image is reductive. It is \emph{elliptic}
if
\[
 C_{\widehat G}(\operatorname{im}\sigma)/Z(\widehat G)
 \quad\text{is finite}.
\]
\end{definition}

The link between cusp forms and parameters is given by excursion
operators. An excursion datum consists of a finite set \(I\), a
function
\(f\in k[\widehat G\backslash\widehat G^I/\widehat G]\), invariant
under simultaneous left and right multiplication, and a tuple
\(\boldsymbol\gamma\in\Gamma_F^I\). Lafforgue's operator
\(S_{I,f,\boldsymbol\gamma}\) acts on cusp forms: it creates an
invariant vector in the cohomology of shtukas with legs indexed by
\(I\), transports it by \(\boldsymbol\gamma\), and contracts with an
invariant covector representing \(f\)
\cite[\S\S0.3--0.5]{VLa18}. These operators commute, and a cusp
form \(\varphi\) has parameter \(\sigma\) when
\begin{equation}\label{fl:eq:main-full-label}
 S^{\mathrm{fun}}_{I,f,\boldsymbol\gamma}\varphi
 =f\bigl((\sigma(\gamma_i))_{i\in I}\bigr)\varphi
 \qquad\text{for every excursion datum }(I,f,\boldsymbol\gamma).
\end{equation}
We write \(\nu_\sigma(I,f,\boldsymbol\gamma)\) for the scalar on the
right. A nonzero function \(\varphi\) which satisfies
\eqref{fl:eq:main-full-label} is called an \emph{automorphic
realization} of \(\sigma\), and we say that it \emph{realizes}
\(\sigma\). The superscript in \(S^{\mathrm{fun}}\) records that we need
these operators on functions that are not yet known to be cuspidal;
on cusp forms \(S^{\mathrm{fun}}\) is Lafforgue's operator, and the
extension is constructed in
Section~\ref{fl:sec:compact-foundations}.

The characteristic enters through the following bound. For a simple
root system put
\[
\begin{array}{c|cccccc}
\Phi&A_n,B_n,C_n,D_n&G_2&F_4&E_6&E_7&E_8\\ \hline
b(\Phi)&0&3&3&5&7&7.
\end{array}
\]
For a reductive group, \(b(G)\) is the maximum of \(b\) over the
simple factors of its derived group, and \(b(G)=0\) for a torus.

\begin{maintheorem}[Elliptic parameters]\label{fl:thm:semisimple-elliptic}
Suppose that \(G\) is split semisimple and \(p>b(G)\). Let
\(\sigma:\Gamma_F\to\widehat G(k)\) be a semisimple elliptic
parameter. Then there are an open compact subgroup
\(K\subset G(\mathbb A_F)\) and a nonzero cusp form
\(\varphi_\sigma\) on \(G(F)\backslash G(\mathbb A_F)/K\) satisfying
\eqref{fl:eq:main-full-label}.
\end{maintheorem}

In the language of Lafforgue's decomposition, the theorem says that
the excursion summand of the space of cusp forms with character
\(\nu_\sigma\) is nonzero at some finite level; a nonzero
finite-dimensional excursion summand always contains a joint
eigenvector, so the two formulations are equivalent. The theorem
covers every group in the isogeny class of a semisimple group, not only the
simply connected and adjoint ones.

In classical terms the theorem has the following consequence. We
write $\Frob_v$ for a geometric Frobenius element at a place $v$.

\begin{corollary}[Hecke eigenvalues]\label{rw:cor:intro-hecke}
In the situation of Theorem~\ref{fl:thm:semisimple-elliptic} there
are a finite set $S$ of places of $F$ and a cuspidal automorphic
representation $\pi$ of $G(\mathbb A_F)$ which is unramified
outside $S$ and whose Satake parameter at each place $v\notin S$ is
the conjugacy class of the semisimple part of $\sigma(\Frob_v)$.
\end{corollary}

The corollary is proved at the end of
Section~\ref{rw:sec:proof-semisimple}. It is weaker than the
theorem: for some groups the
conjugacy classes of the elements $\sigma(\Frob_v)$ do not
determine $\sigma$ up to conjugacy; the excursion operators do.

Two instances may help to fix ideas. If \(G\) is the split group of
type \(E_8\) and \(p>7\), every semisimple elliptic homomorphism
\(\Gamma_F\to E_8(k)\) is the parameter of a cusp form on \(G\).
If \(G=\mathrm{Sp}_{2n}\), the same holds for parameters into
\(\mathrm{SO}_{2n+1}(k)\) in every characteristic, including
\(p=2\). A parameter over \(\mathbb F_5(t)\) with image
\(S_3\subset\mathrm{PGL}_2(k)\) is followed through the first steps
of the geometric construction in
Sections~\xref{II}{fl:sec:low-rank-example}
and~\xref{II}{rw:sec:example-curve}.

\begin{remark}[The characteristic restriction]
\label{rw:rem:characteristic-restriction}
For the types \(G_2,F_4,E_6,E_7,E_8\), the condition \(p>b(G)\) is
the same as \(p\nmid|W_G|\). It is a hypothesis of our geometric
method, in which it supplies nondegenerate invariant forms on the
Lie algebra and its Levi subalgebras and keeps wild inertia inside
tori when the parameter is lifted to characteristic zero. We do not
expect it to reflect an obstruction to automorphy. For classical
groups a second method removes it. For elliptic parameters of the
exceptional types in the excluded characteristics, that is, for
\(G_2\) and \(F_4\) in characteristics two and three, for \(E_6\)
in characteristics two, three and five, and for \(E_7\) and \(E_8\)
in characteristics two, three, five and seven, the theorems make no
assertion in either direction.
\end{remark}

\subsection{The two existence theorems}
\label{rw:sec:coverage}

Theorem~\ref{fl:thm:semisimple-elliptic}, and with it
Theorems~\ref{fl:thm:main}
and~\ref{fl:thm:central-period-realization} below, are deduced in
this paper from two existence theorems for semisimple groups, which
are proved in the two sequels.

\begin{theorem}[Geometric construction
{\cite[Theorem~\xnum{II}{fl:thm:weyl-prime-realization}]{HuII}}]
\label{rw:thm:intro-geometric}
Let \(G\) be split, simply connected and semisimple, and suppose
that \(p\nmid|W_G|\). Then every semisimple elliptic parameter
\(\sigma:\Gamma_F\to\widehat G(k)\) is the parameter of a cusp form
on \(G\): the conclusion of
Theorem~\ref{fl:thm:semisimple-elliptic} holds.
\end{theorem}

\begin{theorem}[Classical groups
{\cite[Theorem~\xnum{III}{fl:thm:classical-realization}]{HuIII}}]
\label{rw:thm:intro-classical}
Let \(G\) be split semisimple with simple factors of types
\(A\), \(B\), \(C\), \(D\), in any positive characteristic. Then
the conclusion of Theorem~\ref{fl:thm:semisimple-elliptic} holds
for every semisimple elliptic parameter of \(G\).
\end{theorem}

\begin{center}
\begin{tabular}{p{0.27\textwidth}p{0.4\textwidth}p{0.22\textwidth}}
\toprule
type of the simple factors & construction & condition on $p$\\
\midrule
$A_n$, $B_n$, $C_n$, $D_n$ &
descent and theta correspondence \cite{HuIII} &
none\\
$G_2$, $F_4$, $E_6$, $E_7$, $E_8$ &
geometric \cite{HuII} &
$p\nmid|W_G|$\\
\bottomrule
\end{tabular}
\end{center}

The proof of Theorem~\ref{rw:thm:intro-geometric} starts from the
geometric Langlands correspondence in characteristic zero. It
provides a sheaf, and the function which the trace of Frobenius
attaches to a nonzero sheaf can vanish identically. The main point
is to arrange that the trace in question is a determinant
\(\det(1-\phi)\) with no zero factor, and that it stays nonzero
under specialization to the given field; an overview is given in
\cite[Section~\xnum{II}{fl:sec:strategy}]{HuII}. The construction
applies to every simply connected group with \(p\nmid|W_G|\), but
it is needed only for the five types in the second row of the
table. The proof of Theorem~\ref{rw:thm:intro-classical} starts
from L.~Lafforgue's theorem for general linear groups and uses
automorphic descent and theta correspondence; an overview is given
in \cite[Section~\xnum{III}{rw:sec:descent-route}]{HuIII}.

The two sequels are independent of each other. Each uses
Sections~\ref{fl:sec:arithmetic-purity}--\ref{fl:sec:arithmetic-central}
of the present paper, and neither uses
Theorems~\ref{fl:thm:semisimple-elliptic}--\ref{fl:thm:central-period-realization}
or
Sections~\ref{fl:sec:minimal-levi}--\ref{fl:sec:reductive-proof}.

\subsection{Parameters which are not elliptic}
\label{rw:sec:general-parameters}

A parameter that is not elliptic factors through a proper Levi
subgroup of \(\widehat G\), and an automorphic form with this
parameter should come from the corresponding Levi subgroup of \(G\)
by Eisenstein series. A minimal dual Levi \(\widehat M\) containing
\(\operatorname{im}\sigma\) exists, and \(\sigma\) is elliptic as a
parameter into \(\widehat M\)
(Proposition~\ref{fl:prop:reductive-centralizer}). The resulting
forms are sought in a space larger than the space of cusp forms,
which we now describe.

By an \emph{automorphic form} we mean a smooth function on
\(G(F)\backslash G(\mathbb A_F)\), invariant under an open compact
subgroup, whose orbit under the spherical Hecke algebra at one
unramified place is finite-dimensional; such a function is
automorphic in the usual sense
(Proposition~\ref{fl:prop:one-place-automorphicity}). Excursion
operators with
arbitrary Galois tuples are not known to act on all such functions.
They do act on those that are invariant under a lattice in the
centre and satisfy an integrality condition for the Hecke action;
integrality is what makes the action of the Weil group extend
continuously to the Galois group. The space
\(\mathcal A^{\mathrm{cont}}(G)\) consists of the
automorphic forms satisfying these two conditions, together with their
twists by characters of the constant field; it is defined in
Definition~\ref{fl:def:parameter-independent-spaces}, and
Theorem~\ref{fl:thm:function-space} gives the action of all
excursion operators on it.

\begin{maintheorem}[Reductive groups]\label{fl:thm:main}
Let \(G/F\) be split connected reductive and let
\(\sigma:\Gamma_F\to\widehat G(k)\) be a semisimple parameter.
Let \(\widehat M\) be a minimal dual Levi containing its image,
and let \(M\) be the corresponding split Levi subgroup of \(G\).
If \(p>b(M)\), there is a nonzero automorphic form
\(\varphi_\sigma\in\mathcal A^{\mathrm{cont}}(G)\) satisfying
\eqref{fl:eq:main-full-label}. If \(\sigma\) is elliptic,
\(\varphi_\sigma\) can be chosen cuspidal; if \(\sigma\) is not
elliptic, \(\varphi_\sigma\) can be chosen not cuspidal.
\end{maintheorem}

When \(\sigma\) is not elliptic, the form \(\varphi_\sigma\) is
not given as a value or a residue of an Eisenstein series induced
from the cuspidal form on \(M\). Its existence is deduced from the
compatibility of the excursion operators with constant terms, by
Nakayama's lemma applied to the functions with compact support
modulo the cuspidal ones (Section~\ref{fl:sec:eisenstein}). The
form obtained in this way is orthogonal to the cuspidal functions
with compact support, and this is why it is not cuspidal.

The twists by characters of the constant field cannot be omitted
from the definition of \(\mathcal A^{\mathrm{cont}}(G)\): when the
centre of \(G\) has positive dimension, the central character of a
continuous parameter need not have finite order.

The bound \(p>b(M)\) depends on the parameter and can be weaker
than \(p>b(G)\). Every proper Levi subgroup of \(E_6\), for example,
has only classical simple factors, so a nonelliptic
\(E_6\)-parameter is realized in every positive characteristic,
while our construction for elliptic \(E_6\)-parameters needs
\(p>5\). Proposition~\ref{fl:prop:levi-root-bound} gives the
comparison for all root systems.

\subsection{A prescribed quotient by the centre}

One often works on the quotient of
\(G(F)\backslash G(\mathbb A_F)\) by a lattice in the centre, and
it is natural to ask which lattices are compatible with a given
parameter. Let \(T_G=(Z_G^\circ)_{\mathrm{red}}\) and
\(C_{T_G}=T_G(F)\backslash T_G(\mathbb A_F)\). A \emph{central lattice} is a free subgroup \(\Xi\subset C_{T_G}\) on which
the adelic degree map
\(\deg:C_{T_G}\to X_*(T_G)\) is injective with image of finite
index. Dual to the inclusion \(T_G\to G\) there is a map
\(\alpha_G:\widehat G\to\widehat T_G\), and class field theory
attaches to \(\sigma\) the character
\[
 \omega_\sigma=\operatorname{CFT}_{T_G}(\alpha_G\sigma)
\]
of \(C_{T_G}\); our normalization sends a parameter \(\chi\) of
\(\mathbb G_m\) to \(t\mapsto\chi(\operatorname{Art}(t)^{-1})\),
with \(\operatorname{Art}\) the arithmetic Artin map
(Section~\ref{fl:sec:parameters-parameters}). Let
\(\mathcal A_\Xi(G)\subset\mathcal A^{\mathrm{cont}}(G)\) be the
subspace of \(\Xi\)-invariant functions satisfying the
integrality condition, without twists
(Definition~\ref{fl:def:parameter-independent-spaces}).

\begin{maintheorem}[Central lattices]
\label{fl:thm:central-period-realization}
Under the hypotheses of Theorem~\ref{fl:thm:main}, let
\(\Xi\subset C_{T_G}\) be a central lattice.
\begin{enumerate}
\item If \(\omega_\sigma|_\Xi=1\), the form \(\varphi_\sigma\) of
Theorem~\ref{fl:thm:main} can be chosen in \(\mathcal A_\Xi(G)\).
\item If \(\sigma\) is elliptic, the excursion summand with
character \(\nu_\sigma\) in the space of cusp forms on
\(G(F)\backslash G(\mathbb A_F)/\Xi\) is nonzero if and only if
\(\omega_\sigma|_\Xi=1\).
\end{enumerate}
\end{maintheorem}

Necessity in (2) is the statement that a cusp form with parameter
\(\sigma\) has central character \(\omega_\sigma\). Sufficiency
rests on the existence theorems. The three theorems are proved
together in Section~\ref{fl:sec:reductive-proof}.

\subsection{Relation to earlier work}
\label{rw:sec:earlier-work}

\emph{General linear groups and excursion operators.}
Laurent Lafforgue proved the correspondence for irreducible
\(\mathrm{GL}_n\)-parameters with finite-order determinant
\cite[Introduction, Theorem \textup{(i)}$_n$--\textup{(ii)}$_n$]{LLa02}.
For a split reductive group, Vincent Lafforgue constructed the
decomposition of cusp forms by parameters \cite[Theorem~0.1]{VLa18}.
That theorem identifies the parameters of the summands that occur;
that the summand of a prescribed parameter is nonzero is a separate
assertion, and it is the subject of this paper. For a split
semisimple group and a parameter which is everywhere unramified
and has Zariski dense image, \cite[Theorem~1.1]{BHKT2019} gives,
after a finite Galois extension of the function field, a cuspidal
representation whose unramified components have the Satake
parameters prescribed by the parameter. Our
conclusion keeps the field fixed.

\emph{Geometric Langlands.}
The geometric input is the restricted geometric Langlands
equivalence of Gaitsgory and Raskin over an algebraically closed
field of characteristic zero \cite[Theorem~1.3.9(ii)]{GR2025},
together with its Whittaker functional. In positive characteristic
their theorem gives an equivalence on a union of connected
components of the spectral side, and whether this union is
everything is their Conjecture~1.3.10. The subsequent preprint
\cite[Theorem~1.4]{OpenAIRestricted2026}, by OpenAI, states that it
is, over \(\overline{\mathbb F}_q\) under the four Lie-theoretic
assumptions of the restricted theory, and over an arbitrary
algebraically closed field when in addition \(p\) is very good and
\(p\nmid|W_G|\); its Corollary~8.2 gives the arithmetic formula for
unramified parameters. We do not use that theorem. Our result
concerns arithmetic parameters over \(F\) that are ramified at
finitely many places and may have finite or disconnected monodromy,
and it produces functions, not sheaves.

Our method does, however, owe several ideas to the three Langlands
preprints
\cite{OpenAIRestricted2026,OpenAIConstructible2026,OpenAIFrobenius2026}
in OpenAI's public \emph{math} repository. Testing a Hecke
eigensheaf against a hypersurface transverse to the nilpotent cone,
letting modification points collide, and working with framed
bundles all appear there and are starting points for
\cite{HuII}. The statements of those preprints have
different hypotheses from ours: the tame eigensheaf theorem
\cite[Theorem~1.1]{OpenAIConstructible2026} assumes geometrically
dense monodromy and one tame regular-unipotent puncture, and the
Frobenius theorem \cite[Theorem~1.1]{OpenAIFrobenius2026} treats
\(\mathrm{SL}_n\) with \(p>n\) and allows a finite extension of
constants. None of them is used as an input.

The steps of the geometric argument which are new, and the further
works on which it draws, are described in
\cite[Section~\xnum{II}{rw:sec:earlier-work}]{HuII}.

\emph{Descent and theta correspondence.}
The second construction starts from automorphy for general linear
groups and uses automorphic descent \cite{GRS1999,GRS2011}, the
theta correspondence \cite{Rallis1984}, and the theory of
automorphic $L$-functions over function fields
\cite{Lomeli2009,Lomeli19}.
The work here is to prove nonvanishing over \(F\), to carry out the
necessary coordinate calculations in small characteristic, in
particular in characteristic two, and to identify the parameter
itself and not only its unramified Hecke eigenvalues.

\emph{What is not proved.}
The theorems assert that the excursion summand of \(\sigma\) is
nonzero at some finite level; the form \(\varphi_\sigma\) is a
nonzero element of it, and it is not unique. We do not determine
multiplicities, the structure of
packets, or compatibility with the local correspondence at the
ramified places.

\subsection{Organization}

Section~\ref{rw:sec:inputs} lists the results we import, and
Section~\ref{fl:sec:parameters-parameters} fixes conventions. The
next seven sections contain the arithmetic used by both
constructions: purity of elliptic parameters
(Section~\ref{fl:sec:arithmetic-purity}); excursion operators on
spaces of functions that are not assumed cuspidal
(Sections~\ref{fl:sec:compact-foundations}
and~\ref{rw:sec:function-spaces-section}); constant terms
(Section~\ref{fl:sec:constant-term}); criteria for cuspidality
(Section~\ref{fl:sec:arithmetic-cuspidality}); irreducible
constituents (Section~\ref{fl:sec:unitary-extraction}); and transfer
between the groups of an isogeny class
(Section~\ref{fl:sec:arithmetic-central}).
Sections~\ref{fl:sec:minimal-levi} and~\ref{fl:sec:eisenstein}
treat Levi subgroups and Eisenstein series, and
Section~\ref{fl:sec:reductive-proof} proves
Theorems~\ref{fl:thm:semisimple-elliptic}--\ref{fl:thm:central-period-realization}
from Theorems~\ref{rw:thm:intro-geometric}
and~\ref{rw:thm:intro-classical}.
Appendix~\ref{rw:app:partial-frobenius} contains the proof of a
lemma on partial Frobenius maps which is used in
Section~\ref{fl:sec:constant-term}. A list of notation is at the
end of the paper.

The elliptic case is shorter than the general one.
Theorem~\ref{fl:thm:semisimple-elliptic} uses from
Section~\ref{fl:sec:constant-term} only the elementary
Lemma~\ref{rw:lem:ct-spherical}, and the elliptic case of
Theorem~\ref{fl:thm:main} uses in addition
Proposition~\ref{fl:prop:reductive-centralizer}.
Theorem~\ref{fl:thm:normalized-ct},
Appendix~\ref{rw:app:partial-frobenius} and
Section~\ref{fl:sec:eisenstein} are needed only for parameters which
are not elliptic.

A reader interested mainly in one of the two constructions needs
only a few statements of this paper. For the geometric
construction it is enough to know the definition of the
operators $S_e^{\mathrm{fun}}$
(Definition~\ref{fl:def:function-excursion-action}) and the
statements of Theorem~\ref{fl:thm:arithmetic-strong-purity}
(elliptic parameters are pure) and of
Proposition~\ref{fl:prop:cuspidal-excursion-character} (a function
whose pairing with functions of compact support has the excursion
eigenvalues of an elliptic parameter is a cusp form with that
parameter). For the construction by descent one needs in addition
the statements of
Lemma~\ref{fl:lem:unitary-admissible-semisimplicity} and of
Theorem~\ref{fl:thm:arithmetic-central-bridge}. Each sequel lists
what it uses from this paper.