\section{Introduction}
\label{rw:sec:III-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.

For general linear groups every irreducible parameter whose
determinant has finite order occurs, by the theorem of Laurent
Lafforgue. We prove that every semisimple
elliptic parameter occurs when the simple factors of \(G\) are of
classical type, in every positive characteristic. The proof
transports cusp forms from general linear groups to \(G\) by
automorphic descent and by the theta correspondence. This is the
third paper of a series of three. The first \cite{HuI} contains
the arithmetic foundations and deduces from the present theorem,
and from its counterpart for the exceptional types \cite{HuII},
the existence of automorphic realizations of semisimple parameters on
all split reductive groups, under the conditions on the
characteristic stated there, which concern only the exceptional
factors.

\subsection{The theorem}

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 cusp form \(\varphi\) 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
\cite[Section~\xnum{I}{fl:sec:compact-foundations}]{HuI}.

\begin{theorem}\label{rw:thm:III-main}
Let \(G/F\) be split semisimple with simple factors of types
\(A\), \(B\), \(C\), \(D\), in any positive characteristic. 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{theorem}

This is Theorem~\ref{fl:thm:classical-realization}. The cusp form
lives over \(F\) itself: neither the field nor its field of
constants is extended. The theorem covers every group in an
isogeny class, not only the simply connected and adjoint ones, and
every characteristic, including \(p=2\). The parameter is recovered
through every excursion operator. This is more than the equality
of Hecke eigenvalues at the unramified places which the descent
method gives directly, and for the even orthogonal groups, where
the standard representation determines a parameter only up to an
outer automorphism, the difference matters.

The function \(\varphi_\sigma\) is a nonzero element of the
excursion summand of \(\sigma\), 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{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 series of papers. 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{Descent and theta correspondence.}
The 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.

In characteristic two the orthogonal groups are defined by
quadratic forms, and one step of the even orthogonal descent, which
in odd characteristic divides a quadratic exponent by two, is
replaced by an argument with the Cartier operator
(Section~\ref{fl:sec:char2-d}).

\subsection{Organization}

Section~\ref{rw:sec:descent-route} explains the construction
without proofs, Section~\ref{rw:sec:inputs} lists the results we
import, including those of \cite{HuI}, and
Section~\ref{fl:sec:parameters-parameters} fixes conventions.
Section~\ref{fl:sec:classical-descent} contains what is common to
all cases: how the parameter is recovered from its standard
representation, and the tools for comparing Fourier coefficients.
Section~\ref{fl:sec:common-descent} constructs the Eisenstein
residue. Sections~\ref{fl:sec:orthogonal-descent},
\ref{fl:sec:odd-symplectic} and~\ref{fl:sec:char2-d} treat the
orthogonal descents, and
Section~\ref{rw:sec:orthogonal-realization} deduces the theorem for
orthogonal and spin groups. Sections~\ref{fl:sec:odd-theta}
and~\ref{fl:sec:classical-double-line} treat the symplectic groups
by the theta correspondence, and Section~\ref{fl:sec:CLA-proof}
assembles the cases. A list of notation is at the end, and
Appendix~\ref{rw:app:computer} lists the identities which were
also checked by computer.