\documentclass[11pt]{article}
\input{preamble}
\hypersetup{
 pdftitle={\seriestitle, III: Classical groups},
 pdfauthor={Boyang Hu}
}
\title{\seriestitle\\[6pt]\Large III. Classical groups}
\author{Boyang Hu\\[3pt]
{\normalsize Tsinghua University}}
\date{12 November 2026}
\begin{document}
\InputIfFileExists{ext-I}{}{}
\InputIfFileExists{ext-II}{}{}
\maketitle
\begin{abstract}
Let \(G\) be a split semisimple group whose simple factors are of
classical type, over a global function field \(F\) of arbitrary
positive characteristic. We prove that every semisimple elliptic
Langlands parameter
\(\Gamma_F\to\widehat G(\overline{\mathbb Q}_\ell)\) is the
parameter of a cusp form on \(G\), in the sense of V.~Lafforgue's
excursion operators; we call such a form an automorphic realization
of the parameter. Neither \(F\) nor its field of constants is
extended.

We start from L.~Lafforgue's theorem for general linear groups and
use automorphic descent and the theta correspondence. The work is
to prove that the descent does not vanish over \(F\), to carry out
the computations with root subgroups in small characteristic, in
particular in characteristic two, and to identify the parameter
itself and not only its unramified Hecke eigenvalues.

This is the third of three papers. Together with the first, which
contains the arithmetic foundations, and the second, which treats
the exceptional types by a geometric method, it gives automorphic
realizations of semisimple parameters on all split reductive groups,
with a restriction on the characteristic for the exceptional types
only.
\end{abstract}

{\small\noindent\textbf{2020 Mathematics Subject Classification.} 11F70, 11R39 (primary); 22E50, 22E55, 11F27 (secondary).

\noindent\textbf{Keywords.} Langlands parameters, function fields, automorphic descent, theta correspondence, Eisenstein series, characteristic two.\par}

\tableofcontents
\bigskip

\input{sections/intro}
\input{sections/overview}
\input{sections/inputs}
\input{sections/normalizations}
\input{sections/cls-parameters}
\input{sections/cls-residues}
\input{sections/cls-orthogonal}
\input{sections/cls-even-descent}
\input{sections/cls-char2-descent}
\input{sections/cls-orthogonal-realization}
\input{sections/cls-quadratic-theta}
\input{sections/cls-symplectic}
\input{sections/cls-assembly}
\appendix
\input{sections/appendix-computer}
\section*{Notation}
\addcontentsline{toc}{section}{Notation}
\markboth{Notation}{}
\input{sections/notation-index}
\input{sections/ai-statement}
\input{bibliography}
\end{document}