\documentclass[11pt]{article}
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\hypersetup{
 pdftitle={\seriestitle, II: The geometric construction},
 pdfauthor={Boyang Hu}
}
\title{\seriestitle\\[6pt]\Large II. The geometric construction}
\author{Boyang Hu\\[3pt]
{\normalsize Tsinghua University}}
\date{12 November 2026}
\begin{document}
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\maketitle
\begin{abstract}
Let \(G\) be a split simply connected semisimple group over a global
function field \(F\) whose characteristic does not divide the order
of the Weyl group of \(G\). 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. The parameter may be ramified at finitely many
places and may have finite monodromy, and neither \(F\) nor its
field of constants is extended.

The starting point is 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 of the construction 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.

This is the second of three papers. Together with the first, which
contains the arithmetic foundations, and the third, which treats
classical groups in every characteristic, 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.} 11R39, 14D24 (primary); 11F70, 14F20, 22E57 (secondary).

\noindent\textbf{Keywords.} Langlands parameters, function fields, geometric Langlands correspondence, Hecke eigensheaves, excursion operators, nearby cycles.\par}

\tableofcontents
\bigskip

\input{sections/intro}
\input{sections/overview}
\input{sections/inputs}
\input{sections/normalizations}
\input{sections/geometry-guide}
\input{sections/geom-coefficients}
\input{sections/example-coefficient}
\input{sections/geom-curves}
\input{sections/example-curves}
\input{sections/geom-extension}
\input{sections/geom-phases}
\input{sections/geom-traces}
\input{sections/geom-excursions}
\input{sections/geom-realization}

\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}