\documentclass[11pt]{article}
\input{preamble}
\hypersetup{
 pdftitle={\seriestitle, I: Arithmetic foundations and the main theorems},
 pdfauthor={Boyang Hu}
}
\title{\seriestitle\\[6pt]\Large I. Arithmetic foundations and the main theorems}
\author{Boyang Hu\\[3pt]
{\normalsize Tsinghua University}}
\date{12 November 2026}
\begin{document}
\InputIfFileExists{ext-II}{}{}
\InputIfFileExists{ext-III}{}{}
\maketitle
\begin{abstract}
Let \(G\) be a split connected reductive group over a global
function field \(F\). V.~Lafforgue decomposed the space of cusp
forms on \(G\) according to semisimple Langlands parameters. This is
the first of three
papers which prove a converse: every semisimple elliptic parameter
is the parameter of a cusp form on \(G\) over \(F\) itself, which
we call an automorphic realization of the parameter. This holds in
every positive characteristic for classical groups, in characteristic
greater than \(3\) for \(G_2\) and \(F_4\), greater than \(5\) for
\(E_6\), and greater than \(7\) for \(E_7\) and \(E_8\). A parameter
that is not elliptic is the parameter of an automorphic form which
is not cuspidal, under
the bound for the smallest Levi subgroup
through which the parameter factors. The form has the prescribed
eigenvalue for every excursion operator, and we determine for which
lattices in the centre it can be taken invariant.

The present paper contains the arithmetic part of the proof. It
defines excursion operators on functions that are not assumed
cuspidal, proves their compatibility with constant terms, shows that
a function on which they act through an elliptic parameter is a
cusp form, and passes from simply connected groups to isogenous
groups and from Levi subgroups to the group. This reduces the
theorems to the existence of a cusp form with a given elliptic
parameter on a simply connected simple group. That is proved in the
second paper by a geometric method, when the characteristic does
not divide the order of the Weyl group, and in the third paper for
classical groups, by descent from general linear groups.
\end{abstract}

{\small\noindent\textbf{2020 Mathematics Subject Classification.} 11R39, 11F70 (primary); 14D24, 22E55, 14G15 (secondary).

\noindent\textbf{Keywords.} Langlands parameters, function fields, excursion operators, shtukas, cusp forms, constant terms.\par}

\tableofcontents
\bigskip

\input{sections/intro}
\input{sections/inputs}
\input{sections/normalizations}
\input{sections/arith-purity}
\input{sections/arith-excursions}
\input{sections/arith-function-spaces}
\input{sections/arith-constant-terms}
\input{sections/arith-cuspidality}
\input{sections/arith-trace-character}
\input{sections/arith-unitary}
\input{sections/arith-central}
\input{sections/arith-levi-eisenstein}
\input{sections/assembly}
\appendix
\input{sections/appendix-partial-frobenius}
\section*{Notation}
\addcontentsline{toc}{section}{Notation}
\markboth{Notation}{}
\input{sections/notation-index}
\input{sections/ai-statement}
\input{bibliography}
\end{document}