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+\chapter{The Global Langlands conjectures}
+
+\section{Overview of the chapter}
+
+In this section we discuss the problem of attempting to state the motivic global Langlands conjectures for connected reductive groups over number fields. More precisely, the goal is to formally state the Buzzard--Gee version of the conjecture, which applies to algebraic automorphic representations. The main difficulties here are in writing down precisely what is meant by phrases such as ``automorphic representation'', ``connected reductive group'', ``local-global compatibility'', ``de Rham Galois representation'', and so on; these words hide a large amount of technical material.
+
+\section{Statement of the conjecture}
+
+Let $R$ be a commutative base ring; it will often be a field but we shall develop the theory in more generality when there is no extra effort needed to do.
+
+\begin{definition}\label{affine_group_scheme_over_affine_base}\lean{???}
+
+  An \emph{affine group scheme over $R$} is a group object in the category of affine schemes over $R$.
+
+  \begin{definition}\label{Hopf_algebra}\lean{TODO}% we have Hopf alegbras in mathlib
+
+    %%%%%%%%%%%%%%%%%%%
+
+    TODO: Connected and reductive 
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