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\begin{document}

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\section*{group of ideles}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra}

[[!include higher algebra - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{ProductFormula}{Product formula}\dotfill \pageref*{ProductFormula} \linebreak
\noindent\hyperlink{StrongApproximationTheorem}{Strong approximation theorem for the idele class group}\dotfill \pageref*{StrongApproximationTheorem} \linebreak
\noindent\hyperlink{automorphic_forms_and_relation_to_dirichlet_characters}{Automorphic forms and Relation to Dirichlet characters}\dotfill \pageref*{automorphic_forms_and_relation_to_dirichlet_characters} \linebreak
\noindent\hyperlink{FunctionFieldAnalogy}{Function field analogy}\dotfill \pageref*{FunctionFieldAnalogy} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \emph{group of ideles} $\mathbb{I}$ is the [[group of units]] in the [[ring of adeles]] $\mathbb{A}$:

\begin{displaymath}
\mathbb{I} = \mathbb{A}^\times \coloneqq GL_1(\mathbb{A})
  \,.
\end{displaymath}
In classical [[algebraic number theory]] one embeds a [[number field]] into the [[cartesian product]] of its [[completions]] at its [[archimedean absolute values]]. This embedding is very useful in the [[proofs]] of several fundamental [[theorems]]. However, it was noticed by [[Claude Chevalley]] and [[André Weil]] that the situation was improved somewhat if the number field is embedded in the cartesian product of its [[formal completions]] at all of its [[absolute values]]. With a few additional restrictions, these objects are known as the \emph{[[adeles]]}, and the [[group of units|units]] of this ring are called the \emph{ideles}.

When considering the adeles and ideles, it is their [[topology]] as much as their algebraic structure that is of interest. Many important results in [[number theory]] translate into simple statements about the topologies of the adeles and ideles. For example, the finiteness of the [[ideal class group]] and the [[Dirichlet unit theorem]] are equivalent to a certain quotient of the ideles being compact and discrete.

(\hyperlink{Weston}{Weston, p. 1})

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{}\hypertarget{}{}
The [[group of units]] of the ring of adeles $\mathbb{A}_{\mathbb{Q}}$ is called the \emph{group of ideles}

\begin{displaymath}
\mathbb{I}_{\mathbb{Q}} \coloneqq GL_1(\mathbb{A}_{\mathbb{Q}}) = \mathbb{A}_{\mathbb{Q}}^\times
  \,.
\end{displaymath}
It is a [[topological group]] via identification with the set $\{(x, x^{-1}) \in \mathbb{A}_\mathbb{Q}^2: \; x \in \mathbb{I}_\mathbb{Q}\}$, seen as a [[subspace]] of $\mathbb{A}_\mathbb{Q}^2$.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
The topology on $\mathbb{I}_\mathbb{Q}$ is strictly finer than the subspace topology inherited from $\mathbb{A}_\mathbb{Q}$. For example, the set $\mathbb{R}^\times \times \prod_p \mathbb{Z}_p^\times$ is a neighborhood of $1$ in $\mathbb{I}_\mathbb{Q}$, but not in the subspace topology. Cf. the discussion \href{https://math.stackexchange.com/a/145452/43208}{here}. Note: multiplicative inversion is not continuous in the subspace topology.

\end{remark}
The same definition holds for the [[ring of adeles]] of any other [[global field]] $K$, here one writes

\begin{displaymath}
\mathbb{I}_K \coloneqq GL_1(\mathbb{A}_K)
\end{displaymath}
or similar. The notation $J_K$ is also common.

\begin{defn}
\label{IdeleClassGroup}\hypertarget{IdeleClassGroup}{}
The [[quotient]]

\begin{displaymath}
K^\times \backslash \mathbb{I}_K
  =
  GL_1(K)\backslash GL_1(\mathbb{A}_K)
\end{displaymath}
is called the \emph{idele class group} of $K$.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
The idele class group, def. \ref{IdeleClassGroup}, appears prominently in the description of the \href{moduli+space+of+bundles#OverCurvesAndTheLanglandsCorrespondence}{moduli space of line bundles} over the [[arithmetic curve]] on which $K$ is the [[rational functions]]. From there it appears in the abelian [[Langlands correspondence]] and in the abelian case of [[Tamagawa measures]].

\end{remark}
The idele class group is a key object in \emph{[[class field theory]]}.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{ProductFormula}{}\subsubsection*{{Product formula}}\label{ProductFormula}

Recall the [[p-adic norm]] ${\vert -\vert}_p$ on $\mathbb{Q}$ for $p$ a [[prime number]], given by

\begin{displaymath}
{\left \vert \frac{a}{b}p^\ell \right \vert}_p
  \coloneqq
  p^{-\ell}
\end{displaymath}
for $a,b$ coprime to $p$. The usual [[absolute value]] [[norm]] one writes

\begin{displaymath}
{\vert -\vert}_\infty
\end{displaymath}
and associates with the ``[[place at infinity|prime at infinity]]''. When an index runs over the set of all primes (``finite primes'') union with the ``prime at infinity'' one usually writes it ``$v$'' instead of $p$.

This induces:

\begin{defn}
\label{IdeleNorm}\hypertarget{IdeleNorm}{}
The \emph{idele norm}

\begin{displaymath}
{\vert -\vert} \colon \mathbb{I}_{\mathbb{Q}} \longrightarrow \mathbb{C}^\times
\end{displaymath}
is the function given by

\begin{displaymath}
{\vert \alpha\vert} \coloneqq \underset{v}{\prod} {\vert \alpha_v\vert}_v
  \,.
\end{displaymath}
\end{defn}
Notice that by construction there is a [[diagonal]] map $\mathbb{Q}^\times \to \mathbb{I}_{\mathbb{Q}}$.

\begin{prop}
\label{ProductFormula}\hypertarget{ProductFormula}{}
\textbf{(product formula)}

The idele norm, def. \ref{IdeleNorm}, is trivial on the diagonal of $\mathbb{Q}^\times$ inside the ideles, in that

\begin{displaymath}
(\alpha \in \mathbb{Q}^\times \to \mathbb{I}_{\mathbb{Q}})
  \;\Rightarrow\;
  {\vert \alpha\vert}
  \coloneqq 
  \underset{v}{\prod} {\vert \alpha_v\vert}_v
  = 1
  \,.
\end{displaymath}
\end{prop}
\begin{remark}
\label{}\hypertarget{}{}
The product formula, prop. \ref{ProductFormula}, says that the idele norm descends to the idele class group, def. \ref{IdeleClassGroup}.

\end{remark}
(e.g. \hyperlink{Garrett11}{Garrett 11, section 1})

\hypertarget{StrongApproximationTheorem}{}\subsubsection*{{Strong approximation theorem for the idele class group}}\label{StrongApproximationTheorem}

\begin{prop}
\label{StrongApproximationTheorem}\hypertarget{StrongApproximationTheorem}{}
\textbf{(strong approximation form ideles)}

The idele class group, def. \ref{IdeleClassGroup}, may be expressed as

\begin{displaymath}
\mathbb{Q}^\times \backslash \mathbb{A}_{\mathbb{Q}}^\times
  \simeq
  (0,\infty)
   \times
  \underset{p}{\prod}
  \mathbb{Z}_p^\times
  \,.
\end{displaymath}
\end{prop}
(e.g. \hyperlink{GoldfeldHundley11}{Goldfeld-Hundley 11, prop. 1.4.5 and below (2.2.7)})

This implies that the [[ring of adeles]] may be decomposed into a rational and an idele class factor as:

\begin{displaymath}
\begin{aligned}
    \mathbb{A}_{\mathbb{Q}}^\times
     & \simeq
    \mathbb{Q}^\times
    \times
    (\mathbb{Q}^\times \backslash \mathbb{A}_{\mathbb{Q}}^\times)
    \\
    & \coloneqq
    \underset{n \in \mathbb{Q}^\times}{\cup}
    n \cdot 
    (\mathbb{Q}^\times \backslash \mathbb{A}_{\mathbb{Q}}^\times)
 \end{aligned}
  \,.
\end{displaymath}
(e.g. \hyperlink{GoldfeldHundley11}{Goldfeld-Hundley 11, prop. 1.4.6 and below (2.2.7)})

This decomposition is crucial in the discussion of the [[Riemann zeta function]] (see there) as an [[adelic integral]].

\hypertarget{automorphic_forms_and_relation_to_dirichlet_characters}{}\subsubsection*{{Automorphic forms and Relation to Dirichlet characters}}\label{automorphic_forms_and_relation_to_dirichlet_characters}

The [[automorphic forms]] of the idele group are essentially [[Dirichlet characters]] in disguise (\hyperlink{GoldfeldHundley11}{Goldfeld-Hundley 11, below def. 2.1.4})

\hypertarget{FunctionFieldAnalogy}{}\subsubsection*{{Function field analogy}}\label{FunctionFieldAnalogy}

Via the [[function field analogy]] one may understand any [[number field]] or [[function field]] $F$ as being the [[rational functions]] on an [[arithmetic curve]] $\Sigma$. Under this identification the [[ring of adeles]] $\mathbb{A}_F$ of $F$ has the interpretation of being the [[ring of functions]] on all punctured [[formal disks]] around all points of $\Sigma$, such that only finitely many of them do not extend to the given point. (\hyperlink{Frenkel05}{Frenkel 05, section 3.2}).

This means for instance that the [[general linear group]] $GL_n(\mathbb{A}_F)$ with [[coefficients]] in the [[ring of adeles]] has the interpretation as being the [[Cech cohomology|Cech cocycles]] for [[algebraic vector bundles]] of [[rank]] $n$ on an [[algebraic curve]] with respect to any [[cover]] of that curve by the complement of a finite number of points together with the [[formal disks]] around these points. Here for $n = 1$ then $GL_1(\mathbb{A}_F)$ is the group of ideles.

This is part of a standard construction of the [[moduli stack of bundles]] on algebraic curves, see at \emph{\href{moduli+space+of+bundles#OverCurvesAndTheLanglandsCorrespondence}{Moduli space of bundles and the Langlands correspondence}}.

[[!include function field analogy -- table]]

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[differential cohesion and idelic structure]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Basics are recalled in

\begin{itemize}%
\item \emph{Adeles} \href{http://wiki.epfl.ch/gant/documents/lecture2-cib2011.pdf}{pdf}


\item Pete Clark, \emph{Adeles and Ideles} (\href{http://math.uga.edu/~pete/8410Chapter6.pdf}{pdf})


\item Erwin Dassen , \emph{Adeles \& Ideles} (\href{http://www.math.leidenuniv.nl/~astolk/monday/notes/dassen-adeles-ideles.pdf}{pdf})


\item Tom Weston, \emph{The idelic approach to number theory} (\href{http://www.math.umass.edu/~weston/oldpapers/idele.pdf}{pdf})


\item  [[Dorian Goldfeld]], [[Joseph Hundley]], chapter 2 of \emph{Automorphic representations and L-functions for the general linear group}, Cambridge Studies in Advanced Mathematics 129, 2011 (\href{https://www.maths.nottingham.ac.uk/personal/ibf/text/gl2.pdf}{pdf})


\item [[Paul Garrett]], \emph{Iwasawa-Tate on $\zeta$-functions and L-functions}, 2011 (\href{http://www-users.math.umn.edu/~garrett/m/mfms/notes_c/Iwasawa-Tate.pdf}{pdf} [[!redirects Poisson formula]]



\end{itemize}
Discussion in the context of the [[geometric Langlands correspondence]] is in

\begin{itemize}%
\item [[Edward Frenkel]], section 3.2 of \emph{Lectures on the Langlands Program and Conformal Field Theory}, in \emph{Frontiers in number theory, physics, and geometry II}, Springer Berlin Heidelberg, 2007. 387-533. (\href{http://arxiv.org/abs/hep-th/0512172}{arXiv:hep-th/0512172})

\end{itemize}
[[!redirects idele]] [[!redirects ideles]]

[[!redirects idele class group]] [[!redirects idele class groups]]

[[!redirects idele group]] [[!redirects idele groups]]

[[!redirects idèle group]] [[!redirects group of idèles]]



\end{document}