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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{ring of adeles} \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{ForTheRationalNumbers}{For the rational numbers}\dotfill \pageref*{ForTheRationalNumbers} \linebreak \noindent\hyperlink{ForAnyNumberField}{For a number field}\dotfill \pageref*{ForAnyNumberField} \linebreak \noindent\hyperlink{ForAGlobalField}{For a global field}\dotfill \pageref*{ForAGlobalField} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{basic_results}{Basic results}\dotfill \pageref*{basic_results} \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{ring of adeles} $\mathbb{A}_k$ of any [[global field]] $k$ -- in particular of the [[rational numbers]] $\mathbb{Q}$ -- is the [[restricted product]] of all [[formal completions]] $k_v$ of $k$ at all its [[places]] $v$, where the restriction is such that only a [[finite number]] of components have [[norm]] greater than 1. (This has a useful geometric interpretation and motivation by the [[function field analogy]], more on which \hyperlink{FunctionFieldAnalogy}{below}). In particular the ring of adeles of the rational numbers is equivalently the [[rationalization]] of the [[product]] of all [[p-adic integers]] (including the ``prime at infinity''). In classical [[algebraic number theory]] one embeds a [[number field]] into the [[cartesian product]] of its [[completion of a ring|completions]] at its [[archimedean absolute values]]. This embedding is very useful in the [[proofs]] of several fundamental [[theorems]]. For example, the [[algebraic integers]] in the number field embed discretely and co-compactly into this [[cartesian product]], i.e., as a [[lattice in a vector space|lattice]], and this opens the way for example to the concrete realization of the [[group of units]] (modulo [[torsion subgroup|torsion]]) as a [[lattice]], and also to the technique of [[Fourier analysis]] where Poisson summation applied to the lattice has classical implications for [[theta functions]] and [[zeta functions]]. It was noticed by [[Claude Chevalley]] and [[André Weil]] that the situation is made even better if the [[number field]] itself 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]]}. Under these restrictions, the given number field embeds discretely and cocompactly into the adeles, i.e., behaves as a lattice where it is again possible to apply Poisson summation. 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} The concept of a \emph{ring of adeles} (older synonym: ring of valuation vectors) makes sense for any [[global field]], hence for any finite-dimensional [[field extension]] of either the [[rational numbers]] or of a [[function field]] over a [[finite field]]. The ring of adeles of the [[rational numbers]] themselves is the classical case which we discuss first in \begin{itemize}% \item \hyperlink{ForTheRationalNumbers}{For the rational numbers} \end{itemize} Then we consider the definition more generally \begin{itemize}% \item \hyperlink{ForAnyNumberField}{For a number field} \end{itemize} and finally in full generality \begin{itemize}% \item \href{}{For a global field}. \end{itemize} \hypertarget{ForTheRationalNumbers}{}\subsubsection*{{For the rational numbers}}\label{ForTheRationalNumbers} We start off very simply with the algebraic description of the adele ring over the [[rational numbers]]. \begin{defn} \label{RingOfAdeles}\hypertarget{RingOfAdeles}{} The ring of \emph{integral adeles} $\mathbb{A}_{\mathbb{Z}}$ is the [[product]] of the [[profinite completion of the integers]] $\widehat{\mathbb{Z}}$, with the [[real numbers]] \begin{displaymath} \mathbb{A}_{\mathbb{Z}} \coloneqq \mathbb{R} \times \widehat{\mathbb{Z}} \,. \end{displaymath} The \emph{ring of adeles} $\mathbb{A}_{\mathbb{Q}}$ (or just $\mathbb{A}$, for short) itself is the [[rationalization]] of the ring of integral adeles, hence its [[tensor product]] with the [[rational numbers]] \begin{displaymath} \mathbb{A}_\mathbb{Q} \coloneqq \mathbb{Q} \otimes_\mathbb{Z} \mathbb{A}_\mathbb{Z} \,. \end{displaymath} \end{defn} This definition has various equivalent reformulations which are often useful. \begin{remark} \label{ProfiniteIntegersAsProductOverpAdics}\hypertarget{ProfiniteIntegersAsProductOverpAdics}{} By \href{profinite+completion+of+the+integers#AsProductOverAlsoPAdicIntegers}{this proposition} we have that the [[profinite completion of the integers]] is equivalently the [[product]] of all [[p-adic integers]] as $p$ ranges over all [[prime numbers]] \begin{displaymath} \hat \mathbb{Z} \simeq \underset{p\;prime}{\prod} \mathbb{Z}_p \end{displaymath} Using this in def. \ref{RingOfAdeles} says that the ring of integral adeles is the product \begin{displaymath} \mathbb{A}_{\mathbb{Z}} \simeq \mathbb{R} \times \underset{p\;prime}{\prod} \mathbb{Z}_p \,. \end{displaymath} \end{remark} From this one obtains the following equivalent characterization: \begin{prop} \label{RationalRingOfAdelesAsRestrictedProduct}\hypertarget{RationalRingOfAdelesAsRestrictedProduct}{} The ring of adeles $\mathbb{A}$, def. \ref{RingOfAdeles}, is equivalently the [[restricted product]] $\prod^\prime$ of the [[p-adic rational numbers]], the restriction being along the inclusion $\mathbb{Z}_p \to \mathbb{Q}_p$: \begin{displaymath} \mathbb{A}_{\mathbb{Q}} = \mathbb{R} \times \underset{p \; prime}{\prod^\prime} \mathbb{Q}_p \end{displaymath} \end{prop} \begin{proof} By remark \ref{ProfiniteIntegersAsProductOverpAdics} the [[tensor product]] to be computed is equivalently \begin{displaymath} \mathbb{A}_{\mathbb{Q}} \simeq \mathbb{Q}\otimes_{\mathbb{Z}} \left( \mathbb{R} \times \underset{p \; prime}{\prod} \mathbb{Z}_p \right) \,. \end{displaymath} Now notice that a [[natural number]] $n$ is a [[group of units|unit]] in $\mathbb{Z}_p$ if $p$ is not a [[prime factor]] of $n$. Therefore for $(a_p) \in \underset{p}{\prod} \mathbb{Z}_p$ and $\frac{c}{d} \in \mathbb{Q}$, then for each of the [[finite number]] of [[prime factors]] $p$ of $d$ the tensor product element $\frac{c}{d} \otimes_{\mathbb{Z}} a_p \in \mathbb{Q}_p$ contains a non-vanishing negative power of $p$ and is hence not in $\mathbb{Z}_p$, whereas for all $p$ that do not appear as prime factors in $d$ it is. \end{proof} Finally notice: \begin{remark} \label{AsProductOverPlaces}\hypertarget{AsProductOverPlaces}{} The [[prime numbers]] correspond to the non-archimedean [[places]] of $\mathbb{Z}$, and under this identification there is one more real [[place at infinity]], ``$p = \infty$'', the [[completion of a ring|completion]] of $\mathbb{Q}$ at which is the real numbers $\mathbb{R}$, which one may therefore write $\mathbb{R} = \mathbb{Q}_\infty$. Using this the characterization of the ring of adeles from prop. \ref{RationalRingOfAdelesAsRestrictedProduct} is equivalently the [[restricted product]] over all real places of the [[formal completion]] of $\mathbb{Q}$ at this place \begin{displaymath} \mathbb{A}_{\mathbb{Q}} \simeq \underset{p \in Places(\mathbb{Z})}{\prod^{\prime}} \mathbb{Q}_p \,. \end{displaymath} Considering this [[restricted product]] not just in bare [[commutative rings]] but in [[topological rings]] yields the right structure of a topological ring on $\mathbb{A}_{\mathbb{Q}}$. This is the content of the following proposition. \end{remark} \begin{prop} \label{}\hypertarget{}{} $\mathbb{A}_\mathbb{Q}$ is a [[locally compact space|locally compact]] [[Hausdorff space|Hausdorff]] commutative ring. In particular, it is complete with respect to its [[uniform space]] structure. \end{prop} \begin{proof} The restricted product is a filtered colimit of a system of \emph{open} inclusions between locally compact Hausdorff rings, and is therefore itself locally compact Hausdorff. If $x_\alpha$ is a Cauchy net, then for all sufficiently large $\alpha, \beta$ the differences $x_\alpha - x_\beta$ lie in a compact neighborhood of the identity. Holding $\beta$ fixed, the limit $\lim_\alpha x_\alpha - x_\beta$ exists by compactness; if $x$ is this limit, then $x + x_\beta$ is the limit of the Cauchy net. \end{proof} If one omits the factor of $\mathbb{R} = \mathbb{Q}_\infty$, then one speaks of the \emph{ring of finite adeles}. \begin{displaymath} \mathbb{A}_{\mathbb{Q}}^f \coloneqq \underset{p \; prime}{\prod^\prime} \mathbb{Q}_p \,. \end{displaymath} All of this generalizes to any [[number field]] $k$. \hypertarget{ForAnyNumberField}{}\subsubsection*{{For a number field}}\label{ForAnyNumberField} Let \begin{itemize}% \item $k$ be a [[number field]]; \item $\mathcal{O} = \mathcal{O}_k$ the [[ring of integers|ring of]] [[algebraic integers]] in $k$: \item $P$ be its [[set]] of [[places]] (equivalence classes of [[absolute values]] on $\mathcal{O}$); \item $S\subset P$ be the set of [[archimedean valuation|archimedean]] places; \item $k_v$ the [[formal completion]] of $k$ at $v\in P$. \end{itemize} Notice that \begin{itemize}% \item for $v \in S$ then $k_v$ is isomorphic to one of the [[local fields]] $\mathbb{R}$ or $\mathbb{C}$; \item for $v \notin S$ then $k_v$ is a [[local field]] with an open compact subring $\mathcal{O}_v$ consisting of elements of [[norm]] $1$ or less. \end{itemize} \begin{defn} \label{}\hypertarget{}{} For $k$ a [[number field]], the ring of \emph{integral adeles} $\mathbb{A}_{\mathcal{O}}$ is the [[product]] of the [[profinite completion]] $\widehat{\mathcal{O}}$ with all the archimedean completions, \begin{displaymath} \mathbb{A}_\mathcal{O} \coloneqq \widehat{\mathcal{O}} \times \prod_{v \in S} k_v \,, \end{displaymath} and the \emph{ring of adeles} over $k$ is the [[tensor product]] $\mathbb{A}_k \coloneqq k \otimes_\mathcal{O} \mathbb{A}_\mathcal{O}$. \end{defn} It may be shown that \begin{displaymath} \widehat{\mathcal{O}} \cong \prod_{v \notin S} \mathcal{O}_v \end{displaymath} where each non-archimedean place $v$ may be identified with a prime ideal $p$. Now, if we view $k$ as the localization of $\mathcal{O}$ obtained by inverting all nonzero elements $x \in \mathcal{O}$, then $k$ may be written as a filtered colimit of a system of inclusions of localizations: \begin{displaymath} k \cong colim_x \; \mathcal{O}[x^{-1}]. \end{displaymath} Componentwise, we may calculate \begin{displaymath} \itexarray{ \mathcal{O}[x^{-1}] \otimes_\mathcal{O} \mathcal{O}_p & = & k_p & \text{if}\; x \in p \\ & = & \mathcal{O}_p & \text{if}\; x \notin p } \end{displaymath} and so \begin{displaymath} \itexarray{ \mathcal{O}[x^{-1}] \otimes_\mathcal{O} \widehat{\mathcal{O}} & \cong & \mathcal{O}[x^{-1}] \otimes_\mathcal{O} \prod_{p \in P \backslash\; S} \mathcal{O}_p \\ & \cong & (\prod_{x \in p} k_p) \times (\prod_{x \notin p} \mathcal{O}_p) } \end{displaymath} Putting these facts together, \begin{displaymath} k \otimes_\mathcal{O} \widehat{\mathcal{O}} \cong colim_x\; (\prod_{x \in p} k_p) \times (\prod_{x \notin p} \mathcal{O}_p) \end{displaymath} whence \begin{displaymath} \begin{array}{lll} k \otimes_\mathcal{O} \mathbb{A}_\mathcal{O} & \cong & k \otimes_\mathcal{O} \left(\prod_{v \in S} k_v \times \widehat{\mathcal{O}}\right) \\ & \cong & colim_x \; \prod_{v \in S} k_v \times (\prod_{p: x \in p} k_p) \times (\prod_{p: x \notin p} \mathcal{O}_p) \end{array} \end{displaymath} where each component of the filtered diagram is locally compact (a product of finitely many locally compact and infinitely many compact spaces) in the [[product topology]]. Taking the filtered colimit in $Top$ over the resulting diagram of open inclusions, the result is again a locally compact ring. In this way the ring of adeles $\mathbb{A}_k$ is topologized. \begin{remark} \label{}\hypertarget{}{} The topology on the adele ring $\mathbb{A}_k$ is strictly finer than the subspace topology inherited from its natural inclusion into $\prod_{v \in P} k_v$ with the product topology. For example, $(\prod_{v \in S} k_v) \times \prod_{p \in P \backslash\; S} \mathcal{O}_p$ is open in the ring of adeles, but not in $\prod_{v \in P} k_v$. \end{remark} \begin{defn} \label{}\hypertarget{}{} The [[group of units]] of the ring of adeles $\mathbb{A}_k$ is called the group of [[ideles]], denoted $\mathbb{I}_k$. \end{defn} Under the subspace topology inherited from $\mathbb{A}_k$, there is no reason for inversion $(-)^{-1}: \mathbb{I}_k \to \mathbb{I}_k$ to be continuous (and in fact it isn't!), so $\mathbb{I}_k$ isn't a topological group when topologized this way. However, we can endow $\mathbb{I}_k$ with the subspace topology given by the embedding $\mathbb{I}_k \to \mathbb{A}_k \times \mathbb{A}_k: x \mapsto (x, x^{-1})$; topologized this way, we get a locally compact topological group. \emph{This is the topology on the ideles.} Alternatively, for each finite $T \subset P$ containing the set of archimedean places $S$, we have a locally compact group \begin{displaymath} \mathbb{I}_{k, T} = (\prod_{v \in T} k_v^\times) \times (\prod_{v \notin T} \mathcal{O}_v^\times) \end{displaymath} (noting that each unit group $\mathcal{O}_v^\times$ is compact), and the idele group can be described as the colimit over a filtered system of open inclusions \begin{displaymath} \mathbb{I}_k = colim_T \; \mathbb{I}_{k, T} \end{displaymath} and indeed the idele topology coincides with the filtered colimit topology. \hypertarget{ForAGlobalField}{}\subsubsection*{{For a global field}}\label{ForAGlobalField} Fully generally, let $k$ be a [[global field]]. Write $P$ for its [[set]] of [[places]] and $k_v$ for its [[formal completion]] at $v \in P$. \begin{defn} \label{AsRestrictedProduct}\hypertarget{AsRestrictedProduct}{} The \emph{ring of adeles} of a [[global field]] $k$ is the [[restricted product]] \begin{displaymath} \mathbb{A}_k \coloneqq \underset{v \in P}{\prod} k_v \end{displaymath} where the restriction is to elements $(x_v)_{v\in P}$ of the actual [[product]] whose components have [[norm]] at most unity -- ${\vert x_v\vert} \leq 1$, except for at most a [[finite number]] of $v$. This is topologized in the same way as discussed \hyperlink{ForAnyNumberField}{above}. \end{defn} Reviews includes (\hyperlink{Mathew10}{Mathew 10}). In the function field case, where $k$ is a finite extension of $\mathbb{F}_p(T)$, the analogous ring of integers $\mathcal{O}(k)$ is the [[integral closure]] in $k$ of the subring $\mathbb{F}_p[T] \hookrightarrow \mathbb{F}_p(T)$. And analogously, the ring of integer adeles $\mathbb{A}_{\mathcal{O}(k)}$ may be defined to be the product of all the completions of $\mathcal{O}(k)$ over all the places of $k$. This is a compact ring. The restricted direct product above map may then, in parallel with the number field case described above, be described as a tensor product \begin{displaymath} \mathbb{A}_k = k \otimes_{\mathcal{O}(k)} \mathbb{A}_{\mathcal{O}(k)} \end{displaymath} where the right side is again interpreted as a colimit in the category of topological rings of a diagram consisting of compact topological rings and open inclusions between them. \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{basic_results}{}\subsubsection*{{Basic results}}\label{basic_results} \begin{prop} \label{}\hypertarget{}{} Under the natural inclusion $i: k \to \mathbb{A}_k$, the subspace topology on $k$ is discrete, and the quotient topology on $\mathbb{A}_k/k$ is compact. \end{prop} As an additive topological group, there is a natural pairing on $\mathbb{A}_k$: \begin{displaymath} \langle -, - \rangle: \mathbb{A}_k \times \mathbb{A}_k \to \mathbb{R}/\mathbb{Z} \cong S^1. \end{displaymath} If $x = (x_v)_{v \in P}$ and $y = (y_v)_{v \in P}$, then \begin{displaymath} \langle x, y \rangle \coloneqq \prod_v \langle x_v, y_v \rangle_v \end{displaymath} where each local pairing $\langle -, - \rangle_v$ is defined to be a composite of the form \begin{displaymath} k_v \times k_v \stackrel{mult}{\to} k_v \stackrel{Tr}{\to} \mathbf{Q}_v \stackrel{\chi_v}{\to} \mathbb{R}/\mathbb{Z} \cong S^1, \end{displaymath} noting that a place $v$ of $k$ restricts to a place on $\mathbb{Q}$. \begin{remarks} \label{}\hypertarget{}{} \begin{itemize}% \item The trace map $Tr$ on the finite algebraic extension $k_v/\mathbf{Q}_v$ is of course defined by $Tr(x) = \sum_\sigma \sigma(x)$ where $\sigma$ ranges over all embeddings of $k_v$ into the algebraic closure of $\mathbb{Q}_v$. \item When $v$ is the archimedean place on $\mathbb{Q}$, we will take the map $\chi_v: \mathbb{R} \to \mathbb{R}/\mathbb{Z}$ to be not the quotient map, but its \emph{additive inverse}. For non-archimedean places $v = p$, the character $\chi_p$ is the composite $\mathbf{Q}_p \to \mathbb{Z}(p^\infty) \hookrightarrow \mathbb{R}/\mathbb{Z}$ as defined \href{http://ncatlab.org/nlab/show/p-adic+number#duality}{here}. \item For each $x, y \in \mathbb{A}_k$, observe that $\langle x_v, y_v \rangle_v = 1 \in S^1$ for all but finitely many places $v$, since $x_v, y_v \in \mathcal{O}_v$ for all but finitely many places. Hence $\langle x, y\rangle$ is well-defined. \end{itemize} \end{remarks} \begin{prop} \label{}\hypertarget{}{} The additive group $\mathbb{A}_k$ is [[Pontrjagin duality|Pontrjagin self-dual]] in the sense that the map $\phi: \mathbb{A}_k \to \mathbb{A}_k^\wedge$ induced from the pairing $\langle-, -\rangle$ is an isomorphism onto the character group. \end{prop} Moreover, define $\pi: \mathbb{A}_k \to k^\wedge$ to be the composite \begin{displaymath} \mathbb{A}_k \stackrel{\phi}{\to} \mathbb{A}_k^\wedge \stackrel{i^\wedge}{\to} k^\wedge. \end{displaymath} \begin{prop} \label{}\hypertarget{}{} The map $\pi \circ i: k \to k^\wedge$ vanishes. The map $\mathbb{A}_k/k \to k^\wedge$ induced by $\pi: \mathbb{A}_k \to k^\wedge$ is an isomorphism of topological groups, so that $\mathbb{A}_k/k$ is Pontrjagin dual to $k$. \end{prop} \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 is most manifest in terms of def. \ref{AsRestrictedProduct} above. 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. (Notice that 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 [[adelic string theory]] \item [[adelic integration]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} \begin{itemize}% \item wikipedia: \href{http://en.wikipedia.org/wiki/Adele_ring}{adele ring}, \href{http://en.wikipedia.org/wiki/Adelic_algebraic_group}{adelic algebraic group} \item D. Goldfeld, J. Hundley, chapter 1 of \emph{Automorphic Representations and L-functions for the General Linear Group}, vol. 1, Cambridge University Press, 2011 (\href{https://www.maths.nottingham.ac.uk/personal/ibf/text/gl1.pdf}{pdf}) \item [[Akhil Mathew]], \emph{\href{http://amathew.wordpress.com/2010/05/21/the-adele-ring/}{The adele ring}}, 2010 \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 \href{http://mathoverflow.net/questions/96137/categorical-description-of-the-restricted-product-adeles}{MO/96137/categorical-description-of-the-restricted-product-adeles} \end{itemize} Discussion in the context of the [[function field analogy]] and the [[geometric Langlands correspondence]] is in \begin{itemize}% \item [[Edward Frenkel]], \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 rings of adeles]] [[!redirects adele]] [[!redirects adeles]] [[!redirects adele ring]] [[!redirects adele rings]] [[!redirects integral adele]] [[!redirects integral adeles]] [[!redirects ring of integral adeles]] [[!redirects rings of integral adeles]] [[!redirects ring of valuation vectors]] [[!redirects valuation vector]] [[!redirects valuation vectors]] [[!redirects ring of finite adeles]] [[!redirects rings of finite adeles]] \end{document}