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The adelics seem counter-intuitive. I wonder how they came up originally, and what was the immediate reward for introducing them. What was the politics of introducing the adelics into mathematical mainstream?

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"Adèles. You put the accent there if you want people to know you speak french." -- Hendrik W. Lenstra, Jr. –  Arturo Magidin Dec 17 '10 at 3:15
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"Politics"? Is that really the best way to put it? –  Pete L. Clark Apr 8 '11 at 3:26
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2 Answers

The ideles came first, and were introduced by Chevalley as part of his program of giving a (then) new proof of class field theory which avoided the analysis of $L$-functions. The adeles came a little later; they were originally called valuation vectors, and were introduced by Artin and Whaples. It took a little time for the details to be sorted out (e.g. Chevalley orginally used a slightly different topology on the ideles to the one used currently; see this MO question for more details); when Tate wrote his thesis in 1950, he used the current definitions (although I think he still used the term valuation vectors, rather than adeles).

As far as I know, the name ideles was invented by Chevalley, as a pun on the word ideals, since from the ideles one could recover all the generalized ideal class groups of traditional algebraic number theory. The term adeles (when it was finally settled on, maybe by Weil?) was then a pun on ideles, together with the fact that the adeles have an additive structure.

The analogue of the adeles in which the number field is replaced by the function field of a smooth projective irreducible algebraic curve (so that the places of the number field are replaced by the points of the curve) appeared in Weil's approach to proving the Riemann--Roch theorem. (In this context the adeles were also called repartitions, although I doubt anyone uses that terminology anymore.)

If one looks in Tate's thesis, there is a result there called the Riemann--Roch theorem, which really does reduce to the usual result with that name in the function field case.

The consideration of adelic points of algebraic groups was first undertaken by Weil, I think, in his reformuation of the work of Tamagawa in the study of what are now called Tamagawa numbers. The use of this point of view in the study of automorphic forms was brought to the fore in the work of Langlands in the late 1960s and early 1970s, although I don't know whether he was the first one who realized that it was the natural way to encode the notion of Hecke eigenform in representation theoretic terms. Chevalley's original concept of idele is subsumed in this general formalism; they are precisely the adelic points of the group $GL_1$.

The overarching reason for introducing the adelic viewpoint, beginning with Chevalley's original work and continuing through to the modern treatment of automorphic forms, has always been the same: it provides the most convenient and flexible framework for studying the interaction of local and global phenomena. All the traditional concepts (e.g. finiteness of the class number, Dirichlet's unit theorem, finite volume of $\Gamma\backslash G$ for semi-simple groups $G$ and congruence subgroups $\Gamma$) are easily expressed in adelic terms in a natural and uniform way, and for studying certain concepts (local reciprocity laws and their interaction with global reciprocity) it is hard to imagine how one could sensibly formulate things in anything other than the adelic manner.

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The pun is also on the old-fashioned female noun Adèle. (Victor Hugo had a daughter named Adèle, about whose sad destiny French director François Truffaut made a film called "L'Histoire d'Adèle H.") –  Georges Elencwajg Dec 17 '10 at 8:23
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And Serre's mother is named Adèle. (See college-de-france.fr/default/EN/all/historique/essai.htm) –  KCd Dec 17 '10 at 17:01
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Recall the process of completing $\mathbb Q$ to obtain $\mathbb R$. Here the metric is coming from the absolute value norm. It turns out that in addition to the usual absolute value, one can put the $p$-adic valuations on $\mathbb Q$. One can do a similar completion and complete $\mathbb Q$ to $\mathbb Q_p$. You can also prove that these are the only valuations on $\mathbb Q$.

More generally, on number fields and function fields(together called global fields), one can similarly classify all the valuations,aka places, and some of them are Archimedean and others non-Archmidean. Now according to Weil in his book Basic Number Theory, the theory of adèles come up with the point of view that from the arithmetician's perspective, no valuation is preferred over the other; and the global field must be seen at once as completing to infinitely many places all at once, with $\mathbb R$ being no better or worse than any other. He uses the theory of adèles to achieve this end. The nomenclature was perhaps influenced by Chevalley's ideles, which can be realized as a subset of adèles. The name is supposed to be a pun on a French girls's name.

If you consult the preface of Weil's Basic Number Theory, you can read exactly what he says. My paraphrase here might be a bit jaded by memory.

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