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Many results in number theory are stated either in a classical language or in an adelic one. I am often impressed of the efficiency and the satisfactory computational properties of the adelic setting, often brief and well-behaved. They seems omnipresent, but barely justified, almost abstruse !

Unfortunately I never found a good reference book for the adeles and ideles definitions and properties : they always are treated in appendix or in a little chapter giving most of the time only the necessary stuff for the self-sufficientness ok the book.

Travelling among tens of lecture notes and books (Weil, Vignéras, Goldfeld, Lang, Milne, Tate, Bump, Gelbart, etc. : all books which have not adeles as main theme !) do not seem to be a good solution in order to have a good idea of adelic objects and properties : what are them ? for what do they exist ? are there examples and computation rules ? what are local and global properties ? splitting properties ? measures ? volumes ? general methods ? approximation theorems ? compactness of adelic groups ? are so many questions always only partially answered, often referring to an other book again...

So here is the question : is there any good reference, the more comprehensive possible, starting from the beginning and treating all the major aspects and properties of adeles, but not being just an arid handbook without intuition nor motivation nor examples ?

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    $\begingroup$ I think the answer might be ... rolling in the deep. $\endgroup$ – Per Alexandersson Oct 18 '14 at 11:27
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    $\begingroup$ Have you looked at Ramakrishnan-Valenza? $\endgroup$ – S. Carnahan Oct 18 '14 at 13:38
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    $\begingroup$ Well I didn't know that you're familiar with his book. Anyways, I think that in general the book will address a lot of your questions. Anyways, it seems to me that your questions are actually biased towards Algebraic groups/Arithmetic groups/Lie groups, more than the adelic structure itself as a restricted product over all completions. I believe you might find some of the answers you are looking for in Margulis' book, and in Platonov-Rapinchuk. $\endgroup$ – Asaf Oct 19 '14 at 15:46
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    $\begingroup$ There is a nice series of comics by Tardi called Adèle Blanc-Sec, but I am not sure that this is what you need. $\endgroup$ – ACL Oct 21 '14 at 11:57
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    $\begingroup$ You already mention Weil as not satisfying you, but I think Weil's <i>Basic Number Theory</i> has a lot of what you ask for. You might also look at the answers to my question mathoverflow.net/q/71727/297 $\endgroup$ – DES-SupportsMonicaAndTransfolk Oct 21 '14 at 15:07
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A short list of references among the answers, with comments, hoping it will be of some use :

  • Ramakrishnan & Valenza, Fourier Analysis on Number Fields, GTM.

There is a short chapter constructing general restricted product of groups, giving them their topology and measures, then applying to obtain adeles and ideles groups, plus approximation theorems and others properties, and class group.General but with no computations. - S. Miller, Adeles, Automorphic Forms and Representations (available there)

30 pages with various elementary calculations, interest in characters, Fourier theory and characters

  • P. Deligne, Formes modulaires et représentations de GL(2) (available there)

Focusing on lattices, and rebuilding automorphic forms on it

  • Goldfeld & Hundley, Automorphic Representations and L-Functions for the General Linear Group, Cambridge University Press.

Just defining adeles and ideles over $\mathbf{Q}$, but developing the theory of automorphic forms for GL in this setting.

  • The two Weil's book, Basic Number Theory & Adeles and Algebraic Groups.

Seems to deal with adeles in great generality.

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    $\begingroup$ Goldfeld and Hundley, "Automorphic Representations and $L$-Functions for the General Linear Group" introduces the adeles (just over $\mathbb{Q}$, unfortunately) in some detail in the first chapter. $\endgroup$ – Peter Humphries Oct 21 '14 at 14:30

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