I am teaching a course leading up to Tate's thesis and I told the students last week, when defining ideles, that the first topology that was put on the ideles was not so good (e.g., it was not Hausdorff; it's basically the profinite topology on the ideles, so archimedean components don't get separated well). You can find this mentioned on the 2nd page of the memorial article on Chevalley by Dieudonne and Tits in AMS Bulletin 17 (1987) (read it here: http://www.ams.org/journals/bull/1987-17-01/S0273-0979-1987-15509-1/S0273-0979-1987-15509-1.pdf), where they also say that Chevalley's introduction of the ideles was "a definite improvement on earlier similar ideas of Prufer and von Neumann, who had only embedded K [the number field] into the product over the finite places" (emphasis theirs). [Edit: Scholl's answer says in a little more detail what Prufer and von Neumann were doing, with references.]

I have two questions:

1) Can anyone point to a specific article where Prufer or von Neumann used a product over just the finite places, or at least indicate whether they were able to do anything with it?

2) Who introduced the restricted product topology on the ideles? (In Chevalley's 1940 paper deriving global class field theory using the ideles and not using complex analysis, Chevalley uses a different topology, as I mentioned above.) I would've guessed it was Weil, but BCnrd told me that he heard it was due to von Neumann. Any answer with some kind of evidence for it is appreciated.

Edit: For those wondering why the usual notation for the ideles is J_K and not I_K, the use of J_K goes right back to Chevalley's papers introducing ideles. (One may imagine I_K could have been taken already for something related to ideals, but in any event it's worth noting the use of "J" wasn't some later development in the subject.)

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    $\begingroup$ In his 1951 report on class field theory (projecteuclid.org/…), Weil is already using the topology prevalent today. He writes : Muni de cette topologie, $I_k$ s'appellera le groupe des idèles de $k$; c'est un groupe abélien séparé, localement compact. $\endgroup$ – Chandan Singh Dalawat Oct 6 '10 at 12:37
  • $\begingroup$ Oh, but Tate's thesis already had the restricted product topology, and that was in 1950. I'm pretty sure the correct topology was known before Tate (and before Matchett, whose thesis I have but I am out of town for a while so I can't take a look so easily). $\endgroup$ – KConrad Oct 6 '10 at 17:27
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    $\begingroup$ You should ask Tate. $\endgroup$ – Felipe Voloch Oct 6 '10 at 19:02
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    $\begingroup$ Who'd have guessed such summoning might actually work! :P $\endgroup$ – Mariano Suárez-Álvarez Oct 6 '10 at 21:13

I know nothing about work of ``idelic nature'' by Von Neumann or Pruefer. Already in the 1930's Weil understood that Chevalley was wrong to ignore the connected component, because Weil understood already then that Hecke's characters were the characters of the idele class group for the right topology on that. I don't know of any place before his paper dedicated to Takagi where he defined the ideles explicitly as a topological group, but he must have understood the situation way before that

When I wrote my thesis I used what seemed to me to be the obvious topology without going into the history of the matter.

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    $\begingroup$ Welcome aboard! $\endgroup$ – Mariano Suárez-Álvarez Oct 6 '10 at 21:10
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    $\begingroup$ Yay! I have no need for further characters. $\endgroup$ – Keenan Kidwell Oct 6 '10 at 22:56
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    $\begingroup$ Yes, I think the answer is Weil. In his short 1936 paper "Remarques sur des resultats recent de C. Chevalley" he complains about Chevalley's topology, and writes down other constructions and topologies (and mentions Grossencharacters) but doesn't write down anything immediately recognizable (to me) as the ideles. In his 1951 paper (J. Math. Soc. Japan) he defines without comment the ideles with the natural (modern) topology. That is also where he wrote "La recherche d'une interpretation pour C_k ... me semble constituer l'un des problemes fondamentaux..." See also his Commentaries in his CW. $\endgroup$ – JS Milne Oct 7 '10 at 0:30
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    $\begingroup$ @John Tate: in your thesis you shift the emphasis so that local compactness is crucial (i.e. so that the abstract Fourier analysis all works), and once you realise this then the topology you want is almost jumping out of the page! $\endgroup$ – Kevin Buzzard Oct 7 '10 at 6:53
  • $\begingroup$ MO has become much more research-level, more of a gallery, and I am counting down as to whom is to join next. $\endgroup$ – Unknown Oct 8 '10 at 17:30

They are too old for Math Reviews, but I think the articles in question are:

  • Von Neumann: "Zur Prüferischen Theorie der idealen Zahlen", Acta Scientiarum Mathematicum (Szeged) 2:4 (1926) (can be read online at the journal's website)

  • Prüfer: "Neue Begründung der algebraischen Zahlentheorie", Math. Annalen 94 (1925), 198-243 (a link to volume 94 of the journal is at the Göttingen archive here)

in both of which one main idea seems to be (in modern language) to consider the embedding of a ring of integers $\frak{o}$ into the product $ \prod_{\frak{p},n}\frak{o}/\frak{p}^n$. The Von Neumann paper even mentions the $p$-adics. That's about all I could extract at a glance, my German being virtually nonexistent - someone with better German will be able do a more thorough job.

EDIT (after further reading):

The aim of both papers appears to be to develop a theory of "Dedekind ideal numbers" in which they appear as elements of an actual ring. The essential difference (in modern language) is that Prüfer uses the algebraic definition of the profinite completion of the ring of integers, whereas Von Neumann takes as his starting point the completion of the number field with respect to the product of the $p$-adic topologies. (So his ring of adeles is simply the product of the finite completions of the number fields, with the product topology). Both authors spend most of the time proving basic algebraic/topological facts about these rings. I could find no significant arithmetic applications in either paper, although Von Neumann appears to promise a sequel (never published) in which he looks at adeles of $\overline{\mathbb{Q}}$ rather than of a fixed number field, and uses them to prove a "unique factorisation" for Dedekind ideal numbers.

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    $\begingroup$ Before the Math Reviews, there was the Jahrbuch. For Hasse's review of Prüfer's paper, see emis.de/cgi-bin/jfmen/MATH/JFM/… $\endgroup$ – Chandan Singh Dalawat Oct 7 '10 at 4:53
  • $\begingroup$ It's a pity Jahrbuch has no entry for von Neumann's paper. Can anyone who reads German comfortably indicate what either of these papers actually achieved with the construction? $\endgroup$ – KConrad Oct 7 '10 at 8:47
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    $\begingroup$ When I asked what the papers actually achieved, I meant besides the idea of embedding the ring of integers into its product of non-archimedean completions. I mean, the point of the papers couldn't have been "Look, we have this embedding", right? $\endgroup$ – KConrad Oct 7 '10 at 10:17
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    $\begingroup$ Wo ist Franz ?? $\endgroup$ – Chandan Singh Dalawat Oct 7 '10 at 13:14
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    $\begingroup$ I have read some more and edited my reply accordingly $\endgroup$ – Tony Scholl Oct 7 '10 at 13:22

Towards the end of his exposé on Groupes de Galois : le cas abélien (27/10/2011), Jean-Pierre Serre says that

In 1936, Chevalley introduced the idèles with a topology which was not separated; in 1936 Weil defined the true (la vraie) topology on the idèles and their relation to Hecke characters — that was important.

Here is a transcript of what he says at 52:20 in the video, reading from his notes :

1936, Chevalley, idèles avec une topologie non séparée ! [J’avais bien un point d’exclamation.]

1936, Weil, les idèles avec leur vraie topologie et la relation avec les caractères de Hecke — ça c’était important.


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