who fixed the topology on ideles?

Question

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 second page of the memorial article Claude Chevalley (1909–1984) by Dieudonné and Tits in Bulletin AMS 17 (1987) (doi:10.1090/S0273-0979-1987-15509-1), where they also say that Chevalley's introduction of the ideles was "a definite improvement on earlier similar ideas of Prüfer 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 Prüfer and von Neumann were doing, with references.]

I have two questions:

1) Can anyone point to a specific article where Prüfer 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.)

Answer 1

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.

Answer 2

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.

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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.

Answer 3

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.

Answer 4

Perhaps it bears noting that, given a commutative topological ring $R$, the group $R^\times$ of units has natural topology given by the subspace topology under the imbedding $x\to (x,x^{-1})\in R\times R$. In particular, this is the coarsest that makes inversion continuous, etc.

(And from the adeles to the ideles this gives the correct topology, unsurprisingly.)

But, yes, this style of characterization was not the norm in those days.