Given a global field $F$, we can construct the ring of adeles. Given a general locally compact ring $R$, when is it isomorphic to the ring of adeles of some global field $F$ and how can I find $F$ in $R$?


Iwasawa gave a characterisation, assuming you are given a subfield F, discrete and such that the quotient is compact. The other conditions are R a semisimple locally compact commutative topological ring with 1 (shared with F). Then R is the adele ring of the global field F.

Edit: I believe it is known that you can't get F from knowledge of R alone. I don't remember details or a reference, but it is something like the fact that the Dedekind zeta function doesn't determine the number field? In other words the ramification degrees e and residue class extension degrees f can be known for each prime, and this will tell you the adele ring R as a restricted product of local fields. But not the field F. Given R, there may be more than one candidate field it contains.

  • 1
    $\begingroup$ Link to Iwasawa's paper (On the rings of valuation vectors, Annals 1953) jstor.org/pss/1969863 it seems like knowing the adele ring as a topological ring is much more information than the zeta function. so it would be interesting to know either an example of different F's with the same R! $\endgroup$ – SGP Apr 9 '11 at 21:39
  • 4
    $\begingroup$ An example of nonisomorphic number fields with isomorphic adele rings is given by de Smit and Perlis in ams.org/journals/bull/1994-31-02/S0273-0979-1994-00520-8/…. The fields are Q((-33)^(1/8)) and Q((-33*16)^(1/8)). They say that PARI shows these fields have different class number, so in paticular the adele ring does not in general determine the class number. $\endgroup$ – KConrad Apr 10 '11 at 3:06
  • $\begingroup$ @KConrad - thanks. Wonders aloud if there is any extra mileage in Tate's thesis in the situation of one R, two Fs. Someone else's thesis topic? $\endgroup$ – Charles Matthews Apr 10 '11 at 10:27
  • $\begingroup$ @KConrad: nice example! $\endgroup$ – SGP Apr 10 '11 at 12:31
  • 1
    $\begingroup$ In Section 2 of de Smit and Perlis' paper there is the following extraordinry sentence, speaking about computations which are in accordance with the main theorem: "Of course this does not qualify as a proof of the theorem. Perhaps rodents in the bowels of the computer center are chewing on wires and altering data". $\endgroup$ – Filippo Alberto Edoardo Jun 3 '18 at 13:51

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.