48
$\begingroup$

I found the claim in the title a bit astonishing when I first read it recently in an interview with Michael Rapoport in the German magazine Spiegel (8 February 2019). And I was wondering how he comes to that conclusion. Here is the article, but the full interview is not available for free, so I will paraphrase the relevant part.

Rapoport talks about dead ends in mathematics and brings up class field theory as an example. He basically says: Class field theory had been proven nearly 100 years ago, and, after that, researchers spent about 70 years to turn it into a satisfactory theory. However, along the way it was realized that the original goal of class field theory had to be abandoned because it did not turn out to be fruitful.

I had only few contacts with people class field theory but never had the impression that number theorists were thinking about it in this way. So I wonder how to interpret Rapoport's claims. I think it boils down to the following questions:

  1. What were the original goals of class field theory?
  2. Why did it not turn out to be fruitful, and is this failure somehow quantifiable?
  3. Are there new ideas the take up the original goal?
  4. Is class field theory rendered obsolete by more general ideas?
| cite | improve this question | | | | |
$\endgroup$
  • 26
    $\begingroup$ I think the original hope was to describe the set of primes splitting completely in non-abelian extensions in a way that would closely resemble the abelian case. The viewpoint that eventually developed into the Langlands program, which is the modern proposed generalization of class field theory to all Galois extensions, looks quite different from how Artin, Hecke, et al. imagined "nonabelian class field theory". At the same time, they couldn't really formulate what they wanted Artin said the main problem was to state what is to be proved, but history has shown that to be an understatement. $\endgroup$ – KConrad Feb 20 '19 at 19:00
  • 9
    $\begingroup$ None of these statements describes class field theory and its history in a way I would agree with. $\endgroup$ – Franz Lemmermeyer Feb 20 '19 at 23:36
  • $\begingroup$ @FranzLemmermeyer: Which comments? Those by Rapoport or those by KConrad ? $\endgroup$ – wood Feb 21 '19 at 18:46
  • $\begingroup$ Rapoport, of course.Sorry for the ambiguity. $\endgroup$ – Franz Lemmermeyer Feb 22 '19 at 5:07
  • 1
    $\begingroup$ Sections 3.10 and 3.11 of ams.org/journals/bull/2018-55-04/S0273-0979-2018-01609-1/… might be of relevance here. $\endgroup$ – ThiKu Feb 24 '19 at 17:19
2
$\begingroup$

The statements ascribed to Rapoport are nonsense --- they must have been garbled in the transmission. I'd guess he may have said that the approaches to nonabelian class field theory before Langlands were a dead end.

The google translate of the original paragraph still makes no sense. Perhaps one could make sense of it in context. The main theorems of abelian class field theory were proved in the 1910s but there were major improvements to the theory in following years (Hasse, Chevalley, Artin, Tate ...). Abelian class field theory remains of fundamental importance.

"don't waste your time with class field theory" by itself doesn't make sense either. The Langlands program incorporates a nonabelian class field theory, and to understand the Langlands program you need to understand abelian class field theory.

| cite | improve this answer | | | | |
$\endgroup$
  • 2
    $\begingroup$ The original quote is as follows: „Neh­men Sie die wich­ti­ge Klas­sen­kör­per­theo­rie, die be­reits An­fang des 20. Jahr­hun­derts be­wie­sen wur­de. 70 Jah­re lang wa­ren die Kol­le­gen da­mit be­schäf­tigt, die­se Theo­rie in eine all­ge­mein ak­zep­ta­ble Form zu brin­gen. Lei­der stell­te sich da­bei her­aus, dass das ur­sprüng­li­che Ziel auf­ge­ge­ben wer­den muss­te, es er­wies sich ein­fach als un­frucht­bar. Trotz­dem gibt es im­mer noch Leu­te, die dar­an for­schen. Aber in der Ge­samt­schau hat das kei­ner­lei Be­deu­tung mehr. Die Ent­wick­lung ging über die­se Grup­pe hin­weg.“ $\endgroup$ – ThiKu Feb 21 '19 at 9:10
  • 5
    $\begingroup$ In the penultimate sentence of the translation, "no importance" would be better than "no meaning". $\endgroup$ – Laurent Moret-Bailly Feb 21 '19 at 15:03
  • 4
    $\begingroup$ I am a native german speaker, and to me the comment from Rapoport really has a connotation of "don't waste your time with class field theory" - of course we cannot know what he said exactly compared to what was written in the magazine instead. $\endgroup$ – wood Feb 21 '19 at 18:49
  • 3
    $\begingroup$ Even in the context of explicit class field theory, it is a very strange statement that it has no significance today. That programme did not get far, but it gave us CM theory, Heegner points, and with that some of the most spectacular successes of 20th century number theory, such as Gauss's class number 1 problem for imaginary quadratic fields and cases of the BSD. $\endgroup$ – Alex B. Feb 22 '19 at 10:18
  • 3
    $\begingroup$ @Alex B.: I very much doubt Rapoport is denying all those advances, because he has contributed to that field himself. Kudla-Rapoport cycles are vast generalizations of Heegner points that have Gross-Zagier type properties, which is key to the BSD advances you mention. I think this is exactly what Rapoport means by "the development went beyond". $\endgroup$ – Zavosh Feb 26 '19 at 17:52

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.