63

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 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?
wood
  • 2,714
  • 35
    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. – KConrad Feb 20 '19 at 19:00
  • 13
    None of these statements describes class field theory and its history in a way I would agree with. – Franz Lemmermeyer Feb 20 '19 at 23:36
  • 1
    @FranzLemmermeyer: Which comments? Those by Rapoport or those by KConrad ? – wood Feb 21 '19 at 18:46
  • Rapoport, of course.Sorry for the ambiguity. – Franz Lemmermeyer Feb 22 '19 at 05:07
  • 1
    Sections 3.10 and 3.11 of http://www.ams.org/journals/bull/2018-55-04/S0273-0979-2018-01609-1/S0273-0979-2018-01609-1.pdf might be of relevance here. – ThiKu Feb 24 '19 at 17:19

3 Answers3

16

Let me address your questions 1. - 4.

  1. What were the original goals of class field theory?

The question is a little bit anachronistic; class field theory describes the splitting of primes in abelian extensions, but that was not the original goal. Kronecker and Weber had studied extensions of complex quadratic number fields generated by values of certain functions (for example the j-function) in the theory of complex multiplication, and Kronecker formulated his youthful dream: these values generate abelian extensions of complex quadratic number fields in the same way that the division points of the exponential function generates abelian extensions of the rationals. Hilbert realized the connection between unramified abelian extensions and class groups (proved by Furtwängler), and then Takagi was able to show that it is possible to describe all abelian extensions of number fields in a similar way (using generalized class groups). Only then did class field theory become the theory of abelian extensions of number fields.

The original questions by Kronecker in connection with his youthful dream are covered in the books on elliptic curves by Silverman, Cox's book on primes of the form $x^2 + ny^2$, and Vladut's book on Kronecker's Jugendtraum.

  1. Why did it not turn out to be fruitful?

This question does not make any sense. Even if you forget class field theory, the struggle for understanding the connection between the local and the global case has redefined modern number theory. And then there's L-functions and Galois cohomology and Galois representations . . .

  1. Are there new ideas the take up the original goal?

Kronecker's ideas were generalized in connection with Hilbert's 12th problem. Let me mention the construction of class fields of real quadratic number fields or Stark's conjectures, to mention but two. See also this article.

Class field theory as a theory of abelian extensions was generalized at least conjecturally in Langlands' program. This is a highly active area of number theory. And then there's Iwasawa theory, geometric class field theory and higher class field theory . . .

  1. Is class field theory rendered obsolete by more general ideas?

Is Euclidean geometry rendered obsolete by Riemannian geometry? From a purely mathematical point: yes. From a pedagogical point the answer is a firm no. We still teach the theorem of Pythagoras long before we define metrics on manifolds, and we do so for a reason.

Franz Lemmermeyer
  • 32,176
7

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.

anon
  • 140
  • 3
    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.“ – ThiKu Feb 21 '19 at 09:10
  • Google Translate: „Take the important class field theory that was already proven at the beginning of the 20th century. For 70 years, colleagues were busy putting this theory into a generally acceptable form. Unfortunately, it turned out that the original goal had to be abandoned, it just proved unfruitful. Still, there are still people doing research. But in the overall view, this has no meaning whatsoever. The development went beyond this group.“ – ThiKu Feb 21 '19 at 09:12
  • 5
    In the penultimate sentence of the translation, "no importance" would be better than "no meaning". – Laurent Moret-Bailly Feb 21 '19 at 15:03
  • 7
    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. – wood Feb 21 '19 at 18:49
  • 2
    I think it should be obvious that he must not mean neither the work of Hilbert, Furtwängler and Takagi nor that of Artin, but that he must refer to attempts to generalize Artin‘s work in other directions than those coming from Langland‘s program. – ThiKu Feb 21 '19 at 22:24
  • Perhaps he meant explicit class field theory in the sense of Hilbert's 12th problem. – Zavosh Feb 22 '19 at 02:38
  • 5
    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. – Alex B. Feb 22 '19 at 10:18
  • 5
    @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". – Zavosh Feb 26 '19 at 17:52
  • 5
    Generating abelian extensions of arbitrary number fields by adjoining special values of automorphic forms is one example of a once-central explicit class field theory problem that has not advanced in a long time. – Zavosh Feb 26 '19 at 17:55
2

I've heard the Langlands programme is one way of describing non-Abelian class field theory (there are others). In which case, given what a huge programme that is, it's alive and kicking.

It seems to me, given your synopsis, that the author was looking to shoot down something. One can after all say that Newtonian Mechanics is a dead theory. It was shot down because the vectorial form of Newton mechanics as thought out originally was too clumsy for the kimd of questions that came up. But it transmuted into Lagrangian and Hamiltonian mechanics, which one can say with some justification, is covariant Newtonian mechanics. Plus it inspired Maxwell's theory as well as Einstein's.

What you want to shoot down depends upon how closely a box you want to tie around it. But that box can be, and is here, mostly in the eye of the beholder. Newtonian theory was so seminal, that it simply refused to stay in boxes that certain practitioners were inclined to draw. Likewise with class field theory.

Mozibur Ullah
  • 2,222
  • 14
  • 21