Class field theory - a "dead end"?

Question

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?

Answer 1

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.

2. 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 . . .

3. 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 . . .

4. 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.

Answer 2

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.

Answer 3

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.