I am aware that the theory of complex multiplication provides us with special functions whose values describe $K^\text{ab}$ where $K$ is an imaginary quadratic extension of the rationals. And that drinfeld modules provide a similarly explicit description of the maximal abelian extension of a global function field.

I am [vaguely] aware that the langlands programme would give us information about nonabelian extensions of global fields.

But what the problem of giving a concrete set of generators for abelian extensions of other fields $K$ when $K$ is not $\mathbb{Q}$, an imaginary quadratic number field, or a global function field? Why can't CM work on say, a real abelian number field? What goes wrong when you try to take the ideas of drinfeld modules back to an arbitrary number field?

Put another way - what is the state of Hilbert's 12th problem? Is it entirely subsumed by the langlands programme? And if not, what are the other proposed lines of attack?

  • $\begingroup$ About this Drinfeld module, what I know is that thanks to the Riemann-Roch theorem, finite extensions of $\mathbb{F}_p(t)$ are much easier than finite extensions of $\mathbb{Q}$ when studying their zeta functions (the Langland program), see this proof of the RH. $\endgroup$ – reuns May 26 '17 at 1:35

As one out of many attempts at constructing class fields let me point out the work by Henri Darmon (see e.g. Elliptic Curves and Class Fields of Real Quadratic Fields: Algorithms and Evidence) on the construction of class fields of real quadratic number fields.


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