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Kronecker's Jugendtraum (Hilbert's 12th problem) asks us to find for any number field $K$ an explicit collection of complex-valued functions whose explicitly described special values generate the maximum abelian extension $K^{ab}$ of $K$.

Its solution has been known for a long time for $\mathbb{Q}$ (exponential function/division points of the circle) and imaginary quadratic fields (elliptic modular functions/division points of CM elliptic curves.) There are at least partial results towards its solution for CM fields (Shimura, Taniyama, Wafa Wei et al. using CM abelian varieties) and totally real fields (Manin using non-commutative tori, Darmon et al.). Please see answers to these related posts for some details: Kronecker's Jugendtraum for real quadratic fields? and CM fields and Hilberts 12th problem

Q. 1: Is there a conceptual reason why tori feature in all these solutions and proposals?

Q. 2: To what extent has Manin's non-commutative tori approach yielded results?

Q. 3: Why could one not dream of an explicit set of transcendental functions whose special values would generate the algebraic closure $\overline{K}$?

The last has been asked before: Why is Kronecker's Jugendtraum only for abelian extensions?, but the answers seem to suggest that "yes, one could, but perhaps not should because first you need to have non-abelian class field theory in place." But do we really need to have non-abelian CFT established before we can even begin to discern what shape a non-abelian Jugendtraum might take? Does the concrete program for realizing non-abelian CFT given by Langlands conjectures not give us any clues to what the non-abelian analogs of tori might be?

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  • $\begingroup$ I am not even a noob for this circle of topics, but naïvely one could say that for non-abelian extensions you should use functions which do not commute, which presumably means that what you are asking for is the non-abelian class field theory. $\endgroup$ – მამუკა ჯიბლაძე 2 days ago

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