Why is Kronecker's Jugendtraum only for abelian and not for more general extensions of number fields?
Wikipedia, Hilbert's Twelfth Problem.
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4$\begingroup$ Do you ask, why Kronecker did not give a more general conjecture about explicit constructions of non-abelian algebraic extensions of a given number field? we already don't have any idea how to construct the class field theory of $\mathbb Q (\sqrt(2))$ explicitly. Also, when Kronecker was alive, there was no "fully developed" class field theory... (even his proof of the Kronecker-Weber theorem (explicit cft for $\mathbb Q$) was given its final (correct) proof only in 1904 (or 5) by Hilbert.) Or are you asking something else? $\endgroup$– user5831Commented Jan 8, 2016 at 15:20
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$\begingroup$ Yes, I am asking why there is no general conjecture for generating extensions of (certain) number fields by adjoining special values of transcendental functions. $\endgroup$– user19475Commented Jan 8, 2016 at 17:57
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3$\begingroup$ It's well known that you can write any algebraic number as special value of some finite combination of theta functions. The nice thing about the known cases of explicit class field theory is that you somehow obtain a geometric interpretation of the action of the Galois group involved. A true geometric picture of the geometry of number fields is (the) a central and wide open problem in number theory. We hardly know anything about that. People in p-adic Hodge are starting to understand to local situation in a geometric way now, but the global setting is still a mystery... $\endgroup$– user5831Commented Jan 9, 2016 at 11:40
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1$\begingroup$ For example, even explicit (abelian) reciprocity laws for local fields in char 0 are not well understood geometrically (although that might change in the near future). So, if you don't even understand the abelian reciprocity laws etc there's little hope to be able to conjecture anything non-trivial about non-abelian things. but maybe I'm mistaken... $\endgroup$– user5831Commented Jan 9, 2016 at 11:44
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$\begingroup$ I'm not sure that we even have a Jugentraum for CM fields that gives ALL abelian extensions. One problem for nonabelian extensions is that we don't have a "listing" of them as we do for abelian extensions, and so the problem is not well-posed.The theory of Shimura varieties does allow you to generate a few nonabelian extensions by special values --- see 5.7 of the 1981 AJM paper by Milne and Shih. $\endgroup$– zenoCommented Jan 9, 2016 at 18:25
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1 Answer
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The bottom line is that in order to have a "Jugendtraum" for a number field $K$, you first want to have a complete class field theory for it.
Kronecker didn't have a general CFT, so his conjecture refered only to $K=\mathbb{Q}(i)$.
After the tools of Kummer theory and class field theory were in place and it was possible to describe $K^{ab}$, it made sense to ask for an explicit version of those existence results. That is basically the modern formulation of the Jugendtraum.
But the problem of finding a non-abelian class field theory is wide open, so asking for an explicit version of it would be premature at best.