Timeline for Why is Kronecker's Jugendtraum only for abelian extensions?
Current License: CC BY-SA 3.0
11 events
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| Aug 20, 2016 at 13:34 | answer | added | Myshkin | timeline score: 8 | |
| Aug 20, 2016 at 13:34 | history | edited | Myshkin | CC BY-SA 3.0 |
+ top level tag (nt.) + cft and open problem tags
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| Jan 9, 2016 at 18:25 | comment | added | zeno | 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. | |
| Jan 9, 2016 at 11:44 | comment | added | user5831 | 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... | |
| Jan 9, 2016 at 11:40 | comment | added | user5831 | 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... | |
| Jan 8, 2016 at 17:57 | comment | added | user19475 | Yes, I am asking why there is no general conjecture for generating extensions of (certain) number fields by adjoining special values of transcendental functions. | |
| Jan 8, 2016 at 15:20 | comment | added | user5831 | 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? | |
| S Jan 8, 2016 at 15:07 | history | suggested | JMP | CC BY-SA 3.0 |
added links
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| Jan 8, 2016 at 15:00 | review | Suggested edits | |||
| S Jan 8, 2016 at 15:07 | |||||
| Jan 8, 2016 at 13:27 | review | Low quality posts | |||
| Jan 8, 2016 at 13:40 | |||||
| Jan 8, 2016 at 13:09 | history | asked | user19475 | CC BY-SA 3.0 |