In the case of a quadratic imaginary number field one can construct its maximal abelian extension using torsion points of an elliptic curve with complex multiplication by this field.

In the case of a local field, Lubin-Tate theory provides an explicit construction of its maximal abelian extension using torsion points of a formal group law on the maximal ideal.

Are there any examples of similar arguments, or even general facts?

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    $\begingroup$ Have you heard of Hilbert's 12th problem? $\endgroup$ – KConrad Mar 1 '15 at 22:37
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    $\begingroup$ Look up "Carlitz-Hayes theory" for more examples. E.g. the abelian extensions of the rational function field $\mathbb{F}_p(T)$ are obtained from torsion of the Carlitz module (except the ones wildly ramified at infinity, which are obtained in a similar way using a parameter $1/T$ at $\infty$). $\endgroup$ – Vesselin Dimitrov Mar 2 '15 at 0:29
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    $\begingroup$ @KConrad Sure, but I meant any generalizations of this particular argument with group law, maybe for smaller class of fields than all number fields. $\endgroup$ – SashaP Mar 2 '15 at 12:26

As mentioned in the comments, this is precisely Hilbert's twelfth problem, for the simple reason that any solution to that problem can be turned into a "group law argument" (or any solution to it is a "group law argument" in disguise in the first place).

For example, you can think of the Kronecker-Weber theorem as saying that abelian extensions of $\mathbb{Q}$ are generated by torsion points of its multiplicative group.

The most general result of this type is Shimura's complete solution for CM-fields (see Complex multiplication of Abelian varieties and its applications to number theory).

You can complete the global field picture as mentioned in the comments (Carlitz-Hayes-Drinfeld) and the local one with Lubin-Tate.

PS. I corrected the question: "In the case of quadratic number field...". Only the imaginary case is know. For real quadratic fields this is wide open (the best way to attack it seems to be via the Stark conjectures).

  • $\begingroup$ Could you precise what you have in mind by "any solution to that problem can be turned into a "group law argument" " ? I don't really see why something like that should be true. $\endgroup$ – Simon Henry Mar 4 '15 at 8:52
  • $\begingroup$ @SimonHenry. Mmmm. I think any construction of explicit class field theory could be expressed from the point of view of arithmetic geometry as "built from torsion points of X". For example, if I understand correctly, some (including Langlands) expect that all maximal abelian extensions come from torsion points of Shimura varieties. $\endgroup$ – Myshkin Mar 4 '15 at 17:25
  • $\begingroup$ @SimonHenry. So, all known solution either come from group laws (imaginary quadratic, CM) or can be turned into one (Kronecker-Weber), and the same goes for the only conjectured solution to the full problem. $\endgroup$ – Myshkin Mar 4 '15 at 17:32
  • $\begingroup$ Ok, I agree that all the known examples of the Hilbert 12th problem are of this form, and I didn't know there was such conjecture. Now there is examples of other sort, for example one can give a complete description of the maximal abelian extension of $\mathbb{C}(X)$ (and of its Galois group) which I don't think fit into this picture of torsion points of some sort of algebraic or formal group. $\endgroup$ – Simon Henry Mar 4 '15 at 19:55
  • $\begingroup$ (I misunderstood your original message and thought that you were talking about "any possible conjectural solution to the Hilbert 12th problem") $\endgroup$ – Simon Henry Mar 4 '15 at 20:01

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