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?



Group laws in class field theory
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](http://books.google.es/books/about/Complex_multiplication_of_Abelian_variet.html?id=-eHuAAAAMAAJ&redir_esc=y)).


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).


