Kronecker's Jugendtraum (or Hilbert's 12'th problem) is to find abelian extensions of arbitrary number fields by adjoining `special' values of transcendental functions. The Kronecker-Weber theorem was the first realisation of this: i.e. $\mathbb{Q}^{ab}=\mathbb{Q}^{cycl}=\mathbb{Q}(e^{2\pi i \mathbb{Q}})$.
If $K$ is an imaginary quadratic field, then the theory of complex multiplication realises Kronecker's dream: roughly $K^{ab}=K(j(\tau), \wp(\tau,z))$, where $\tau$ is a `special' value of the upper half plane corresponding to an elliptic curve $E\cong\mathbb{C} / \Lambda$ with complex multiplication by the ring of integers of $K$, and $z \in \Lambda \otimes \mathbb{Q}$ (this is like adjoining $j(E)$ and the $x$-coordinates of the torsion points of $E$).
Is there some class of special analytic functions, and some special kind of objects which conjecturally play the role of the $j$-function, the $\wp$-function and CM elliptic curves for real quadratic fields?
Best Answer
For totally real quadratic fields, Stark's conjecture (still a conjecture) gives an answer. I quote:
In the case that $k$ is totally real, Tate (1984, 3.8) determines the subfield they generate; for example, when $[K:Q]=2$, they generate the abelian closure of $k$ in $\mathbb{R}$. This has implications for Hilbert's 12th problem. To paraphrase Tate (ibid. p.95):
"If the conjecture $\mathrm{St}(K/k,S)$ is true in this situation, then the formula $\varepsilon=\exp(-2\zeta^{\prime}(0,1))$ gives generators of abelian extensions of $k$ that are special values of transcendental functions. Finding generators of class fields of this shape is the vague form of Hilbert's 12th problem, and the Stark conjecture represents an important contribution to this problem. However, it is a totally unexpected contribution: Hilbert asked that we discover the functions that play, for an arbitrary number field, the same role as the exponential function for $\mathbb{Q}{}$ and the elliptic modular functions for a quadratic imaginary field. In contrast, Stark's conjecture, by using $L$-functions directly, bypasses the transcendental functions that Hilbert asked for. Perhaps a knowledge of these last functions will be necessary for the proof of Stark's conjecture."
Remarkably, $St(K/k,S)$ is useful for the explicit computation of class fields, and has even been incorporated into the computer algebra system PARI/GP.
That was the situation in 1985, but, as Lemmermeyer notes, Darmon and others have been studying these questions.