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?