Hilbert's 12th problem

Let $K$ be an algebraic number field (a finite extension of $\mathbb{Q}$) with a fixed algebraic closure $\overline{K}$, and let

$$K^{\mathrm{ab}}=\bigcup\{\,L:\;K\subseteq L\subseteq\overline{K},\;[L:K]<\infty,\;\operatorname{Gal}(L/K)\ \text{abelian}\,\}$$

be the maximal abelian extension of $K$.

For every such $K$ there exist analytic (transcendental) functions $f_i$ of one or several complex variables, given by explicit analytic expressions attached to $K$ alone, and an explicitly described set $S_i$ of arguments in the domain of each $f_i$, arising from $K$-rational data (elements of $K$, lattices or period data attached to $K$, and torsion points of the associated analytic groups), such that every value $f_i(s)$ with $s\in S_i$ is algebraic over $K$ and

$$K^{\mathrm{ab}}=K\bigl(f_i(s)\;:\;i\in I,\;s\in S_i\bigr),$$

with, in addition, an explicit description of the action of $\operatorname{Gal}(K^{\mathrm{ab}}/K)$ on these special values, so that for each finite abelian extension $L/K$ a finite subset of the values generating $L$ over $K$ can be exhibited.

The shape of the required data is fixed by the following two instances. For $K=\mathbb{Q}$ one may take $f(z)=e^{2\pi i z}$ and $S=\mathbb{Q}$, so that $\mathbb{Q}^{\mathrm{ab}}=\mathbb{Q}\bigl(e^{2\pi i/n}:n\ge 1\bigr)$. For $K=\mathbb{Q}(\tau)$ imaginary quadratic with $\operatorname{Im}\tau>0$, one may take the elliptic modular function $j$, the Weierstrass function $z\mapsto\wp(\tau,z)$ associated with the lattice $\mathbb{Z}+\mathbb{Z}\tau$, and the exponential function: $K^{\mathrm{ab}}$ is generated over $K$ by $j(\tau)$, by the values $\wp(\tau,z)$ at the torsion points $z$ of $\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau)$, and by the roots of unity.