Hilbert's original statement of his 12th problem is rather misleading: he seems to imply that the abelian extensions of imaginary quadratic fields are generated by special values of elliptic modular functions, which is not correct.
One particularly appealing way to state the Kronecker–Weber theorem is by saying that the maximal abelian extension of Q can be obtained by adjoining the special values exp(2πi/n) of the exponential function.
One interpretation of Hilbert's twelfth problem asks to provide a suitable analogue of exponential, elliptic, or modular functions, whose special values would generate the maximal abelian extension Kab of a general number field K. In this form, it remains unsolved.
Before that Hecke (1912) in his dissertation used Hilbert modular forms to study abelian extensions of real quadratic fields.
Complex multiplication of abelian varieties was an area opened up by the work of Shimura and Taniyama.
Robert Langlands argued in 1973 that the modern version of the Jugendtraum should deal with Hasse–Weil zeta functions of Shimura varieties.
While he envisaged a grandiose program that would take the subject much further, more than thirty years later serious doubts remain concerning its import for the question that Hilbert asked.
A separate development was Stark's conjecture (in the abelian rank-one case), which in contrast dealt directly with the question of finding particular units that generate abelian extensions of number fields and describe leading coefficients of Artin L-functions.
In 2021, Dasgupta and Kakde announced a p-adic solution to finding the maximal abelian extension of totally real fields by proving the integral Gross–Stark conjecture for Brumer–Stark units.