Modern Development
Developments since around 1960 have certainly contributed. 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. This gives rise to abelian extensions of CM-fields in general. The question of which extensions can be found is that of the Tate modules of such varieties, as Galois representations. Since this is the most accessible case of l-adic cohomology, these representations have been studied in depth.
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 (Harold Stark), which in contrast dealt directly with the question of finding interesting, particular units in number fields. This has seen a large conjectural development for L-functions, and is also capable of producing concrete, numerical results.
Read more about this topic: Hilbert's Twelfth Problem
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