Hilbert's twelfth problem

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Hilbert's twelfth problem, of the 23 Hilbert's problems, is the extension of Kronecker's theorem on abelian extensions of the rational numbers, to any base number field. The classical theory of complex multiplication does this for any imaginary quadratic field. The more general cases, now often known as the Kronecker Jugendtraum (although not so accurately), are still open as of 2005. Leopold Kronecker is supposed to have described the complex multiplication issue as his 'liebster Jugendtraum' or dearest dream of his youth.

Developments since around 1960 have certainly contributed; before that, only Erich Hecke's dissertation on the real quadratic field case was considered substantive, and that remained isolated. 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 wrote an important report (Ein Märchen) in relation to automorphic representation theory and extension of the Jugendtraum. The presumable reason for his lapse into German can be explained: Märchen is fairy tale, and his conclusion in respect of extensions came down on the negative side, at least from his viewpoint or Langlands philosophy. The main thrust, on this view, of the higher-dimensional theory is to get control of Shimura varieties, their special points (CM-points), and automorphic L-functions. Evaluation of transcendental functions at given points to create specific algebraic numbers is not dealt with.

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.