Hilbert's twelfth problem

Hilbert's twelfth problem, of the 23 Hilbert's problems, is the extension of Kronecker-Weber 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 _de. Jugendtraum" (although not so accurately), are still open as of 2005. Leopold Kronecker is supposed to have described the complex multiplication issue as his " _de. liebster Jugendtraum" or “dearest dream of his youth”.

Description of the problem

The "ur"-problem of algebraic number theory is to describe the fields of algebraic numbers. The work of Galois made it clear that field extensions are controlled by certain groups, the Galois groups. The simplest situation, which is already at the boundary of what we can do, is when the group in question is abelian. All quadratic extensions, obtained by adjoining the roots of a quadratic equation, are abelian, and their study was commenced by Gauss. Another type of abelian extension of the field Q of rational numbers is given by adjoining the "n"th roots of unity, resulting in the cyclotomic fields. Already Gauss had shown that, in fact, every quadratic field is contained in a larger cyclotomic field. Kronecker and Weber showed that, in fact, any finite abelian extension of Q is contained a suitably chosen cyclotomic field. Kronecker's (and Hilbert's) question addresses the situation of a more general algebraic number field K: what are the algebraic numbers necessary to construct all abelian extensions of K? The complete answer to this question has been completely worked out only when K is an imaginary quadratic field or its generalization, a CM-field.

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(2π"i"/"n") of the exponential function. Similarly, the theory of complex multiplication shows that the maximal abelian extension of Q(τ), where τ is an imaginary quadratic irrationality, can be obtained by adjoining the special values of more complicated modular functions. One, somewhat narrow, interpretation of Hilbert's twelfth problem asks to provide a suitable analogue of exponential or modular functions, whose special values would generate the maximal abelian abelian extension K^ab of a general number field K. In this form, it remains unsolved. A considerably more abstract description of the field K^ab was obtained in the class field theory, developed by Hilberthimself, Emil Artin, and others in the first half of the 20th century.

Modern development

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 argued in 1973 that the modern version of the " _de. 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.

References

*cite book |chapter=Some contemporary problems with origins in the Jugendtraum |title=Mathematical developments arising from Hilbert problems |last=Langlands |first=R. P. |authorlink= |coauthors= |year=1976 |series=Proc. Sympos. Pure Math. |volume=28 |publisher=Amer. Math. Soc. |location=Providence, RI |isbn= |pages=pp. 401–418 |url=
*cite book |title=Kronecker's Jugendtraum and modular functions |last=Vladut |first=S. G. |authorlink= |coauthors= |year=1991 |series=Studies in the Development of Modern Mathematics |volume=2 |publisher=Gordon and Breach Science Publishers |location=New York |isbn=2881247547 |pages= |url=