Es handelt sich um meinen liebsten Jugendtraum, nämlich um den Nachweis, dass die Abel’schen Gleichungen mit Quadratwurzeln rationaler Zahlen durch die Transformations-Gleichungen elliptischer Functionen mit singularen Moduln grade so erschöpft werden, wie die ganzzahligen Abel’schen Gleichungen durch die Kreisteilungsgleichungen.
Hilbert's twelfth problem is the extension of theKronecker–Weber theorem onabelian extensions of therational numbers, to any basenumber field. It is one of the 23 mathematicalHilbert problems and asks for analogues of theroots of unity that generate a whole family of further number fields, analogously to thecyclotomic fields and their subfields.Leopold Kronecker described the complex multiplication issue as hisliebster Jugendtraum, or "dearest dream of his youth", so the problem is also known asKronecker's Jugendtraum.
The classical theory ofcomplex multiplication, now often known as theKronecker Jugendtraum, does this for the case of anyimaginary quadratic field, by usingmodular functions andelliptic functions chosen with a particularperiod lattice related to the field in question.Goro Shimura extended this toCM fields. In the special case of totally real fields,Samit Dasgupta andMahesh Kakde provided a construction of the maximal abelian extension of totally real fields using theBrumer–Stark conjecture.
The general case of Hilbert's twelfth problem is still open.
The fundamental problem ofalgebraic number theory is to describe thefields of algebraic numbers. The work ofGalois made it clear that field extensions are controlled by certaingroups, theGalois groups. The simplest situation, which is already at the boundary of what is well understood, is when the group in question isabelian. All quadratic extensions, obtained by adjoining the roots of a quadratic polynomial, are abelian, and their study was commenced byGauss. Another type of abelian extension of the fieldQ ofrational numbers is given by adjoining thenth roots of unity, resulting in thecyclotomic fields. Already Gauss had shown that, in fact, everyquadratic field is contained in a larger cyclotomic field. TheKronecker–Weber theorem shows that any finite abelian extension ofQ is contained in a cyclotomic field. Kronecker's (and Hilbert's) question addresses the situation of a more general algebraic number fieldK: what are the algebraic numbers necessary to construct all abelian extensions ofK? The complete answer to this question has been completely worked out only whenK is animaginary quadratic field or its generalization, aCM-field.
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. (It is hard to tell exactly what Hilbert was saying, one problem being that he may have been using the term "elliptic function" to mean both the elliptic function ℘ and the elliptic modular functionj.)First it is also necessary to use roots of unity, though Hilbert may have implicitly meant to include these. More seriously, while values of elliptic modular functions generate theHilbert class field, for more general abelian extensions one also needs to use values of elliptic functions. For example, the abelian extension is not generated by singular moduli and roots of unity.
One particularly appealing way to state the Kronecker–Weber theorem is by saying that the maximal abelian extension ofQ can be obtained by adjoining the special values exp(2πi/n) of theexponential function. Similarly, the theory ofcomplex multiplication shows that the maximal abelian extension ofQ(τ), whereτ is an imaginary quadratic irrationality, can be obtained by adjoining the special values of ℘(τ,z) andj(τ) ofmodular functionsj and elliptic functions ℘, and roots of unity, whereτ is in the imaginary quadratic field andz represents a torsion point on the corresponding elliptic curve. 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 extensionKab of a general number fieldK. In this form, it remains unsolved. A description of the fieldKab was obtained in theclass field theory, developed byHilberthimself,Emil Artin, and others in the first half of the 20th century.[note 1] However the construction ofKab in class field theory involves first constructing larger non-abelian extensions usingKummer theory, and then cutting down to the abelian extensions, so does not really solve Hilbert's problem which asks for a more direct construction of the abelian extensions.
Developments since around 1960 have certainly contributed. Before thatHecke (1912) in his dissertation usedHilbert modular forms to study abelian extensions ofreal quadratic fields.Complex multiplication of abelian varieties was an area opened up by the work ofShimura andTaniyama. This gives rise to abelian extensions ofCM-fields in general. The question of which extensions can be found is that of theTate modules of such varieties, asGalois representations. Since this is the most accessible case ofℓ-adic cohomology, these representations have been studied in depth.
Robert Langlands argued in 1973 that the modern version of theJugendtraum should deal withHasse–Weil zeta functions ofShimura varieties. While he envisaged agrandiose 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 wasStark'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 ofArtinL-functions. In 2021, Dasgupta and Kakde announced ap-adic solution to finding the maximal abelian extension of totally real fields by proving the integral Gross–Stark conjecture for Brumer–Stark units.[1][2]