View source for Hilbert's twelfth problem

{{Short description|Problem about mathematical number fields}}
{{quotebox|align=right|width=33%
|quote={{lang|de|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.}}
|source=Kronecker in a letter to Dedekind in 1880 reproduced in volume V of his collected works, page 455}}

'''Hilbert's twelfth problem''' is the extension of the [[Kronecker–Weber theorem]] on [[abelian extension]]s of the [[rational number]]s, to any base [[number field]]. It is one of the 23 mathematical [[Hilbert problems]] and asks for analogues of the [[roots of unity]] that generate a whole family of further number fields, analogously to the [[cyclotomic field]]s and their subfields. [[Leopold Kronecker]] described the complex multiplication issue as his {{lang|de|liebster Jugendtraum}}, or "dearest dream of his youth", so the problem is also known as '''Kronecker's Jugendtraum'''.

The classical theory of [[complex multiplication]], now often known as the {{lang|de|Kronecker Jugendtraum}}, does this for the case of any [[imaginary quadratic field]], by using [[modular form|modular functions]] and [[elliptic function]]s chosen with a particular [[period lattice]] related to the field in question. [[Goro Shimura]] extended this to [[CM field]]s. In the special case of totally real fields, [[Samit Dasgupta]] and [[Mahesh Kakde]] provided a construction of the maximal abelian extension of totally real fields using the [[Brumer–Stark conjecture]].

The general case of Hilbert's twelfth problem is still open.

== Description of the problem ==

The fundamental problem of [[algebraic number theory]] is to describe the [[algebraic number field|fields of algebraic numbers]]. The work of [[Évariste Galois|Galois]] made it clear that field extensions are controlled by certain [[group (mathematics)|groups]], the [[Galois group]]s. The simplest situation, which is already at the boundary of what is well understood, is when the group in question is [[abelian group|abelian]]. All quadratic extensions, obtained by adjoining the roots of a quadratic polynomial, are abelian, and their study was commenced by [[Carl Friedrich Gauss|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 field]]s. Already Gauss had shown that, in fact, every [[quadratic field]] is contained in a larger cyclotomic field. The [[Kronecker–Weber theorem]] shows that any finite abelian extension of '''Q''' is contained in a 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]].

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 function ''j''.)
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 the [[Hilbert class field]], for more general abelian extensions one also needs to use values of elliptic functions. For example, the abelian extension <math>\mathbf{Q}(i,\sqrt[4]{1+2i})/\mathbf{Q}(i)</math> 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 of '''Q''' can be obtained by adjoining the special values exp(2{{pi}}''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  ℘(''τ'',''z'') and ''j''(''τ'') of [[modular function]]s ''j'' and  elliptic functions ℘, and roots of unity, where ''τ'' is in the imaginary quadratic field and ''z'' 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 extension '''K'''<sup>ab</sup> of a general number field '''K'''. In this form, it remains unsolved. A description of the field '''K'''<sup>ab</sup> was obtained in the [[class field theory]], developed by [[David Hilbert|Hilbert]]
himself, [[Emil Artin]], and others in the first half of the 20th century.<ref group="note">In particular, [[Teiji Takagi]] proved the existence of the absolute abelian extension as the well-known [[Takagi existence theorem]].</ref> However the construction of '''K'''<sup>ab</sup> in class field theory involves first constructing larger non-abelian extensions using [[Kummer 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.

== Modern developments ==

Developments since around 1960 have certainly contributed. Before that {{harvs|txt|authorlink=Erich Hecke|last=Hecke|year=1912}} in his dissertation used [[Hilbert modular form]]s to study abelian extensions of [[real quadratic field]]s. [[Complex multiplication of abelian varieties]] was an area opened up by the work of [[Goro Shimura|Shimura]] and [[Yutaka Taniyama|Taniyama]]. This gives rise to abelian extensions of [[CM-field]]s in general. The question of which extensions can be found is that of the [[Tate module]]s of such varieties, as [[Galois representation]]s. Since this is the most accessible case of [[l-adic cohomology|ℓ-adic cohomology]], these representations have been studied in depth.

[[Robert Langlands]] argued in 1973 that the modern version of the ''{{lang|de|Jugendtraum}}'' should deal with [[Hasse–Weil zeta function]]s of [[Shimura variety|Shimura varieties]]. While he envisaged a [[Langlands program|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-function|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.<ref>{{cite arXiv|last1=Dasgupta|first1=Samit|last2=Kakde|first2=Mahesh|date=2021-03-03|title=Brumer-Stark Units and Hilbert's 12th Problem|class=math.NT|eprint=2103.02516}}</ref><ref>{{Cite web|last=Houston-Edwards|first=Kelsey|date=2021-05-25|title=Mathematicians Find Long-Sought Building Blocks for Special Polynomials|url=https://www.quantamagazine.org/mathematicians-find-polynomial-building-blocks-hilbert-sought-20210525/|access-date=2021-05-28|website=Quanta Magazine|language=en}}</ref>

==Notes==
{{Reflist|group=note}}

== References ==
===Footnotes===
{{Reflist}}

===Sources===
*{{cite book |chapter=Some contemporary problems with origins in the Jugendtraum |title=Mathematical developments arising from Hilbert problems |last=Langlands |first=R. P. |author-link=Robert Langlands |year=1976 |series=Proc. Sympos. Pure Math. |volume=28 |publisher=[[American Mathematical Society]] | location=Providence, RI | isbn=0-8218-1428-1 |pages=401–418 | editor-first=Felix E. | editor-last=Browder | editor-link=Felix Browder | zbl=0345.14006 |url=http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/pdf/jugend-ps.pdf}}
*{{Cite book | last1=Schappacher | first1=Norbert | title=Matériaux pour l'histoire des mathématiques au XX<sup>e</sup> siècle (Nice, 1996) | url=http://www.emis.de/journals/SC/1998/3/html/smf_sem-cong_3_243-273.html | publisher=[[Société Mathématique de France]] | location=Paris | series=Sémin. Congr. | isbn=978-2-85629-065-1  | mr=1640262 | year=1998 | volume=3 | chapter=On the history of Hilbert's twelfth problem: a comedy of errors | pages=243–273 | zbl=1044.01530 }}
*{{cite book |title=Kronecker's Jugendtraum and modular functions |last=Vlǎduţ |first=S. G. |year=1991 |series=Studies in the Development of Modern Mathematics |volume=2 |publisher=Gordon and Breach Science Publishers |location=New York |isbn=2-88124-754-7 | zbl=0731.11001 }}

{{Hilbert's problems}}

[[Category:Algebraic number theory]]
[[Category:Conjectures]]
[[Category:Hilbert's problems|#12]]