231:
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
34:
Es handelt sich um meinen liebsten
Jugendtraum, nĂ€mlich um den Nachweis, dass die Abelâschen Gleichungen mit Quadratwurzeln rationaler Zahlen durch die Transformations-Gleichungen elliptischer Functionen mit singularen Moduln grade so erschöpft werden, wie die ganzzahligen Abelâschen Gleichungen
385:
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
314:
503:(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
236:.) 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
747:
418:, 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.
443:
496:
that would take the subject much further, more than thirty years later serious doubts remain concerning its import for the question that
Hilbert asked.
515:-adic solution to finding the maximal abelian extension of totally real fields by proving the integral GrossâStark conjecture for BrumerâStark units.
665:
673:
208:
is contained in a cyclotomic field. Kronecker's (and
Hilbert's) question addresses the situation of a more general algebraic number field
587:
740:
232:
being that he may have been using the term "elliptic function" to mean both the elliptic function â and the elliptic modular function
883:
872:
176:. All quadratic extensions, obtained by adjoining the roots of a quadratic polynomial, are abelian, and their study was commenced by
708:
640:
877:
867:
243:
905:
847:
857:
852:
832:
827:
915:
240:, for more general abelian extensions one also needs to use values of elliptic functions. For example, the abelian extension
862:
842:
837:
733:
817:
632:
485:
797:
172:. The simplest situation, which is already at the boundary of what is well understood, is when the group in question is
319:
One particularly appealing way to state the
KroneckerâWeber theorem is by saying that the maximal abelian extension of
201:
138:
51:
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772:
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703:. Studies in the Development of Modern Mathematics. Vol. 2. New York: Gordon and Breach Science Publishers.
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42:
Kronecker in a letter to
Dedekind in 1880 reproduced in volume V of his collected works, page 455
658:
Schappacher, Norbert (1998). "On the history of
Hilbert's twelfth problem: a comedy of errors".
161:
354:
is an imaginary quadratic irrationality, can be obtained by adjoining the special values of â(
704:
669:
636:
565:
Dasgupta, Samit; Kakde, Mahesh (2021-03-03). "Brumer-Stark Units and
Hilbert's 12th Problem".
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114:
79:
55:
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371:
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provided a construction of the maximal abelian extension of totally real fields using the
59:
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71:
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in class field theory involves first constructing larger non-abelian extensions using
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173:
532:
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134:
122:
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212:: what are the algebraic numbers necessary to construct all abelian extensions of
459:
427:
410:, and others in the first half of the 20th century. However the construction of
216:? The complete answer to this question has been completely worked out only when
725:
407:
17:
74:
that generate a whole family of further number fields, analogously to the
618:(1976). "Some contemporary problems with origins in the Jugendtraum". In
588:"Mathematicians Find Long-Sought Building Blocks for Special Polynomials"
535:
proved the existence of the absolute abelian extension as the well-known
458:
in general. The question of which extensions can be found is that of the
455:
225:
126:
426:
Developments since around 1960 have certainly contributed. Before that
661:
Matériaux pour l'histoire des mathématiques au XX siÚcle (Nice, 1996)
571:
88:, or "dearest dream of his youth", so the problem is also known as
394:. In this form, it remains unsolved. A description of the field
729:
164:
made it clear that field extensions are controlled by certain
309:{\displaystyle \mathbf {Q} (i,{\sqrt{1+2i}})/\mathbf {Q} (i)}
144:
The general case of
Hilbert's twelfth problem is still open.
631:. Proc. Sympos. Pure Math. Vol. 28. Providence, RI:
316:
is not generated by singular moduli and roots of unity.
625:
Mathematical developments arising from Hilbert problems
377:
and elliptic functions â, and roots of unity, where
323:
can be obtained by adjoining the special values exp(2
246:
470:, these representations have been studied in depth.
308:
82:described the complex multiplication issue as his
180:. Another type of abelian extension of the field
200:is contained in a larger cyclotomic field. The
196:. Already Gauss had shown that, in fact, every
30:
476:argued in 1973 that the modern version of the
129:. In the special case of totally real fields,
741:
701:Kronecker's Jugendtraum and modular functions
8:
478:
466:. Since this is the most accessible case of
342:shows that the maximal abelian extension of
100:
83:
32:
454:. This gives rise to abelian extensions of
444:Complex multiplication of abelian varieties
204:shows that any finite abelian extension of
748:
734:
726:
511:. In 2021, Dasgupta and Kakde announced a
570:
292:
287:
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261:
247:
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381:is in the imaginary quadratic field and
27:Problem about mathematical number fields
557:
524:
586:Houston-Edwards, Kelsey (2021-05-25).
446:was an area opened up by the work of
431:
7:
664:. SĂ©min. Congr. Vol. 3. Paris:
192:th roots of unity, resulting in the
66:. It is one of the 23 mathematical
35:durch die Kreisteilungsgleichungen.
121:related to the field in question.
25:
293:
248:
105:, does this for the case of any
438:to study abelian extensions of
666:Société Mathématique de France
303:
297:
284:
252:
70:and asks for analogues of the
1:
633:American Mathematical Society
499:A separate development was
434:) in his dissertation used
338:. Similarly, the theory of
158:fields of algebraic numbers
152:The fundamental problem of
932:
390:of a general number field
188:is given by adjoining the
148:Description of the problem
763:
486:HasseâWeil zeta functions
224:or its generalization, a
222:imaginary quadratic field
117:chosen with a particular
107:imaginary quadratic field
99:, now often known as the
48:Hilbert's twelfth problem
537:Takagi existence theorem
95:The classical theory of
50:is the extension of the
906:Algebraic number theory
492:. While he envisaged a
202:KroneckerâWeber theorem
154:algebraic number theory
139:BrumerâStark conjecture
90:Kronecker's Jugendtraum
52:KroneckerâWeber theorem
699:VlÇduĆŁ, S. G. (1991).
479:
464:Galois representations
462:of such varieties, as
340:complex multiplication
310:
101:
97:complex multiplication
84:
39:
33:
440:real quadratic fields
436:Hilbert modular forms
311:
102:Kronecker Jugendtraum
78:and their subfields.
668:. pp. 243â273.
635:. pp. 401â418.
398:was obtained in the
336:exponential function
244:
85:liebster Jugendtraum
422:Modern developments
238:Hilbert class field
156:is to describe the
916:Hilbert's problems
757:Hilbert's problems
501:Stark's conjecture
400:class field theory
306:
115:elliptic functions
56:abelian extensions
893:
892:
675:978-2-85629-065-1
620:Browder, Felix E.
494:grandiose program
490:Shimura varieties
484:should deal with
468:â-adic cohomology
372:modular functions
282:
194:cyclotomic fields
125:extended this to
111:modular functions
80:Leopold Kronecker
76:cyclotomic fields
16:(Redirected from
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186:rational numbers
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68:Hilbert problems
60:rational numbers
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531:In particular,
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160:. The work of
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131:Samit Dasgupta
119:period lattice
72:roots of unity
62:, to any base
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595:. Retrieved
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533:Teiji Takagi
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460:Tate modules
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135:Mahesh Kakde
123:Goro Shimura
94:
89:
64:number field
47:
46:
40:
31:
911:Conjectures
480:Jugendtraum
109:, by using
18:Jugendtraum
900:Categories
719:0731.11001
692:1044.01530
651:0345.14006
597:2021-05-28
572:2103.02516
547:References
509:-functions
408:Emil Artin
552:Footnotes
456:CM-fields
406:himself,
350:), where
334:) of the
127:CM fields
452:Taniyama
226:CM-field
684:1640262
622:(ed.).
609:Sources
448:Shimura
430: (
404:Hilbert
174:abelian
58:of the
717:
707:
690:
682:
672:
649:
639:
505:Artin
362:) and
220:is an
168:, the
166:groups
162:Galois
629:(PDF)
567:arXiv
519:Notes
428:Hecke
370:) of
178:Gauss
705:ISBN
670:ISBN
637:ISBN
450:and
432:1912
133:and
113:and
715:Zbl
688:Zbl
647:Zbl
488:of
184:of
54:on
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884:24
878:23
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853:18
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686:.
680:MR
678:.
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