218:
Stark units in the abelian rank-one case have been computed in specific examples, allowing verification of the veracity of his refined conjecture. These also provide an important computational tool for generating abelian extensions of number fields, forming the basis for some standard algorithms for
346:, which gives a refinement of the abelian rank-one Stark conjecture at totally split finite primes (for totally complex extensions of totally real base fields). The function field analogue of the Brumer–Stark conjecture was proved by John Tate and
276:. This provides a conceptual framework for studying the conjectures, although at the moment it is unclear whether Manin's techniques will yield the actual proof.
1284:
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Modular
Functions of One Variable V: Proceedings International Conference, University of Bonn, Sonderforschungsbereich Theoretische Mathematik, July 1976
158:
The Stark conjectures, in the most general form, predict that the leading coefficient of an Artin L-function is the product of a type of regulator, the
230:
of totally real number fields, and the conjectures provide one solution to
Hilbert's twelfth problem, which challenged mathematicians to show how
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254:, which were covered in the work of Stark, the abelian Stark conjectures is still unproved for number fields. More progress has been made in
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833:(2004), "Real multiplication and noncommutative geometry (ein Alterstraum)", in Piene, Ragni; Laudal, Olav Arnfinn (eds.),
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250:-function takes on only rational values. Except when the base field is the field of rational numbers or an imaginary
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206:, as Kummer theory implies). As such, this refinement of his conjecture has theoretical implications for solving
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Stark's principal conjecture has been proven in a few special cases, such as when the character defining the
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327:, and Robert Pollack in 2011. The proof was completed and made unconditional by Dasgupta,
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Dasgupta, Samit; Kakde, Mahesh (2023). "On the Brumer-Stark
Conjecture and Refinements".
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447:. Encyclopaedia of Mathematical Sciences. Vol. 49 (Second ed.). p. 171.
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proposed a function field analogue of Rubin's conjecture and proved it in some cases.
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Popescu, Cristian D. (1999), "On a refined Stark conjecture for function fields",
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186: = 0 is one, Stark's refined conjecture predicts the existence of
1097: = 1. III. Totally real fields and Hilbert's twelfth problem",
709:"A Stark conjecture "over Z" for abelian L-functions with multiple zeros"
1051:
Stark, H. M. (1977), "Class fields and modular forms of weight one", in
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921:"A Stark conjecture over Z for abelian L-functions with multiple zeros"
800:
111:
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847:
725:
708:
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Gross, Benedict H. (1988). "On the values of abelian L-functions at
1011: = 1. II. Artin L-functions with rational characters",
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refinement of the Stark conjecture in the abelian case. In 1999,
1190:"Les conjectures de Stark sur les fonctions L d'Artin en s=0"
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The first rank-zero cases are used in recent versions of the
785:
Burns, David; Sands, Jonathan; Solomon, David, eds. (2004),
791:, Contemporary Mathematics, vol. 358, Providence, RI:
234:
may be constructed over any number field by the methods of
1059:, Lecture Notes in Math, vol. 601, Berlin, New York:
632:
Rosen, Michael (2002), "15. The Brumer-Stark conjecture",
331:, and Kevin Ventullo in 2018. A further refinement of the
299:
analogue of the Stark conjectures relating derivatives of
548:
Dasgupta, Samit; Kakde, Mahesh; Ventullo, Kevin (2018).
515:
Darmon, Henri; Dasgupta, Samit; Pollack, Robert (2011).
967: = 1. I. L-functions for quadratic forms.",
588:
Journal of the
Faculty of Science, University of Tokyo
517:"Hilbert Modular Forms and the Gross-Stark Conjecture"
498:
Journal of the
Faculty of Science, University of Tokyo
145:, which generate abelian extensions of number fields.
617:. Progress in Mathematics. Vol. 47. Boston, MA:
963:Stark, Harold M. (1971), "Values of L-functions at
788:
Stark's conjectures: recent work and new directions
742:"On a refined Stark conjecture for function fields"
1251:, archived from the original on February 4, 2012
350:in 1984. In 2023, Dasgupta and Kakde proved the
335:-adic conjecture was proposed by Gross in 1988.
1200:(1–3), Boston, MA: Birkhäuser Boston: 143–153,
219:computing abelian extensions of number fields.
8:
607:Les conjectures de Stark sur les fonctions
98:expressing the leading coefficient of the
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1141: = 1. IV. First derivatives at
1137:Stark, Harold M. (1980), "L-functions at
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1093:Stark, Harold M. (1976), "L-functions at
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1007:Stark, Harold M. (1975), "L-functions at
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894:
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390: = 1. IV. First derivatives at
386:Stark, Harold M. (1980), "L-functions at
319:-units. This was proved conditionally by
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256:function fields of an algebraic variety
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106:of a number field as the product of a
268:) related Stark's conjectures to the
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198:that are abelian over the base field
52:
48:
44:
40:
7:
445:Introduction to Modern Number Theory
60:
1285:Unsolved problems in number theory
94:. The conjectures generalize the
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634:Number theory in function fields
311:(for totally even characters of
1248:Lectures on Stark's Conjectures
835:The legacy of Niels Henrik Abel
640:, vol. 210, New York, NY:
550:"On the Gross-Stark Conjecture"
224:PARI/GP computer algebra system
1:
984:10.1016/S0001-8708(71)80009-9
926:Annales de l'Institut Fourier
793:American Mathematical Society
740:Popescu, Cristian D. (1999).
713:Annales de l'Institut Fourier
638:Graduate Texts in Mathematics
443:; Panchishkin, A. A. (2007).
96:analytic class number formula
1162:10.1016/0001-8708(80)90049-3
1114:10.1016/0001-8708(76)90138-9
1028:10.1016/0001-8708(75)90087-0
534:10.4007/annals.2011.174.1.12
484:Gross, Benedict H. (1982). "
411:10.1016/0001-8708(80)90049-3
1196:, Progress in Mathematics,
686:10.4007/annals.2023.197.1.5
569:10.4007/annals.2018.188.3.3
202:(and not just abelian over
71:of the leading term in the
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313:totally real number fields
1261:: CS1 maint: unfit URL (
208:Hilbert's twelfth problem
55:) and later expanded by
1245:Hayes, David R. (1999),
1194:Mathematical Programming
1055:; Zagier, D. B. (eds.),
367:Cristian Dumitru Popescu
1295:Algebraic number theory
1148:Advances in Mathematics
1100:Advances in Mathematics
1014:Advances in Mathematics
970:Advances in Mathematics
896:10.1023/A:1000833610462
759:10.1023/A:1000833610462
397:Advances in Mathematics
354:away from the prime 2.
352:Brumer–Stark conjecture
344:Brumer–Stark conjecture
270:noncommutative geometry
190:, whose roots generate
92:algebraic number fields
18:Stark's conjecture
883:Compositio Mathematica
746:Compositio Mathematica
290:Gross–Stark conjecture
174:When the extension is
104:Dedekind zeta function
67:information about the
831:Manin, Yuri Ivanovich
674:Annals of Mathematics
557:Annals of Mathematics
521:Annals of Mathematics
170:Abelian rank-one case
137:of the L-function at
1300:Zeta and L-functions
1063:, pp. 277–287,
919:Rubin, Karl (1996),
841:, pp. 685–727,
837:, Berlin, New York:
707:Rubin, Karl (1996).
228:Hilbert class fields
182:of an L-function at
1290:Field (mathematics)
857:2002math......2109M
605:Tate, John (1984).
114:of the field and a
1206:10.1007/BF01580857
1069:10.1007/BFb0063951
1053:Serre, Jean-Pierre
180:order of vanishing
135:order of vanishing
79:associated with a
1215:978-0-8176-3188-8
1145: = 0",
1078:978-3-540-08348-1
866:978-3-540-43826-7
810:978-0-8218-3480-0
454:978-3-540-20364-3
394: = 0",
192:Kummer extensions
131:abelian extension
35:, introduced by
33:Stark conjectures
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1280:Conjectures
611:d'Artin en
492:-series at
226:to compute
214:Computation
188:Stark units
149:Formulation
143:Stark units
110:related to
69:coefficient
65:conjectural
1274:Categories
1186:Tate, John
779:References
660:1043.11079
619:Birkhäuser
594:: 177–197.
504:: 979–994.
471:1079.11002
359:Karl Rubin
309:-functions
280:Variations
162:, with an
1171:0001-8708
1123:0001-8708
1037:0001-8708
993:0001-8708
949:0373-0956
905:0010-437X
719:: 33–62.
694:219557526
463:0938-0396
420:0001-8708
357:In 1996,
340:John Tate
338:In 1984,
284:In 1980,
108:regulator
1257:citation
1232:13291194
1188:(1984),
768:15198245
363:integral
242:Progress
178:and the
133:and the
102:for the
63:), give
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1179:0563924
1131:0437501
1087:0450243
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1001:0289429
957:1385509
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853:Bibcode
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