Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



The Brumer-Stark conjecture in some families of extensions of specified degree

Authors: Cornelius Greither, Xavier-François Roblot and Brett A. Tangedal
Journal: Math. Comp. 73 (2004), 297-315
MSC (2000): Primary 11R42; Secondary 11R29, 11R80, 11Y40
Published electronically: June 19, 2003
Corrigendum: Math. Comp. 84 (2015), 955--957
MathSciNet review: 2034123
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: As a starting point, an important link is established between Brumer's conjecture and the Brumer-Stark conjecture which allows one to translate recent progress on the former into new results on the latter. For example, if $K/F$ is an abelian extension of relative degree $2p$, $p$ an odd prime, we prove the $l$-part of the Brumer-Stark conjecture for all odd primes $l\ne p$ with $F$ belonging to a wide class of base fields. In the same setting, we study the $2$-part and $p$-part of Brumer-Stark with no special restriction on $F$ and are left with only two well-defined specific classes of extensions that elude proof. Extensive computations were carried out within these two classes and a complete numerical proof of the Brumer-Stark conjecture was obtained in all cases.

References [Enhancements On Off] (What's this?)

  • [Ba] D. BARSKY, Fonctions zêta $p$-adiques d'une classe de rayon des corps de nombres totalement réels. Groupe d'étude d'analyse ultramétrique 1977-78. Errata, idem 1978-79. MR 80g:12009
  • [CN] PIERRETTE CASSOU-NOGU`ES, Valeurs aux entiers négatifs des fonctions zêta et fonctions zêta $p$-adiques, Invent. Math. 51 (1979), 29-59. MR 80h:12009b
  • [Co] H. COHEN, Advanced Topics in Computational Number Theory, GTM 193, Springer-Verlag, New York, 2000. MR 2000k:11144
  • [DR] P. DELIGNE and K. RIBET, Values of abelian $L$-functions at negative integers over totally real fields, Invent. Math. 59 (1980), 227-286. MR 81m:12019
  • [FT] A. FRÖHLICH and M. TAYLOR, Algebraic Number Theory, Cambridge University Press, Cambridge, 1991. MR 94d:11078
  • [Gr1] C. GREITHER, Some cases of Brumer's conjecture for abelian CM extensions of totally real fields, Math. Zeitschrift 233 (2000), 515-534. MR 2001c:11124
  • [Gr2] -, An equivariant version of an index formula of Fröhlich and McCulloh, J. Number Th. 69 (1998), 1-15. MR 99c:11135
  • [Ha] DAVID R. HAYES, Base change for the conjecture of Brumer-Stark, J. Reine Angew. Math. 497 (1998), 83-89. MR 99h:11131
  • [Ja] G. JANUSZ, Algebraic Number Fields, 2nd Ed., American Mathematical Society, Providence, 1996. MR 96j:11137
  • [Kl] H. KLINGEN, Über die Werte der Dedekindsche Zetafunktion, Math. Ann. 145 (1962), 265-272. MR 24:A3138
  • [La] S. LANG, Cyclotomic Fields I and II, GTM 121, Springer-Verlag, New York, 1990. MR 91c:11001
  • [PARI] C. BATUT, K. BELABAS, D. BERNARDI, H. COHEN, M. OLIVIER, The Computer Number Theory System PARI/GP,
  • [Ro] X. ROBLOT, Stark's conjectures and Hilbert's twelfth problem, Experiment. Math. 9 (2000), 251-260. MR 2001g:11174
  • [RT] X. ROBLOT and B. TANGEDAL, Numerical verification of the Brumer-Stark conjecture, in: Algorithmic Number Theory, Proceedings of ANTS-IV Leiden (ed. W. Bosma), Lecture Notes in Computer Science 1838, Springer-Verlag, Berlin, (2000), 491-503. MR 2002e:11158
  • [Sa1] J. SANDS, Galois groups of exponent two and the Brumer-Stark conjecture, J. Reine Angew. Math. 349 (1984), 129-135. MR 85i:11098
  • [Sa2] -, Abelian fields and the Brumer-Stark conjecture, Compos. Math. 53 (1984), 337-346. MR 86c:11102
  • [Si] C. SIEGEL, Über die Fourierschen Koeffizienten von Modulformen, Nachr. Akad. Wiss. Göttingen 3 (1970), 15-56. MR 44:2706
  • [So] D. SOLOMON, On the classgroups of imaginary abelian fields, Ann. Inst. Fourier, Grenoble 40 (1990), 467-492. MR 92a:11133
  • [Ta1] J. TATE, Brumer-Stark-Stickelberger, Séminaire de Théorie des Nombres Univ. Bordeaux I, Talence (1980-81) Exposé n$^\circ$ 24. MR 83m:12018b
  • [Ta2] -, Les Conjectures de Stark sur les Fonctions L d'Artin en $s=0$, Notes d'un cours à Orsay rédigées par D. Bernardi et N. Schappacher, Birkhäuser-Verlag, Boston, 1984. MR 86e:11112
  • [Wi] A. WILES, On a conjecture of Brumer, Ann. Math. 131 (1990), 555-565. MR 91i:11164

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 11R42, 11R29, 11R80, 11Y40

Retrieve articles in all journals with MSC (2000): 11R42, 11R29, 11R80, 11Y40

Additional Information

Cornelius Greither
Affiliation: Institut für theoretische Informatik und Mathematik, Fakultät für Informatik, Universität der Bundeswehr München, 85577 Neubiberg, F. R. Germany

Xavier-François Roblot
Affiliation: Institut Girard Desargues, Université Claude Bernard (Lyon I), 69622 Villeurbanne, France

Brett A. Tangedal
Affiliation: Department of Mathematics, College of Charleston, Charleston, South Carolina 29424-0001

Keywords: Algebraic number fields, Brumer-Stark conjecture
Received by editor(s): December 20, 2001
Published electronically: June 19, 2003
Article copyright: © Copyright 2003 American Mathematical Society