previous ::
next
Bloch-Kato conjecture and main conjecture of Iwasawa theory for Dirichlet characters
Annette Huber and Guido Kings
Source: Duke Math. J. Volume 119, Number 3 (2003), 393-464.
Abstract
The Tamagawa number conjecture proposed by S. Bloch and K. Kato describes the "special values" of L-functions in terms of cohomological data. The main conjecture of Iwasawa theory describes a p-adic L-function in terms of the structure of modules for the Iwasawa algebra. We give a complete proof of both conjectures (up to the prime 2) for L-functions attached to Dirichlet characters.
We use the insight of Kato and B. Perrin-Riou that these two conjectures can be seen as incarnations of the same mathematical content. In particular, they imply each other. By a bootstrapping process using the theory of Euler systems and explicit reciprocity laws, both conjectures are reduced to the analytic class number formula. Technical problems with primes dividing the order of the character are avoided by using the correct cohomological formulation of the main conjecture.
Primary Subjects: 11G55
Secondary Subjects: 11R23, 19F27
Full-text: Access denied (no subscription detected)
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text
Alternatively, the document is available for a cost of $25. Select the "buy article" button below to purchase this document from a secured VeriSign, Inc. site.
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.dmj/1082744770
Mathematical Reviews number (MathSciNet):
MR2002643
Digital Object Identifier: doi:10.1215/S0012-7094-03-11931-6
Zentralblatt MATH identifier:
1044.11095
References
T. M. Apostol, Introduction to Analytic Number Theory, Springer, New York, 1976.
Mathematical Reviews (MathSciNet):
MR55i:7892
A. Beilinson, Higher regulators and values of L-functions, J. Soviet. Math. 30 (1985), 2036--2070.
Mathematical Reviews (MathSciNet):
MR86h:11103
A. Beilinson and P. Deligne, Motivic polylogarithm and Zagier conjecture, unpublished preprint, 1992.
J.-R. Belliard and T. Nguyen Quang Do, Formules de classes pour les corps abéliens réels, Ann. Inst. Fourier (Grenoble) 51 (2001), 903--937.
Mathematical Reviews (MathSciNet):
MR2002j:11134
D. Benois and T. Nguyen Quang Do, Les nombres de Tamagawa locaux et la conjecture de Bloch et Kato pour les motifs $\mathbbQ(m)$ sur un corps abélien, Ann. Sci. École Norm. Sup. (4) 35 (2002), 641--672. \CMP1 951 439
Mathematical Reviews (MathSciNet):
MR1951439
Digital Object Identifier: doi:10.1016/S0012-9593(02)01104-7
S. Bloch and K. Kato, "$L$-functions and Tamagawa numbers of motives" in The Grothendieck Festschrift, Vol. I, Progr. Math. 86, Birkhäuser, Boston, 1990, 333--400.
Mathematical Reviews (MathSciNet):
MR92g:11063
A. Borel, Cohomologie de $\rm SL\sbn$ et valeurs de fonctions zeta aux points entiers, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4 (1977), 613--636.
Mathematical Reviews (MathSciNet):
MR58:22016
D. Burns and M. Flach, Tamagawa numbers for motives with (non-commutative) coefficients, Doc. Math. 6 (2001), 501--570.
Mathematical Reviews (MathSciNet):
MR2002m:11055
D. Burns and C. Greither, On the equivariant Tamagawa number conjecture for Tate motives, Invent. Math. 153 (2003), 303--359.
Mathematical Reviews (MathSciNet):
MR1992015
Digital Object Identifier: doi:10.1007/s00222-003-0291-x
W. G. Dwyer and E. M. Friedlander, Algebraic and etale $K$-theory, Trans. Amer. Math. Soc. 292 (1985), 247--280.
Mathematical Reviews (MathSciNet):
MR87h:18013
Digital Object Identifier: doi:10.2307/2000179
JSTOR: links.jstor.org
H. Esnault, On the Loday symbol in the Deligne-Beilinson cohomology, $K$-theory 3 (1989), 1--28.
Mathematical Reviews (MathSciNet):
MR90j:14025
Digital Object Identifier: doi:10.1007/BF00534143
J.-M. Fontaine, Valeurs spéciales des fonctions $L$ des motifs, Astérisque 206 (1992), 4, 205--249., Séminaire Bourbaki, Vol. 1991/92, exp. 751.
Mathematical Reviews (MathSciNet):
MR94f:11055
J.-M. Fontaine and B. Perrin-Riou, "Autour des conjectures de Bloch et Kato: cohomologie galoisienne et valeurs de fonctions $L$" in Motives (Seattle, Wash., 1991), Proc. Sympos. Pure Math. 55, Part 1, Amer. Math. Soc., Providence, 1994, 599--706.
Mathematical Reviews (MathSciNet):
MR95j:11046
A. Huber and G. Kings, Degeneration of $l$-adic Eisenstein classes and of the elliptic polylog, Invent. Math. 135 (1999), 545--594.
Mathematical Reviews (MathSciNet):
MR2000b:11068
Digital Object Identifier: doi:10.1007/s002220050295
A. Huber and J. Wildeshaus, Classical motivic polylogarithm according to Beilinson and Deligne, Doc. Math. 3 (1998), 27--133., ; Correction, Doc. Math. 3 (1998), 297--299.
Mathematical Reviews (MathSciNet):
MR99k:19001a
Mathematical Reviews (MathSciNet):
MR99k:19001b
U. Jannsen, "On the $l$-adic cohomology of varieties over number fields and its Galois cohomology" in Galois Groups over $\mathbbQ$ (Berkeley, Calif., 1987), Math. Sci. Res. Inst. Publ. 16, Springer, New York, 1989, 315--360.
Mathematical Reviews (MathSciNet):
MR90i:11064
K. Kato, Iwasawa theory and $p$-adic Hodge theory, Kodai Math. J. 16 (1993), 1--31.
Mathematical Reviews (MathSciNet):
MR94f:11056
Digital Object Identifier: doi:10.2996/kmj/1138039701
Project Euclid: euclid.kmj/1138039701
--. --. --. --., "Lectures on the approach to Iwasawa theory for Hasse-Weil $L$-functions via $B\sb \rm dR$, I" in Arithmetic Algebraic Geometry (Trento, Italy, 1991), Lecture Notes in Math. 1553, Springer, Berlin, 1993, 50--163.
Mathematical Reviews (MathSciNet):
MR96f:11067
Digital Object Identifier: doi:10.1007/BFb0084729
--------, Lectures on the approach to Iwasawa theory of Hasse-Weil $L$-functions via $B_\DR$, II, unpublished preprint, 1993.
--. --. --. --., Euler systems, Iwasawa theory, and Selmer groups, Kodai Math. J. 22 (1999), 313--372.
Mathematical Reviews (MathSciNet):
MR2000k:11076
Digital Object Identifier: doi:10.2996/kmj/1138044090
Project Euclid: euclid.kmj/1138044090
M. Kolster and T. Nguyen Quang Do, Universal distribution lattices for abelian number fields, preprint, 2000.
M. Kolster, T. Nguyen Quang Do, and V. Fleckinger, Twisted $S$-units, $p$-adic class number formulas, and the Lichtenbaum conjectures, Duke Math. J. 84 (1996), 679--717., ; Correction, Duke Math. J. 90 (1997), 641--643.
Mathematical Reviews (MathSciNet):
MR97g:11136
Mathematical Reviews (MathSciNet):
MR98k:11172
Digital Object Identifier: doi:10.1215/S0012-7094-96-08421-5
Project Euclid: euclid.dmj/1077244040
S. Lichtenbaum, "Values of zeta-functions, étale cohomology, and algebraic $K$-theory" in Algebraic $K$-Theory II: "Classical" Algebraic $K$-Theory and Connections with Arithmetic (Seattle, Wash., 1972), Lecture Notes in Math. 342, Springer, Berlin, 1973, 489--501.
Mathematical Reviews (MathSciNet):
MR53:10765
B. Mazur and A. Wiles, Class fields of abelian extensions of $\mathbbQ$, Invent. Math. 76 (1984), 179--330.
Mathematical Reviews (MathSciNet):
MR85m:11069
Digital Object Identifier: doi:10.1007/BF01388599
J. Milne, Arithmetic Duality Theorems, Perspect. Math. 1, Academic Press, Boston, 1986.
Mathematical Reviews (MathSciNet):
MR88e:14028
J. Neukirch, "The Beilinson conjecture for algebraic number fields" in Beilinson's Conjectures on Special Values of L-Functions, Perspect. Math. 4, Academic Press, Boston, 1988, 193--242.
Mathematical Reviews (MathSciNet):
MR90f:11042
B. Perrin-Riou, Théorie d'Iwasawa et hauteurs $p$-adiques, Invent. Math. 109 (1992), 137--185.
Mathematical Reviews (MathSciNet):
MR93g:11109
Digital Object Identifier: doi:10.1007/BF01232022
--. --. --. --., Théorie d'Iwasawa des représentations $p$-adiques sur un corps local, Invent. Math. 115 (1994), 81--169.
Mathematical Reviews (MathSciNet):
MR95c:11082
Digital Object Identifier: doi:10.1007/BF01231755
--------, Fonctions $L$ p-adiques des représentations p-adiques, Astérisque 229, Math. Soc. France, Montrouge, 1995.
Mathematical Reviews (MathSciNet):
MR96e:11062
--. --. --. --., Syst èmes d'Euler p-adiques et théorie d'Iwasawa, Ann. Inst. Fourier (Grenoble) 48 (1998), 1231--1307.
Mathematical Reviews (MathSciNet):
MR99m:11124
M. Rapoport, "Comparison of the regulators of Beilinson and of Borel" in Beilinson's Conjectures on Special Values of $L$-Functions, Perspect. Math. 4, Academic Press, Boston, 1988, 169--192.
Mathematical Reviews (MathSciNet):
MR89j:11114
K. Rubin, Euler Systems, Ann. of Math. Stud. 147, Princeton Univ. Press, Princeton, 2000.
Mathematical Reviews (MathSciNet):
MR2001g:11170
P. Schneider, Über gewisse Galoiscohomologiegruppen, Math. Z. 168 (1979), 181--205.
Mathematical Reviews (MathSciNet):
MR81i:12010
Digital Object Identifier: doi:10.1007/BF01214195
C. Soulé, "On higher $p$-adic regulators" in Algebraic $K$-Theory (Evanston, Ill., 1980), Lecture Notes in Math. 854, Springer, Berlin, 1981.
Mathematical Reviews (MathSciNet):
MR83b:12013
--. --. --. --., "Operations onétale $K$-theory: Applications" in Algebraic $K$-Theory (Oberwolfach, Germany, 1980), Part I, Lecture Notes in Math. 966, Springer, Berlin, 1982, 271--303.
Mathematical Reviews (MathSciNet):
MR85b:18011
L. C. Washington, Introduction to Cyclotomic Fields, 2d ed., Grad. Texts in Math. 83, Springer, Berlin, 1997.
Mathematical Reviews (MathSciNet):
MR97h:11130
previous ::
next
Duke Mathematical Journal
