Duke Mathematical Journal

Bloch-Kato conjecture and main conjecture of Iwasawa theory for Dirichlet characters

Annette Huber and Guido Kings

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


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.

Article information

Duke Math. J., Volume 119, Number 3 (2003), 393-464.

First available in Project Euclid: 23 April 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G55: Polylogarithms and relations with $K$-theory
Secondary: 11R23: Iwasawa theory 19F27: Étale cohomology, higher regulators, zeta and L-functions [See also 11G40, 11R42, 11S40, 14F20, 14G10]


Huber, Annette; Kings, Guido. Bloch-Kato conjecture and main conjecture of Iwasawa theory for Dirichlet characters. Duke Math. J. 119 (2003), no. 3, 393--464. doi:10.1215/S0012-7094-03-11931-6. https://projecteuclid.org/euclid.dmj/1082744770

Export citation


  • T. M. Apostol, Introduction to Analytic Number Theory, Springer, New York, 1976.
  • A. Beilinson, Higher regulators and values of L-functions, J. Soviet. Math. 30 (1985), 2036--2070.
  • A. Beilinson and P. Deligne, Motivic polylogarithm and Zagier conjecture, unpublished preprint, 1992.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.
  • D. Benois and T. Nguyen Quang Do, Les nombres de Tamagawa locaux et la conjecture de Bloch et Kato pour les motifs $\mathbb{Q}(m)$ sur un corps abélien, Ann. Sci. École Norm. Sup. (4) 35 (2002), 641--672. \CMP1 951 439
  • 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.
  • A. Borel, Cohomologie de ${\rm SL}\sb{n}$ et valeurs de fonctions zeta aux points entiers, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4 (1977), 613--636.
  • D. Burns and M. Flach, Tamagawa numbers for motives with (non-commutative) coefficients, Doc. Math. 6 (2001), 501--570.
  • D. Burns and C. Greither, On the equivariant Tamagawa number conjecture for Tate motives, Invent. Math. 153 (2003), 303--359.
  • W. G. Dwyer and E. M. Friedlander, Algebraic and etale $K$-theory, Trans. Amer. Math. Soc. 292 (1985), 247--280.
  • H. Esnault, On the Loday symbol in the Deligne-Beilinson cohomology, $K$-theory 3 (1989), 1--28.
  • J.-M. Fontaine, Valeurs spéciales des fonctions $L$ des motifs, Astérisque 206 (1992), 4, 205--249., Séminaire Bourbaki, Vol. 1991/92, exp. 751.
  • 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.
  • A. Huber and G. Kings, Degeneration of $l$-adic Eisenstein classes and of the elliptic polylog, Invent. Math. 135 (1999), 545--594.
  • 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.
  • U. Jannsen, "On the $l$-adic cohomology of varieties over number fields and its Galois cohomology" in Galois Groups over $\mathbb{Q}$ (Berkeley, Calif., 1987), Math. Sci. Res. Inst. Publ. 16, Springer, New York, 1989, 315--360.
  • K. Kato, Iwasawa theory and $p$-adic Hodge theory, Kodai Math. J. 16 (1993), 1--31.
  • --. --. --. --., "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.
  • --------, 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.
  • 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.
  • 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.
  • B. Mazur and A. Wiles, Class fields of abelian extensions of ${\mathbb{Q}}$, Invent. Math. 76 (1984), 179--330.
  • J. Milne, Arithmetic Duality Theorems, Perspect. Math. 1, Academic Press, Boston, 1986.
  • 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.
  • B. Perrin-Riou, Théorie d'Iwasawa et hauteurs $p$-adiques, Invent. Math. 109 (1992), 137--185.
  • --. --. --. --., Théorie d'Iwasawa des représentations $p$-adiques sur un corps local, Invent. Math. 115 (1994), 81--169.
  • --------, Fonctions $L$ p-adiques des représentations p-adiques, Astérisque 229, Math. Soc. France, Montrouge, 1995.
  • --. --. --. --., Syst èmes d'Euler p-adiques et théorie d'Iwasawa, Ann. Inst. Fourier (Grenoble) 48 (1998), 1231--1307.
  • 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.
  • K. Rubin, Euler Systems, Ann. of Math. Stud. 147, Princeton Univ. Press, Princeton, 2000.
  • P. Schneider, Über gewisse Galoiscohomologiegruppen, Math. Z. 168 (1979), 181--205.
  • C. Soulé, "On higher $p$-adic regulators" in Algebraic $K$-Theory (Evanston, Ill., 1980), Lecture Notes in Math. 854, Springer, Berlin, 1981.
  • --. --. --. --., "Operations onétale $K$-theory: Applications" in Algebraic $K$-Theory (Oberwolfach, Germany, 1980), Part I, Lecture Notes in Math. 966, Springer, Berlin, 1982, 271--303.
  • L. C. Washington, Introduction to Cyclotomic Fields, 2d ed., Grad. Texts in Math. 83, Springer, Berlin, 1997.
  • A. Wiles, Higher explicit reciprocity laws, Ann. Math. (2) 107 (1978), 235--254.