Duke Mathematical Journal

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

Annette Huber and Guido Kings

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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.

Article information

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

Dates
First available in Project Euclid: 23 April 2004

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1082744770

Digital Object Identifier
doi:10.1215/S0012-7094-03-11931-6

Mathematical Reviews number (MathSciNet)
MR2002643

Zentralblatt MATH identifier
1044.11095

Subjects
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]

Citation

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. http://projecteuclid.org/euclid.dmj/1082744770.


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