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



Duke Mathematical Journal
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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

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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

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