Duke Mathematical Journal
- Duke Math. J.
- Volume 119, Number 3 (2003), 393-464.
Bloch-Kato conjecture and main conjecture of Iwasawa theory for Dirichlet characters
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.
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