Bloch-Kato Conjecture and Main Conjecture of Iwasawa Theory for Dirichlet Characters, by Annette Huber, Guido Kings

Abstract: This is a revised version of preprint 0279. The Tamagawa number conjecture proposed by Bloch and Kato describes the ``special values'' of $L$-functions of motives in terms of cohomological data. The main conjecture of Iwasawa theory describes a $p$-adic $L$-function in terms of the structure of modules under the Iwasawa algebra. In this paper we show:

Theorem: Let $\chi$ be a Dirichlet character. Then the Main Conjecture of Iwasawa theory holds for $\chi$ and $p\neq 2$ and the Bloch-Kato conjecture holds for the Dirichlet L-function $L(\chi,r)$ with $r\in\Z$ up to powers of $2$.

What we want to advocate strongly is the insight, due to Kato, that the Bloch-Kato conjecture and the Main Conjecture are two incarnations of the same mathematical content. Note also that the Main Conjecture is proved by Mazur and Wiles for the $p$-adic L-function under the condition that $p^2$ does not divide the conductor of the character. This does only lead to a result for cyclotomic units under the condition that $p$ does not divide $\Phi(N)$. We remove these conditions and in fact give a formulation of the Main Conjecture using the Bloch-Kato conjecture in which all problems with bad primes disappear. The proof uses the Euler system machinery as developed by Rubin, Kato and Perrin-Riou and avoids computations with cyclotomic units as much as possible.

Annette Huber, Guido Kings <,>