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

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.

Hyperbolic tessellations and generators of for imaginary quadratic fields

“…If is an abelian field, then the Bloch-Kato conjecture for ℎ 0 (Spec( )) (1 − ) is known to be valid. Up to the 2-primary part, this was verified independently by Huber and Kings [32] and by Greither and the first author [17], and the 2-primary component was subsequently resolved by Flach [25]. Theorem 2.3 thus leads directly to the following result.…”

“…For = 2 we can make this conjectural formula completely explicit by using a result of Levine [41]. Using results of Huber and Kings [32], of Greither and the first author [17] and of Flach [25], relating to the Tamagawa number conjecture, we can then prove the (unconditional) validity of (1.1) for all if is abelian over Q, with a precise expression for , .…”

“…(iii) Up to sign and an unknown power of 2, Theorem 2.3 is proved by Huber and Kings in [32,Th. 1.4.1] under the assumption that c S , , , is bijective for all odd primes , as conjectured by Quillen and Lichtenbaum.…”

“…contains certain erroneous Euler factors, but the necessary correction is provided by Benois and Nguyen Quang Do in [3, §A.3]). The general approach of [38] also provided motivation for the subsequent work of Huber and Kings in [32].…”

On the p-adic Beilinson conjecture and the equivariant Tamagawa number conjecture

“…The case r > 0 is due to Burns and Flach [19]. A slightly weaker variant of the ETNC, where the integral group ring Z[G] is essentially replaced by a maximal order containing it, has been studied earlier by Huber and Kings [44].…”