The Equivariant Tamagawa Number Conjecture: A survey

We give a survey of the equivariant Tamagawa number (a.k.a. Bloch-Kato) conjecture with particular emphasis on proven cases. The only new result is a proof of the 2-primary part of this conjecture for Tate-motives over abelian fields. This article is an expanded version of a survey talk given at the conference on Stark’s conjecture, Johns Hopkins University, Baltimore, August 5-9, 2002. We have tried to retain the succinctness of the talk when covering generalities but have considerably expanded the section on examples. Most of the following recapitulates well known material due to many people. Section 3 is joint work with D. Burns (for which [14], [15], [16], [17] are the main references). In section 5.1 we have given a detailed proof of the main result which also covers the prime l = 2 (unavailable in the literature so far).