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.