Iwasawa theory and fitting ideals

By studying the Fitting ideals of the minus parts of the ideal class groups of CM fields, we give a more precise relationship than the usual main conjecture between the analytic side and the algebraic side. In particular, for the cyclotomic ℤ_P-extension F of an abelian field F, we determine the initial Fitting ideal of the minus part of the Galois group of the maximal unramified abelian pro-p-extension of F, over F as a Z _p[[Ga[(F∞/ℚ)]]-module. We also study the Fitting ideals of the Selmer groups of an elliptic curve and certain Galois cohomology groups.