General SectionIwasawa theory for class groups of CM fields with p = 2
Review articleOpen access

AbstractIn this paper, we study Iwasawa theory for p=2. First of all, we show that the classical Iwasawa main conjecture holds true even for p=2 over a totally real field k assuming μ=0 and Leopoldt's conjecture. Using the Iwasawa main conjecture, we study the 2-component of the ideal class group of a CM-field K of finite degree as a Galois module. More precisely, for a CM-field K which is cyclic over the base field k, we determine the Fitting ideal of the minus quotient of the 2-component of the ideal class group. In particular, when k=Q and K/Q is imaginary and cyclic, we prove that the Fitting ideal coincides with the Stickelberger ideal.

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