## Class groups of abelian fields, and the main conjecture.(English)Zbl 0729.11053

The first part of this paper gives a proof of the main conjecture of Iwasawa theory for abelian base fields, including the case $$p=2$$, by Kolyvagin’s method of Euler systems. On the way, one obtains a general result on local units modulo circular units. This is then used to deduce theorems on the order of $$\chi$$-parts of p-class groups of abelian number fields: first for relative class groups as has been done recently by Solomon for odd p, and second for class groups of real fields (again including the case $$p=2)$$. As a consequence, a generalization of the Gras conjecture is stated and proved.

### MSC:

 11R23 Iwasawa theory 11R29 Class numbers, class groups, discriminants 11R42 Zeta functions and $$L$$-functions of number fields 11R20 Other abelian and metabelian extensions 11R18 Cyclotomic extensions 11R27 Units and factorization
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