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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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