Greither, Cornelius Class groups of abelian fields, and the main conjecture. (English) Zbl 0729.11053 Ann. Inst. Fourier 42, No. 3, 449-500 (1992). 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. Reviewer: C.Greither (München) Cited in 1 ReviewCited in 36 Documents 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 Keywords:cyclotomic extensions; p-adic L-functions; main conjecture of Iwasawa theory; units; abelian number fields; relative class groups; class groups of real fields; Gras conjecture PDF BibTeX XML Cite \textit{C. Greither}, Ann. Inst. Fourier 42, No. 3, 449--500 (1992; Zbl 0729.11053) Full Text: DOI Numdam EuDML