Some cases of Brumer’s conjecture for abelian CM extensions of totally real fields. (English) Zbl 0965.11047

Greither, Cornelius

Math. Z. 233, No. 3, 515-534 (2000).

Brumer’s conjecture on annihilators of class groups for abelian CM extensions \(K\) of a totally real field \(F\), is a direct generalization of Stickelberger’s theorem over \(\mathbb{Q}\). The author proves this conjecture for so-called “nice” extensions (extensions of prime power conductor are nice, but they are not the only ones). More precisely, using Iwasawa theory, he calculates the Fitting ideal of the minus class group of \(K\). The main difficulty consists in “avoiding the trivial zeros” of \(p\)-adic \(L\)-functions, by extending a method of A. Wiles [Ann. Math. (2) 131, 555-565 (1990; Zbl 0719.11082)]. The result can also be applied to Chinburg’s third conjecture, as in a previous paper of the author [Math. Z. 229, 107-136 (1998; Zbl 0919.11072)].

Reviewer: T.Nguyen Quang Do (Besançon)

MSC:

11R80	Totally real fields
11R23	Iwasawa theory
11R42	Zeta functions and \(L\)-functions of number fields

Keywords:

Brumer’s conjecture; annihilators of class groups; abelian CM extensions; totally real field; Iwasawa theory; Fitting ideal; minus class group; \(p\)-adic \(L\)-functions; Chinburg’s third conjecture

Citations:

Zbl 0719.11082; Zbl 0919.11072