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

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