## Computing Fitting ideals of Iwasawa modules.(English)Zbl 1067.11067

Let $$K/F$$ be a finite Abelian Galois extension of number fields, with $$F$$ totally real and $$K$$ CM, and denote, for a given odd prime number $$p$$, by $$K_\infty$$, $$F_\infty$$ the cyclotomic $$\mathbb{Z}_p$$-extensions of $$K$$, $$F$$, respectively. The paper is concerned with computing the (initial) Fitting ideals of the minus parts of the Iwasawa modules $$X_{std}$$, $$X_{du}$$ over the Iwasawa algebra $$\mathbb{Z}_l[[G_{K_\infty/F}]]$$. Here, $$X_{std}$$ is the ‘standard’ Iwasawa module, i.e., the projective limit of the $$p$$-parts of the class groups in the cyclotomic tower over $$K$$, and $$X_{du}$$ is a certain ‘dual’ module, namely the Galois group of the maximal Abelian $$p$$-extension of $$K_\infty$$ over $$HK_\infty$$, where $$H$$ is the $$p$$-class field of the minimal field $$K_n$$ in the tower $$K_\infty/K$$ so that all $$p$$-adic places become totally ramified in $$K_\infty/K_n$$.
The first result describing the Fitting ideal of a closely related module $$Y_S$$ (compare e.g. [J. Ritter and A. Weiss, Mem. Am. Math. Soc. 748 (2002; Zbl 1002.11082)]) has Brumer-Stickelberger type consequences on the size of the annihilator of $$X_{du}^-$$ and of the $$p$$-part of the minus class group of $$K$$. The next sections are devoted to the actual calculation of the Fitting ideals of $$X_{du}^-$$ and $$X_{std}^-$$ (outside the Teichmüller character when $$\zeta_p\in K$$, but for $$X^-_{std}$$ also at the Teichmüller character of the $$p$$-regular part of $$G_{K/F}$$); it is assumed here that the $$\mu$$-invariant of $$X^-_{std}$$ vanishes. The overlap between these and recent results of Kurihara is discussed. Under appropriate assumptions the validity of the $$p$$-part of the Brumer-Stark conjecture is a consequence of the shown computations (the main assumption is the nontrivial zeroes condition).

### MSC:

 11R23 Iwasawa theory 11R42 Zeta functions and $$L$$-functions of number fields 12G10 Cohomological dimension of fields 13D02 Syzygies, resolutions, complexes and commutative rings

Zbl 1002.11082
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