Computing Fitting ideals of Iwasawa modules.

*(English)*Zbl 1067.11067Let \(K/F\) 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).

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).

Reviewer: Jürgen Ritter (Augsburg)

##### 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 |