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Iwasawa theory and Fitting ideals. (English) Zbl 1056.11063

Let \(F/{\mathbb Q}\) be an imaginary abelain extension of finite degree and let \(\text{Cl}'(F)\) denote the class group of \(F\) considered over the ring \({\mathbb Z}':={\mathbb Z}[1/2]\), so that it is viewed as a \({\mathbb Z}'[\text{Gal}(F/{\mathbb Q}]\)-module. For any module \(M\) over this group ring, \(M^-\) denotes the submodule on which the complex conjugation acts by \(-1\). Conjectures which go under the general rubric of Main Conjectures, relate algebraic invariants associated to such Galois modules with analytic or arithmetic invariants related to \(p\)-adic \(L\)-functions. The author considers the minus part of the initial Fitting ideal of \(\text{Cl}'(F)^-\) (which is an algebraic invariant) and the minus part of a ‘Stickelberger element’ \(\Theta_{F/{\mathbb Q}} \in {\mathbb Z}'[\text{Gal}(F/{\mathbb Q})]\), which is to be viewed as an arithmetic or analytic object as it is related to zeta functions. He conjectures that these two invariants are actually equal, in the spirit of the Main Conjectures. Of course, this conjecture is equivalent to the two elements being equal when extended to \({\mathbb Z}_p\) for each odd prime \(p\).
The author proves the latter conjecture in the following three cases:
(i) When no prime of \(F^+\) above \(p\) splits in \(F/F^+\).
(ii) When \(F=K_n\), where \(K_n\) is the \(n\)-th cyclotomic layer of the cyclotomic \({\mathbb Z}_p\)-extension of an abelian field extension \(K\) of \(F\) whose degree is prime to \(p\).
(iii) When \(p\) is tamely ramified in \(F/{\mathbb Q}\), and \(F\) does not contain a primitive \(p\)-th root of unity.

MSC:

11R23 Iwasawa theory
11R42 Zeta functions and \(L\)-functions of number fields
13C20 Class groups
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