Stickelberger ideals and Fitting ideals of class groups for abelian number fields

Abstract

In this paper, we determine completely the initial Fitting ideal of the minus part of the ideal class group of an abelian number field over Q up to the 2-component.This answers an open question of Mazur and Wiles (Invent Math 76:179-330, 1984) up to the 2-component, and proves Conjecture 0.

Correction to: Stickelberger ideals and Fitting ideals of class groups for abelian number fields

“…Lemma 6.3 in [2] was also used in the proofs of Theorems 0.4, 0.6 in [2], so we also give in this article a correct proof of Theorems 0.4, 0.6 in [2]. Simply speaking, the proof in [3] (and in [2]) showing that the Fitting ideal is contained in the Stickelberger ideal is correct. Since Lemma 6.3 was used to show the other inclusion, our goal in this article is to give a correct argument for this inclusion.…”

“…In this correction, we give, without using Lemma 6.3 in [2], a proof of Theorems 3.5, 3.6 in [3]. Lemma 6.3 in [2] was also used in the proofs of Theorems 0.4, 0.6 in [2], so we also give in this article a correct proof of Theorems 0.4, 0.6 in [2].…”

“…In the proofs of Theorems 3.5, 3.6 in [3], which give equality of the Fitting ideal of the class group of a CM-field and the Stickelberger ideal, we used Lemma 6.3 in [2], which bounds the index of the Fitting ideal in terms of the class number. But the proof of the lemma is not correct, more precisely, an exact sequence in the proof of Lemma 6.4 in [2] (on page 64 line 19), does not hold in general.…”

“…For an R-module M, length R (M) denotes the length of M as an R-module. Let K /k be an abelian extension as in §3 in [3]. Recall that we decomposed Gal(K /k) = K × K where # K is prime to p, and K is a p-group.…”

Tate sequences and Fitting ideals of Iwasawa modules

“…Let G = Gal(L/k), p be a fixed odd prime number, and let A L be the minus part of the p-part of the classical ideal class group cl(L). For example, if k = Q, the second author showed with T. Miura in [KM1] that the Fitting ideal of A L over Z p [G] equals the "Stickelberger ideal" (tensored with Z p ). For general k, we know from earlier work (see [Gr2], [Ku2]) that the Pontrjagin dual of the class group (in the usual sense) works better than the class group itself, and the first author proved in [Gr2] that the Fitting ideal of the Pontrjagin dual A ∨ L over Z p [G] equals the "Stickelberger ideal" (tensored with Z p ), assuming the equivariant Tamagawa number conjecture and that the group µ p ∞ (L) of the roots of unity in L with p-power order is cohomologically trivial.…”