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. 1 in Kurihara (J Reine Angew Math 561:39-86, 2003). We also study Brumer's conjecture and prove a stronger version for a CM-field, assuming certain conditions, in particular on the Galois group.
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