Abstract

For an abelian number field $K$ containing a primitive $p^{{\rm th}}$ root of unity ($p$ an odd prime) and satisfying certain technical conditions, we parametrize the ${\Bbb Z}_p[{\rm G}(K/\Bbb{Q})]$-annihilators of the minus'' part $A_{K}^{-}$ of the $p$-class group by means of modules of Jacobi sums. Using a reflection theorem and Bloch-Kato's reciprocity law, we then determine the Fitting ideal of the plus'' part $A_{K}^{+}$ in terms of twisted'' Gauss sums.