Nombres de Weil, sommes de Gauss et annulateurs Galoisiens

Abstract

For an abelian number field K containing a primitive p-th root of unity (p an odd prime) and satisfying certain technical conditions, we parametrize the Z_p[G(K/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.