Unités semi-locales modulo sommes de Gauss

For an odd prime number p and an abelian number field k, let k∞/k be the cyclotomic Zp-extension. Let X∞ be the projective limit of the p-parts of the ideal class groups of each intermediate field of k∞/k. It is conjectured (Greenberg's Conjecture) that X∞ is finite when k is totally real. In this paper we give an interpretation of the characteristic polynomial of X∞ in terms of certain Gauss sums. We also give analogous results at finite level. Our results generalize those obtained by Ichimura (J. Number Theory 68 (1998) 36) and Hachimori (Manuscripta Math. 95 (1998) 377) in the semi-simple case.