Higher analogues of Stickelberger's theorem

Let l be an odd prime number, F denote any totally real number field and E/F be an Abelian CM extension of F of conductor f. In this paper we prove that for every n odd and almost all prime numbers I we have S-n(E/F, I) C Ann(Z1[G(E/F)])H(2) (O-E[1/l]; Z(l)(n + 1)) where S-n(E/F, l) is the Stiekelberger ideal (Ann. of Math. 135 (1992) 325-360; J. Coates, p-adic L-functions and Iwasawa's theory, in: Algebraic Number Fields by A. Frohlich, Academic Press, London, 1977). In addition if we assume the Quillen-Lichtenbaum conjecture then S-n(E/F, l) subset of A(nnZl[G(E/F)]) K-2n(OE)(l). (C) 2003 Academie des sciences. Published by Editions scientifiques et medicales Elsevier SAS. All rights reserved.