Higher analogues of Stickelberger's theorem

Abstract

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 l we have S n(E/F,l)⊂Ann Z l[G(E/F)] H 2( O E[1/l]; Z l(n+1)) where S n ( E/ F, l) is the Stickelberger ideal (Ann. of Math. 135 (1992) 325–360; J. Coates, p-adic L-functions and Iwasawa's theory, in: Algebraic Number Fields by A. Fröhlich, Academic Press, London, 1977). In addition if we assume the Quillen–Lichtenbaum conjecture then S n(E/F,l)⊂Ann Z l[G(E/F)] K 2n( O E) l. To cite this article: G. Banaszak, C. R. Acad. Sci. Paris, Ser. I 337 (2003).