PUB - Publikationen an der Universität Bielefeld

This paper is an announcement of the following result, whose proof will be forthcoming. Let F be a totally real number field, and let be a tower of fields with L / F a finite abelian extension. Let I denote the kernel of the natural projection from to . Let denote the Stickelberger element encoding the special values at zero of the partial zeta functions of L / F, taken relative to sets S and T in the usual way. Let r denote the number of places in S that split completely in K. We show that , unless K is totally real in which case we obtain and . This proves a conjecture of Gross up to the factor of 2 in the case that K is totally real and . In this article we sketch the proof in the case that K is totally complex.