The Eisenstein cocycle and Gross’s tower of fields conjecture

Abstract

This paper is an announcement of the following result, whose proof will be forthcoming. Let F be a totally real number field, and let \(F \subset K \subset L\) be a tower of fields with L / F a finite abelian extension. Let I denote the kernel of the natural projection from \(\mathbf {Z}[\mathrm{Gal}(L/F)]\) to \(\mathbf {Z}[\mathrm{Gal}(K/F)]\). Let \(\Theta \in \mathbf {Z}[\mathrm{Gal}(L/F)]\) 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 \(\Theta \in I^{r}\), unless K is totally real in which case we obtain \(\Theta \in I^{r-1}\) and \(2\Theta \in I^r\). This proves a conjecture of Gross up to the factor of 2 in the case that K is totally real and \(\#S \ne r\). In this article we sketch the proof in the case that K is totally complex.