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The Eisenstein cocycle and Gross’s tower of fields conjecture. (English. French summary) Zbl 1415.11170
Summary: 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 \(\mathbb {Z}[\mathrm{Gal}(L/F)]\) to \(\mathbb {Z}[\mathrm{Gal}(K/F)]\). Let \(\Theta \in \mathbb {Z}[\mathrm{Gal}(L/F)]\) 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 \(\#S \neq r\). In this article we sketch the proof in the case that \(K\) is totally complex.

11R42 Zeta functions and \(L\)-functions of number fields
11R80 Totally real fields
Full Text: DOI
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