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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.

##### MSC:
 11R42 Zeta functions and $$L$$-functions of number fields 11R80 Totally real fields
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##### References:
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