# The Eisenstein cocycle and Gross’s tower of fields conjecture

Published

Journal Article

© 2016, Fondation Carl-Herz and Springer International Publishing Switzerland. 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⊂ K⊂ L be a tower of fields with L / F a finite abelian extension. Let I denote the kernel of the natural projection from Z[ Gal (L/ F) ] to Z[ Gal (K/ F) ]. Let Θ ∈ Z[ 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 Θ ∈ Ir, unless K is totally real in which case we obtain Θ ∈ Ir-1 and 2 Θ ∈ Ir. This proves a conjecture of Gross up to the factor of 2 in the case that K is totally real and # S≠ r. In this article we sketch the proof in the case that K is totally complex.

### Full Text

### Duke Authors

### Cited Authors

- Dasgupta, S; Spieß, M

### Published Date

- August 1, 2016

### Published In

### Volume / Issue

- 40 / 2

### Start / End Page

- 355 - 376

### Electronic International Standard Serial Number (EISSN)

- 2195-4763

### International Standard Serial Number (ISSN)

- 2195-4755

### Digital Object Identifier (DOI)

- 10.1007/s40316-015-0046-2

### Citation Source

- Scopus