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The Eisenstein cocycle and Gross’s tower of fields conjecture

Publication ,  Journal Article
Dasgupta, S; Spieß, M
Published in: Annales Mathematiques du Quebec
August 1, 2016

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.

Duke Scholars

Published In

Annales Mathematiques du Quebec

DOI

EISSN

2195-4763

ISSN

2195-4755

Publication Date

August 1, 2016

Volume

40

Issue

2

Start / End Page

355 / 376
 

Citation

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Dasgupta, S., & Spieß, M. (2016). The Eisenstein cocycle and Gross’s tower of fields conjecture. Annales Mathematiques Du Quebec, 40(2), 355–376. https://doi.org/10.1007/s40316-015-0046-2
Dasgupta, S., and M. Spieß. “The Eisenstein cocycle and Gross’s tower of fields conjecture.” Annales Mathematiques Du Quebec 40, no. 2 (August 1, 2016): 355–76. https://doi.org/10.1007/s40316-015-0046-2.
Dasgupta S, Spieß M. The Eisenstein cocycle and Gross’s tower of fields conjecture. Annales Mathematiques du Quebec. 2016 Aug 1;40(2):355–76.
Dasgupta, S., and M. Spieß. “The Eisenstein cocycle and Gross’s tower of fields conjecture.” Annales Mathematiques Du Quebec, vol. 40, no. 2, Aug. 2016, pp. 355–76. Scopus, doi:10.1007/s40316-015-0046-2.
Dasgupta S, Spieß M. The Eisenstein cocycle and Gross’s tower of fields conjecture. Annales Mathematiques du Quebec. 2016 Aug 1;40(2):355–376.
Journal cover image

Published In

Annales Mathematiques du Quebec

DOI

EISSN

2195-4763

ISSN

2195-4755

Publication Date

August 1, 2016

Volume

40

Issue

2

Start / End Page

355 / 376