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Annales mathématiques du Québec

, Volume 40, Issue 2, pp 355–376 | Cite as

The Eisenstein cocycle and Gross’s tower of fields conjecture

  • Samit DasguptaEmail author
  • Michael Spieß
Article
  • 104 Downloads

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.

Keywords

Stickelberger elements Eisenstein cocycle Gross’s conjecture 

Résumé

Ce papier est une annonce du résultat suivant, dont la preuve est imminente. Soit F un corps de nombres totalement réel, et soit \(F \subset K \subset L\) une tour d’extensions, où l’extension L / F est abélienne finie. Soit I le noyau de la projection naturelle de \(\mathbf {Z}[\mathrm{Gal}(L/F)]\) vers \(\mathbf {Z}[\mathrm{Gal}(K/F)]\). Soit \(\Theta \in \mathbf {Z}[\mathrm{Gal}(L/F)]\) l’élément de Stickelberger qui encode les valeurs spéciales en zéro des fonctions zêta partielles de L / F, prise par rapport à des ensembles S et T de places de F de la manière usuelle. Soit r le nombre de places dans S qui sont totalement déployées dans K. Nous démontrons que \(\Theta \in I^r\), à moins que K ne soit totalement réel auquel cas nous obtenons \(\Theta \in I^{r-1}\) et \(2 \Theta \in I^r\). Ceci démontre une conjecture de Gross, à un facteur de 2 près dans le cas où K est totalement réel et \(\#S \ne r\). Dans cet article, nous esquissons une preuve dans le cas où l’extension K est totalement complexe.

Mathematics Subject Classification

11R42 

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Copyright information

© Fondation Carl-Herz and Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.University of California Santa CruzSanta CruzUSA
  2. 2.Universität BielefeldBielefeldGermany

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