Equivariant Weierstrass preparation and values of \(L\)-functions at negative integers
type: Item
instance of: scholarly article
title: Equivariant Weierstrass preparation and values of \(L\)-functions at negative integers
zbMATH Open document ID: 1142.11371
published in: Documenta Mathematica
publication date: 2004-01-19T00:00:00Z
full work available at URL: https://eudml.org/doc/54396
full work available at URL: burns greither.dm.html
review text: Summary: We apply an equivariant version of the \(p\)-adic Weierstrass preparation theorem in the context of possible non-commutative generalizations of the power series of Deligne and Ribet. We then consider CM abelian extensions of totally real fields and by combining our earlier considerations with the known validity of the Main Conjecture of Iwasawa theory we prove, modulo the conjectural vanishing of certain \(\mu\)-invariants, a (corrected version of a) conjecture of Snaith and the `rank zero component' of Kato's Generalized Iwasawa Main Conjecture for Tate motives of strictly positive weight. We next use the validity of this case of Kato's conjecture to prove a conjecture of Chinburg, Kolster, Pappas and Snaith and also to compute explicitly the Fitting ideals of certain natural étale cohomology groups in terms of the values of Dirichlet \(L\)-functions at negative integers. This computation improves upon results of Cornacchia and Østvær, of Kurihara and of Snaith, and, modulo the validity of a certain aspect of the Quillen-Lichtenbaum Conjecture, also verifies a finer and more general version of a well known conjecture of Coates and Sinnott.
Mathematics Subject Classification ID: 11R42
Mathematics Subject Classification ID: 11R33
Mathematics Subject Classification ID: 11R70
zbMATH DE Number: 2028834
zbMATH Keywords: Iwasawa theory
zbMATH Keywords: values of \(L\)-functions
zbMATH Keywords: Euler characteristics
zbMATH Keywords: Fitting ideals
author: David Burns
author: Cornelius Greither
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2028834