Motivic \(L\)-functions and Galois module structures
type: Item
instance of: scholarly article
title: Motivic \(L\)-functions and Galois module structures
zbMATH Open document ID: 0867.11081
DOI: 10.1007/BF01444212
author: Matthias Flach
author: David Burns
published in: Mathematische Annalen
publication date: 1996-07-09T00:00:00Z
full work available at URL: https://eudml.org/doc/165434
review text: During the last decades, many results about varieties over global fields were (at least conjecturally) generalized to motives, which today occupy a prominent position in arithmetic algebraic geometry. Using perfect complexes and their determinants, the conjectures of Bloch and Kato about \(L\)-functions were recently extended to motives with coefficients by \textit{J.-M. Fontaine} and \textit{B. Perrin-Riou} [Proc. Symp. Pure Math. 55, 599--706 (1994; Zbl 0821.14013)] and, independently, by \textit{K. Kato} [Kodai Math. J. 16, 1--31 (1993; Zbl 0798.11050)]. The paper under review discusses applications of these extended conjectures to Galois module theory, namely by defining invariants which describe the Galois module structure of various cohomology groups arising from motives defined over an algebraic number field and admitting an action of a finite abelian Galois group. In the case of Tate motives of arbitrary weight, the existence of these invariants is established independently of conjectures in Proposition 1.42, and connections to Chinburg's invariants, as introduced by \textit{T. Chinburg} [Invent. Math. 74, 321--349 (1983; Zbl 0564.12016) and Ann. Math. (2) 121, 351--376 (1985; Zbl 0567.12010)], are given in Proposition 1.48. In the last section, relations between conjectures about the value of motivic \(L\)-functions at zero and results of \textit{A. Fröhlich} [J. Reine Angew. Math. 397, 42--99 (1989; Zbl 0693.12012)] as well as a conjecture of Chinburg are established.
reviewed by: Günter Lettl
Mathematics Subject Classification ID: 11R33
Mathematics Subject Classification ID: 11G40
Mathematics Subject Classification ID: 14G10
zbMATH DE Number: 892579
zbMATH Keywords: Chinburg invariants
zbMATH Keywords: Galois module theory
zbMATH Keywords: cohomology groups
zbMATH Keywords: Tate motives
zbMATH Keywords: value of motivic \(L\)-functions at zero
zbMATH Keywords: conjecture of Chinburg
MaRDI profile type: Publication
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OpenAlex ID: W2034420302
description: scientific article; zbMATH DE number 892579