Motivic \(\varepsilon\)-factors at infinity and regularized dimensions
type: Item
instance of: scholarly article
title: Motivic \(\varepsilon\)-factors at infinity and regularized dimensions
zbMATH Open document ID: 0821.14004
DOI: 10.1016/0019-3577(94)90015-9
author: Christopher Deninger
published in: Indagationes Mathematicae. New Series
publication date: 1995-10-03T00:00:00Z
review text: Using results of other work of the author [cf. Motives, Proc. Summer Res. Conf. Motives, Univ. Washington 1991, Proc. Symp. Pure Math. 55, Part I, 707-743 (1994; Zbl 0816.14010)] one can rewrite the non-archimedean \(\varepsilon\)-factors of the \(L\)-function of a motive in terms of regularized superdimensions. The formula thus obtained suggests a similar result for the archimedean \(\varepsilon\)-factors. To obtain such a formula consider the categories \({\mathcal M} {\mathcal H}_ \mathbb{C}\) and \({\mathcal M} {\mathcal H}_ \mathbb{R}\) of real mixed Hodge structures and of real mixed Hodge structures with an involution \(F_ \infty\) respecting the weight filtration and such that its \(\mathbb{C}\)-linear extension maps \(F^ \bullet H_ \mathbb{C}\) into \(\overline F^ \bullet H_ \mathbb{C}\) for an object \(H\) of \({\mathcal M} {\mathcal H}_ \mathbb{R}\), respectively. Also, for \(K = \mathbb{R}\) or \(\mathbb{C}\) and a number field \(E\), write \({\mathcal M} {\mathcal H}_ K (E)\) for the \(E\)-linear category of objects of \({\mathcal M} {\mathcal H}_ K\) with multiplication by \(E\). Then on \({\mathcal M} {\mathcal H}_ K (E)\) one defines an \(E \otimes \mathbb{C}\)-invariant filtration \(\gamma^ \nu H_ \mathbb{C} = F^ \nu H_ \mathbb{C} \cap \overline F^ \nu H_ \mathbb{C}\). This filtration is used (by Fontaine and Perrin-Riou) to define the local \(\varepsilon\)- factors at infinity. On the other hand, let \({\mathcal F} il_ K\) denote the exact additive category of finite dimensional complex vector spaces with finite decreasing filtration and, in case \(K = \mathbb{R}\), there should exist an involution \(F_ \infty\) respecting this filtration and inducing \((- 1)^ \bullet\) on \(\text{Gr}^ \bullet V\) for \(V \in {\mathcal F} il_ \mathbb{R}\). In his paper cited above, the author defined functors \(\mathbb{D}\) (the Riemann-Hilbert correspondence) for the finite places and \(\mathbb{D}^ +\) from \({\mathcal F} il_ K\) to the category of \(\mathbb{L}_ K…
Mathematics Subject Classification ID: 14A20
Mathematics Subject Classification ID: 14G10
zbMATH DE Number: 749790
zbMATH Keywords: \(\varepsilon\)-factors of the \(L\)-function of a motive
zbMATH Keywords: superdimensions
reviewed by: W. W. J. Hulsbergen
MaRDI profile type: Publication
Wikidata QID: Q56813088
full work available at URL: 0019-3577(94)90015-9
OpenAlex ID: W2054294562
cites work: Les Constantes des Equations Fonctionnelles des Fonctions L
cites work: Q4297075
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cites work: Q3855289
description: scientific article; zbMATH DE number 749790