The sign of the functional equation of the \(L\)-function of an orthogonal motive
type: Item
instance of: scholarly article
title: The sign of the functional equation of the \(L\)-function of an orthogonal motive
zbMATH Open document ID: 0821.14011
DOI: 10.1007/BF01241124
author: Takeshi Saito
published in: Inventiones Mathematicae
publication date: 1995-10-03T00:00:00Z
full work available at URL: https://eudml.org/doc/144271
review text: Let \(K\) and \(E\) be fields of characteristic \(\neq 2\). A quadratic \(K\)- vector space is a finite dimensional \(K\)-vector space \(D\) with a nondegenerate quadratic form \(q : D \to K\). With \(q\) is associated a symmetric bilinear form \(b : D \times D \to K\), and one has the notion of orthogonality. Write \[ H(K, \mathbb{Z}/2)^* : = \biggl\{ 1 + a_ 1 + a_ 2 \in \bigoplus_{i = 0}^ 2 H^ i (K, \mathbb{Z}/2);\;a_ i \in H^ i (K, \mathbb{Z}/2) \biggr\}. \] The cup product induces an abelian group structure on \(H(K, \mathbb{Z}/2)^*\). A quadratic \(K\)-vector space \((D,q)\) determines its Hasse-Witt class hw\((D) \in H (K, \mathbb{Z}/2)^*\). The Hasse-Witt class is also defined for virtual quadratic vector spaces by hw\(([D_ 1] - [D_ 2]) =\text{hw}([D_ 1])\text{hw}([D_ 2])^{-1}\). On the other hand, for a profinite group \(G\) and a quadratic \(E\)-vector space \(V\) one defines an orthogonal \(E\)-representation of \(G\) as a continuous representation \(\rho : G \to O(V)\). For such a representation one can define its Stiefel-Whitney class sw\((\rho) \in H (G, \mathbb{Z}/2)^*\). Several properties of orthogonal representations, Hasse-Witt classes and Stiefel-Whitney classes are recalled. In particular, for \(K = E\) and \(G = \text{Gal} (K^ s/K)\) an elegant short proof is given of Fröhlich's theorem giving the relation between the Stiefel-Whitney class and the Hasse-Witt class of an orthogonal \(K\)-representation \(\rho : G \to O(v)\) with associated quadratic \(K\)-vector space \(D\) corresponding to the cohomology class of the \(I\)-cocycle \(\rho\): sw\( (\rho) =\text{hw}([D] - [V_ K]) +\text{sp}(\rho)\) in \(H(K, \mathbb{Z}/2)^*\), where sp\((\rho) \in H^ 2 (K, \mathbb{Z}/2)\) denotes the spinor class of \(\rho\). Fröhlich's theorem is generalized for \(p\)-adic representations of Hodge-Tate type. One has to generalize the notion of Stiefel-Whitney class to \(p\)-adic orthogonal representations of profinite groups such as the absolute Galois groups of l…
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 14A20
Mathematics Subject Classification ID: 11S40
zbMATH DE Number: 746071
zbMATH Keywords: characteristic \(p\)
zbMATH Keywords: \(\varepsilon\)-factor
zbMATH Keywords: functional equation of \(L\)- function
zbMATH Keywords: Stiefel-Whitney class
zbMATH Keywords: Hasse-Witt class
zbMATH Keywords: orthogonal representations
zbMATH Keywords: motive
reviewed by: W. W. J. Hulsbergen
MaRDI profile type: Publication
cites work: Iwasawa theory for the symmetric square of an elliptic curve.
cites work: Les constantes locales de l'équation fonctionnelle de la fonction L d'Artin d'une représentation orthogonale
cites work: Construction de représentations $p$-adiques
cites work: Orthogonal representations of Galois groups, Stiefel-Whitney classes and Hasse-Witt invariants.
cites work: On the functional equation of the Artin L-function for characters of real representations
cites work: Conducteurs d'Artin des caracteres reels. (Artin conductors of real characters)
cites work: Relations between \(K_2\) and Galois cohomology
description: scientific article; zbMATH DE number 746071