The sign of the functional equation of the L-function of an orthogonal motive

Abstract

We prove that the sign of the functional equation of the L-function of an orthogonal motive of even weight is always positive, if we admit the conjecture of the product formula for the epsilon factor of Deligne [D3] (5.2). More precisely, we define the sign independently of any conjecture by using the product formula as the definition and by working with one chosen "good" prime f and we prove its positivity. The positivity has been established for the Artin motives [Fr-Q][D2], for the symmetric square of a modular elliptic curve [C-Sc] and in positive characteristic by Serre [Fr-Q]. As in the proof in [D2], we consider the second Stiefel-Whitney class in zBr(K) of an orthogonal representation of the absolute Galois group G~: of an algebraic number field K. The positivity of the sign is then a consequence of the reciprocity law of the square residue symbol. Fixing a "good" prime f, we work with an f-adic representation and compare the local epsilon factor and the local Stiefel-Whitney class. For a prime vXr it is a theorem of Deligne [D2]. For a prime vie , we compute the Stiefel-Whitney class assuming the representation is cristalline (Theorem 2) using a generalization of a theorem of Fr6hlich [Fr] to f-adic representation of Hodge-Tate type (Theorem 1). By combining them with an elementary computation for the Hodge structures at r iot , we conclude that the sign is positive (Theorem 3). The crucial step in the proof of Theorem 2, given in section 1, is the determination by Fontaine and Lafaille [Fo-L] of the action of the tame inertia on the semisimplification of reduction modulo p of a cristalline representation. The theory of orthogonal representations of Galois group including a proof of Fr6hlich's theorem is reviewed in a preliminary section 0.