The sign of the functional equation of the \(L\)-function of an orthogonal motive. (English) Zbl 0821.14011

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 local fields. So now let \(K\) be a complete discrete valuation field of characteristic zero with perfect residue field of characteristic \(p > 2\). Let \((V, \rho)\) be a continuous orthogonal \(\mathbb{Q}_ p\)-representation of \(G_ K\) which is assumed to be crystalline. Let \(D = (B_{\text{cris}} \otimes_{\mathbb{Q}_ p} V)^{G_ K}\) be the corresponding filtered quadratic \(K\)-vector space (in the language of Fontaine) supposed to satisfy \(F^{{p - 1 \over 2}} D = 0\), and write \(h(D) : = \sum_{q > 0} q \dim_ K \text{Gr}_ F^ qD \pmod 2\), then, among other things, assuming \(K\) is an unramified finite extension of \(\mathbb{Q}_ p\), one finds sw\(_ 2 (\rho) = (- 1)^{h(D) \cdot [k : \mathbb{Q}_ p]} \in H^ 2 (K, \mathbb{Z}/2)\), where one identifies \(H^ 2 (K, \mathbb{Z}/2) = \{\pm 1\}\).
Turning to the global case, let \(K\) and \(E\) denote algebraic number fields, and let \(M\) be a motive over \(K\) with \(E\)-coefficients in the sense of Deligne. \(M\) gives rise to a system of realizations, also written \(M\), over \(K\) with \(E\)-coefficients, a certain integer \(n \in \mathbb{Z}\), the weight, and a \(r \in \mathbb{N}\), the rank, satisfying a list of conditions, the first of which says that one has a free \(K \otimes_ \mathbb{Q} E\)-module \(D\) of rank \(r\) with a finite decreasing filtration \(F\) by sub-\(K \otimes_ \mathbb{Q} E\)-modules, not necessarily free. For such a system \(M\) of realizations (coming from the motive \(M \dots)\) one defines an \(\varepsilon\)-factor \(\varepsilon (M)\) (as a product of local \(\varepsilon\)-factors, and conjectured to be well-defined) and its completed \(L\)-function \(\Lambda (M,s)\). One has the usual conjectures on its analytic continuation and its functional equation involving \(\varepsilon (M)\), a discriminant and a conductor. If now \(M\) is an orthogonal system of realizations, i.e. there exists a non-degenerate symmetric bilinear form \(M \otimes M \to 1 (-n)\), then one shows that the functional equation (for suitable (half)integer values of the argument) involves a sign \(w(M) = \pm 1\), implying the functional equation to become \(\Lambda (M,s) = w(M) \Lambda (M^*,1 - s)\) for \(s = {n + 1 \over 2}\).
The main result of the paper concerns this sign. It says that for an orthogonal system of realizations \(M\) of even weight \(n = 2m\) (and such that for primes \(\ell \neq 2\) unramified in \(K\) with \(F^{m + {\ell - 1} \over 2} D = 0)\) one has \(w(M) = + 1\). The proof uses the expression of the second Stiefel-Whitney class in the local case. In particular, for a smooth projective variety \(X\) over \(K\) and even \(n\) one has \(w(H^ n_ \ell (X)) = + 1\) for almost all \(\ell\).


14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14A20 Generalizations (algebraic spaces, stacks)
11S40 Zeta functions and \(L\)-functions
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