## 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 $$\ell$$.

### MSC:

 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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### References:

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