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Tannakian twists of quadratic forms and orthogonal Nori motives

We revisit classical results of Serre, Fröhlich and Saito in the theory of quadratic forms. Given a neutral Tannakian category $(\mathcal{T},ω)$ over a field $k$ of characteristic $\neq 2$, another fiber functor $η$ over a $k$-scheme $X$ and an orthogonal object $(M,q)$ in $\mathcal{T}$, we show formulas relating the torsor $\bf{Isom}^{\otimes}(ω,η)$ to Hasse-Witt invariants of the quadratic space $ω(M,q)$ and the symmetric bundle $η(M,q)$. We apply this result to various neutral Tannakian categories arising in different contexts. We first consider Nori's Tannakian category of essentially finite bundles over an integral proper $k$-scheme $X$ with a rational point, in order to study an analogue of the Serre-Fröhlich embedding problem for Nori's fundamental group scheme. Then we consider Fontaine's Tannakian categories of $B$-admissible representations, in order to obtain a generalization of both the classical Serre-Fröhlich formula and Saito's analogous result for Hodge-Tate $p$-adic representations. Finally we consider Nori's category of mixed motives over a number field. These last two examples yield formulas relating the torsor of periods of an orthogonal motive to Hasse-Witt invariants of the associated Betti and de Rham quadratic forms and to Stiefel-Withney invariants of the associated local $l$-adic orthogonal representations. We give some computations for Artin motives and for the motive of a smooth hypersurface.