Tannakian twists of quadratic forms and orthogonal Nori motives

TL;DR

This paper extends classical results on quadratic forms within Tannakian categories, relating torsors to Hasse-Witt invariants, and applies these to motives, Galois representations, and fundamental group schemes.

Contribution

It introduces formulas connecting torsors of fiber functors to Hasse-Witt invariants in Tannakian categories, generalizing classical results to motives and Galois representations.

Findings

Formulas relating torsors to Hasse-Witt invariants in various Tannakian contexts

Applications to Nori's fundamental group scheme and motives over number fields

Computations for Artin motives and hypersurface motives

Abstract

We revisit classical results of Serre, Fr\"ohlich and Saito in the theory of quadratic forms. Given a neutral Tannakian category $(\mathcal{T},\omega)$ over a field $k$ of characteristic $\neq 2$, another fiber functor $\eta$ over a $k$-scheme $X$ and an orthogonal object $(M,q)$ in $\mathcal{T}$, we show formulas relating the torsor $\bf{Isom}^{\otimes}(\omega,\eta)$ to Hasse-Witt invariants of the quadratic space $\omega(M,q)$ and the symmetric bundle $\eta(M,q)$. 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-Fr\"ohlich 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-Fr\"ohlich 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.