Galois descent for motives: the K3 case

Abstract

A theorem of Grothendieck tells us that if the Galois action on the Tate module of an abelian variety factors through a smaller field, then the abelian variety, up to isogeny and finite extension of the base, is itself defined over the smaller field. Inspired by this, we give a Galois descent datum for a motive $H$ over a field by asking that the Galois action on an $\ell$-adic realisation factor through a smaller field. We conjecture that this descent datum is effective, that is if a motive $H$ satisfies the above criterion, then it must itself descend to the smaller field. We prove this conjecture for K3 surfaces, under some hypotheses. The proof is based on Madapusi-Pera's extension of the Kuga-Satake construction to arbitrary characteristic.