Effective Galois descent for motives: the K3 case
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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 l-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 fields.