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arXiv · 2210.13675

Galois descent for motives: the K3 case

Angus McAndrew
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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 $\ell$-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.

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Angus McAndrew. 2022-10-25. Galois descent for motives: the K3 case. https://arxiv.org/abs/2210.13675

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