论文标题
动机的Galois下降：K3案例
Galois descent for motives: the K3 case
论文作者
论文摘要
Grothendieck的一个定理告诉我们，如果Galois对Abelian品种因子的泰特模块的作用通过较小的田地，则在较小的田地上定义了底座的Abelian品种，直至基础的同等基础和有限的扩展。受此启发的启发，我们通过要求Galois在$ \ ell $ -ADIC实现因子上通过较小的字段提供了一个动机$ h $的Galois下降数据。我们猜想该下降基准是有效的，也就是说，如果动机$ h $满足上述标准，则它本身必须下降到较小的字段。 在某些假设下，我们证明了K3表面的这种猜想。证明是基于Madapusi-Pera将Kuga-Satake结构扩展到任意特征的。
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.
