Local to global principle for the moduli space of K3 surfaces

TL;DR

This paper extends the finite descent obstruction concept from abelian varieties to K3 surfaces, showing that certain Hodge-theoretical data can determine a K3 surface over a number field.

Contribution

It establishes an analogous result for K3 surfaces under technical conditions, linking Hodge theory and $ ext{l}$-adic representations to the moduli space.

Findings

Finite descent obstruction holds for K3 surfaces under certain conditions.

Hodge-theoretical data determines a K3 surface from $ ext{l}$-adic representations.

Results connect Galois representations with geometric structures of K3 surfaces.

Abstract

Recently S. Patrikis, J.F. Voloch and Y. Zarhin have proven, assuming several well known conjectures, that the finite descent obstruction holds on the moduli space of principally polarised abelian varieties. We show an analogous result for K3 surfaces, under some technical restrictions. This is possible since abelian varieties and K3s are quite well described by `Hodge-theoretical' results. In particular the theorem we present can be interpreted as follows: a family of $\ell$-adic representations that looks like the one induced by the transcendental part of the $\ell$-adic cohomology of a K3 surface (defined over a number field) determines a Hodge structure which in turn determines a K3 surface (which may be defined over a number field).