Under the assumption of the Hodge, Tate and Fontaine–Mazur conjectures we give a criterion for a compatible system of \(\ell \)-adic representations of the absolute Galois group of a number field to be isomorphic to the second cohomology of a K3 surface. This is achieved by producing a motive M realizing the compatible system, using a local to global argument for quadratic forms to produce a K3 lattice in the Betti realization of M and then applying surjectivity of the period map for K3 surfaces to obtain a complex K3 surface. Finally we use a very general descent argument to show that the complex K3 surface admits a model over a number field.
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It is a pleasure to thank Stefan Patrikis, for suggested this problem to me, for his patient guidance and for the many helpful discussions we had. I would also like to thank Domingo Toledo for some helpful discussions. The author was partially supported by NSF DMS 1246989.
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Klevdal, C. Recognizing Galois representations of K3 surfaces. Res. number theory 5, 16 (2019). https://doi.org/10.1007/s40993-019-0154-1