Recognizing Galois representations of K3 surfaces

Abstract

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.