Potential Automorphy of K3 Surfaces with Large Picard Rank

The first part of this paper studied ${\mathrm{G}\mathrm{S}\mathrm{p}}_4$-type abelian varieties and the corresponding compatible systems of ${\mathrm{G}\mathrm{S}\mathrm{p}}_4$ representations. Techniques in \cite{BCGP} are applied to show that one can prove the potential modularity of these abelian varieties and compatible systems under some conditions that guarantee a sufficient amount of good primes. Then, in the second part, we use the potential modularity theorems to prove that K3 surfaces over totally real field $F$ with Picard rank $\ge 17$ are potentially modular.