Potential Modularity of K3 Surfaces with Large Picard Rank

The first part of this thesis studied GSp4-type abelian varieties and the correspondingcompatible systems of GSp4 representations. Techniques in [BCGP21] 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 ≥ 17 are potentially modular.