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Abstract
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.