Potential Automorphy of K3 Surfaces with Large Picard Rank

TL;DR

This paper proves that K3 surfaces over totally real fields with high Picard rank are potentially modular by leveraging potential modularity theorems and techniques from GSp4-type abelian varieties.

Contribution

It establishes the potential modularity of certain K3 surfaces with large Picard rank, extending modularity results to new classes of algebraic surfaces.

Findings

K3 surfaces with Picard rank ≥ 17 are potentially modular.

Application of techniques from GSp4 abelian varieties to K3 surfaces.

Use of potential modularity theorems to connect Galois representations and automorphic forms.

Abstract

The first part of this paper studied $\mathrm{GSp}_4$-type abelian varieties and the corresponding compatible systems of $\mathrm{GSp}_4$ 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.