A conjectural uniform construction of many rigid Calabi-Yau threefolds

Given a rational Hecke eigenform $f$ of weight $2$, Eichler-Shimura theory gives a construction of an elliptic curve over ${\mathbb Q}$ whose associated modular form is $f$. Mazur, van Straten, and others have asked whether there is an analogous construction for Hecke eigenforms $f$ of weight $k>2$ that produces a variety for which the Galois representation on its etale ${\mathrm H}^{k-1}$ (modulo classes of cycles if $k$ is odd) is that of $f$. In weight $3$ this is understood by work of Elkies and Sch"utt, but in higher weight it remains mysterious, despite many examples in weight $4$. In this talk I will present a new construction based on families of K3 surfaces of Picard number $19$ that recovers many existing examples in weight $4$ and produces almost $20$ new ones.