Schütt’s question on geometric realizations of weight 3 CM newforms

Determine which weight 3 CM newforms with rational Fourier coefficients admit geometric realizations in a smooth projective variety X over Q with h^{2,0}(X)=dim H^{0}(X, Ω^{2}_{X})=1.

Background

The authors prove modularity results for rank-21 cubic fourfolds and show that their Fano varieties furnish additional geometric realizations of weight-3 CM newforms with rational coefficients.

Schütt posed a classification question about which weight-3 CM newforms with rational coefficients can be realized geometrically in smooth projective varieties over Q with h{2,0}=1; Elkies and Schütt showed under ERH that every such newform is associated to a singular K3 surface, but the unconditional classification remains open.

References

Question. Which weight 3 CM newforms with rational Fourier coefficients have geometric realizations in a smooth projective variety $X$ over $\mathbb{Q}$ with $h{2, 0}(X)={\rm dim}H{0}(X, \Omega{2}_{X})=1$ ?

— Arithmetic Period Map and Complex Multiplication for Cubic Fourfolds  (2512.11355 - Ito, 12 Dec 2025) in Introduction, before Acknowledgements