Shioda–Artin supersingularity conjectures for K3 surfaces

A K3 surface is a smooth projective surface with trivial canonical bundle and $H^1(X,\mathcal O_X)=0$. It is Shioda-supersingular if its Néron–Severi rank equals its second Betti number, and Artin-supersingular if its formal Brauer group has infinite height. A surface is unirational if it admits a dominant rational map from a projective space.

Shioda–Artin supersingularity conjectures. For K3 surfaces, Shioda-supersingularity implies unirationality; Artin-supersingularity implies unirationality; and Artin-supersingularity implies Shioda-supersingularity.

Unirational K3 surfaces are known to be both Shioda-supersingular and Artin-supersingular, and Shioda-supersingularity is known to imply Artin-supersingularity. The three converses listed here are conjectured; for elliptic K3 surfaces, the two supersingularity notions coincide.