5 problems

Artin–Rudakov–Shafarevich–Shioda conjecture. A K3 surface over $k$ is supersingular if and only if it is unirational.

Fully supersingular Bloch--Beilinson conjecture. Numerical and algebraic equivalence coincide on ${\rm CH}^{i}(X)_{\mathbb{Q}}$, so $\operatorname{Griff}^{i}(X)_{\mathbb{Q}}=0$; th…

Artin--Shioda--Rudakov--Safarevic conjecture. A K3 surface is supersingular if and only if it is unirational.

Artin's Frobenius-pullback conjecture. The elliptic fibration on $X$ arises via Frobenius pullback from a rational elliptic surface.

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