Supersingular K3 Surfaces are Unirational

We show that supersingular K3 surfaces in characteristic $p\geq5$ are related by purely inseparable isogenies. This implies that they are unirational, which proves conjectures of Artin, Rudakov, Shafarevich, and Shioda. As a byproduct, we exhibit the moduli space of rigidified K3 crystals as an iterated $\mathbb{P}^1$-bundle over $\mathbb{F}_{p^2}$. To complete the picture, we also establish Shioda-Inose type isogeny theorems for K3 surfaces with Picard rank $\rho\geq19$ in positive characteristic.Comment: 31 pages; many details added, final versio