Supersingular K3 Surfaces are Unirational

We show that supersingular K3 surfaces in characteristic p≥5 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 P1-bundle over Fp2. To complete the picture, we also establish Shioda-Inose type isogeny theorems for K3 surfaces with Picard rank ρ≥19 in positive characteristic.