Supersingular K3 surfaces are unirational

We show that supersingular K3 surfaces in characteristic $$p\ge 5$$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 $${{\mathbb P}}^1$$^P1-bundle over $${{\mathbb F}}_{p^2}$$F^p2. To complete the picture, we also establish Shioda–Inose type isogeny theorems for K3 surfaces with Picard rank $$\rho \ge 19$$ρ≥19 in positive characteristic.