Supersingular K3 surfaces are unirational

We show that supersingular K3 surfaces in characteristic p≥5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\ge 5$$\end{document} 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb P}}^1$$\end{document}-bundle over Fp2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb F}}_{p^2}$$\end{document}. To complete the picture, we also establish Shioda–Inose type isogeny theorems for K3 surfaces with Picard rank ρ≥19\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho \ge 19$$\end{document} in positive characteristic.