Shioda’s Conjecture on Unirationality

In characteristic zero, Castelnuovo proved that every unirational surface is rational. In positive characteristic, this fails dramatically: there exist many non-rational, often even general-type, surfaces that are nevertheless unirational. In 1977, Shioda conjectured that such phenomena are completely explained by the Galois representations—i.e., a simply-connected surface is unirational if and only if it is supersingular. In this talk, I will present a counterexample to this conjecture. The construction requires a new obstruction, inspired by ideas from the study of the hyperbolicity of complex varieties.