Rational points of rationally simply connected varieties over global function fields
arXiv:1703.08334
Abstract
A complex projective manifold is rationally connected, resp. rationally simply connected, if finite subsets are connected by a rational curve, resp. the spaces parameterizing these connecting rational curves are themselves rationally connected. We prove that a projective scheme over a global function field with vanishing "elementary obstruction" has a rational point if it deforms to a rationally simply connected variety in characteristic 0. This gives new, uniform proofs over these fields of the Period-Index Theorem, the quasi-split case of Serre's "Conjecture II", and Lang's property.
14 pages; second draft shorter, incorporates work of Yi Zhu, implies quasi-split case of Serre's "Conjecture II"