For a complex projective manifold that is rationally connected, resp. rationally simply connected, every finite subset is 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 has a rational point if it deforms to a rationally simply connected variety in characteristic 0 with vanishing elementary obstruction . 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 C 2 property.