Weak approximation over function fields of curves over large or finite fields

Abstract

Let K = k(C) be the function field of a curve over a field k and let X be a smooth, projective, separably rationally connected K-variety with $${X(K)\neq\emptyset}$$ . Under the assumption that X admits a smooth projective model $${\pi: \mathcal{X}\to C}$$ , we prove the following weak approximation results: (1) if k is a large field, then X(K) is Zariski dense; (2) if k is an infinite algebraic extension of a finite field, then X satisfies weak approximation at places of good reduction; (3) if k is a nonarchimedean local field and R-equivalence is trivial on one of the fibers $${\mathcal{X}_p}$$ over points of good reduction, then there is a Zariski dense subset $${W\subseteq C(k)}$$ such that X satisfies weak approximation at places in W. As applications of the methods, we also obtain the following results over a finite field k: (4) if |k| > 10, then for a smooth cubic hypersurface X/K, the specialization map $${X(K)\longrightarrow \prod_{p\in P}\mathcal{X}_p(\kappa(p))}$$ at finitely many points of good reduction is surjective; (5) if $${\mathrm{char}\,k\neq 2,\,3}$$ and |k| > 47, then a smooth cubic surface X over K satisfies weak approximation at any given place of good reduction.