Weak Approximation over Function Fields of Curves over Large or Finite Fields

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.