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

TL;DR

This paper establishes weak approximation results for rationally connected varieties over function fields of curves over various types of fields, including large, finite, and local fields, with applications to cubic hypersurfaces and surfaces.

Contribution

It proves new weak approximation theorems for rationally connected varieties over function fields over large, finite, and local fields, extending previous results and providing explicit applications.

Findings

Zariski density of rational points over large fields

Weak approximation at places of good reduction over infinite algebraic extensions

Surjectivity of specialization maps for cubic hypersurfaces over 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.