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

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

2009-07-15

arxiv.org/abs/0907.2529v4

Math. Ann. 348 (2010), No. 2, 357-377

Mathematics

Algebraic Geometry

numbering style changed; Theorem 2 in Section 1 strengthens its early version; many subsequent changes in Section 5

Scientific paper

10.1007/s00208-010-0481-y

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.

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