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

作者：Hu Yong^*

来源：Mathematische Annalen, 2010, 348(2): 357-377.

DOI：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) not equal empty set. Under the assumption that X admits a smooth projective model pi : chi -> 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 chi(p) over points of good reduction, then there is a Zariski dense subset W subset of 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 vertical bar k vertical bar > 10, then for a smooth cubic hypersurface X/K, the specialization map X(K) -> Pi(p is an element of P) chi(p)(kappa(p)) at finitely many points of good reduction is surjective; (5) if char k not equal 2, 3 and vertical bar k vertical bar > 47, then a smooth cubic surface X over K satisfies weak approximation at any given place of good reduction.

• 出版日期2010-10