Geometric families of constant reductions and the Skolem property

Abstract

Let F | K F|K be a function field in one variable and V \mathcal V be a family of independent valuations of the constant field K . K. Given v ∈ V , v\in \mathcal V , a valuation prolongation v \mathrm v to F F is called a constant reduction if the residue fields F v | K v F\mathrm v |Kv again form a function field of one variable. Suppose t ∈ F t\in F is a non-constant function, and for each v ∈ V v\in \mathcal V let V t V_{t} be the set of all prolongations of the Gauß valuation v t v_{t} on K ( t ) K(t) to F . F. The union of the sets V t V_{t} over all v ∈ V v\in \mathcal V is denoted by V t . \mathbfit {V}_{t}. The aim of this paper is to study families of constant reductions V \mathbfit {V} of F F prolonging the valuations of V \mathcal V and the criterion for them to be principal, that is to be sets of the type V t . \mathbfit {V}_{t}. The main result we prove is that if either V \mathcal V is finite and each v ∈ V v\in \mathcal V has rational rank one and residue field algebraic over a finite field, or if V \mathcal V is any set of non-archimedean valuations of a global field K K satisfying the strong approximation property, then each geometric family of constant reductions V \mathbfit {V} prolonging V \mathcal V is principal. We also relate this result to the Skolem property for the existence of V \mathcal V -integral points on varieties over K , K, and Rumely’s existence theorem. As an application we give a birational characterization of arithmetic surfaces X / S \mathcal X /S in terms of the generic points of the closed fibre. The characterization we give implies the existence of finite morphisms to P S 1 . \mathbb P ^{1}_{S}.