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Geometric Families of Constant Reductions and the Skolem Property

Barry Green
Transactions of the American Mathematical Society
Vol. 350, No. 4 (Apr., 1998), pp. 1379-1393
Stable URL: http://www.jstor.org/stable/117622
Page Count: 15
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Geometric Families of Constant Reductions and the Skolem Property
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Abstract

Let F|K be a function field in one variable and V be a family of independent valuations of the constant field K. Given v∈ V, a valuation prolongation v to F is called a constant reduction if the residue fields Fv|Kv again form a function field of one variable. Suppose t∈ F is a non-constant function, and for each v∈ V let Vt be the set of all prolongations of the [Note: Equation omitted. See the image of page 1379 for this equation.] valuation vt on K(t) to F. The union of the sets Vt over all v∈ V is denoted by Vt. The aim of this paper is to study families of constant reductions V of F prolonging the valuations of V and the criterion for them to be principal, that is to be sets of the type Vt. The main results we prove is that if either V is finite and each v∈ V has rational rank one and residue field algebraic over a finite field, or if V is any set of non-archimedean valuations of a global field K satisfying the strong approximation property, then each geometric family of constant reductions V prolonging V is principal. We also relate this result to the Skolem property for the existence of V-integral points on varieties over K, and Rumely's existence theorem. As an application we give a birational characterization of arithmetic surfaces χ /S in terms of the generic points of the closed fibre. The characterization we give implies the existence of finite morphisms to PS 1.

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