You are not currently logged in.
Access your personal account or get JSTOR access through your library or other institution:
If You Use a Screen ReaderThis content is available through Read Online (Free) program, which relies on page scans. 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
Transactions of the American Mathematical Society
Vol. 350, No. 4 (Apr., 1998), pp. 1379-1393
Published by: American Mathematical Society
Stable URL: http://www.jstor.org/stable/117622
Page Count: 15
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.
Preview not available
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.
Transactions of the American Mathematical Society © 1998 American Mathematical Society