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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Geometric families of constant reductions and the Skolem property
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by Barry Green PDF
Trans. Amer. Math. Soc. 350 (1998), 1379-1393 Request permission

Abstract:

Let $F|K$ be a function field in one variable and $\mathcal V$ be a family of independent valuations of the constant field $K.$ Given $v\in \mathcal V ,$ a valuation prolongation $\mathrm v$ to $F$ is called a constant reduction if the residue fields $F\mathrm v |Kv$ again form a function field of one variable. Suppose $t\in F$ is a non-constant function, and for each $v\in \mathcal V$ let $V_{t}$ be the set of all prolongations of the Gauß valuation $v_{t}$ on $K(t)$ to $F.$ The union of the sets $V_{t}$ over all $v\in \mathcal V$ is denoted by $\mathbfit {V}_{t}.$ The aim of this paper is to study families of constant reductions $\mathbfit {V}$ of $F$ prolonging the valuations of $\mathcal V$ and the criterion for them to be principal, that is to be sets of the type $\mathbfit {V}_{t}.$ The main result we prove is that if either $\mathcal V$ is finite and each $v\in \mathcal V$ has rational rank one and residue field algebraic over a finite field, or if $\mathcal V$ is any set of non-archimedean valuations of a global field $K$ satisfying the strong approximation property, then each geometric family of constant reductions $\mathbfit {V}$ prolonging $\mathcal V$ is principal. We also relate this result to the Skolem property for the existence of $\mathcal V$-integral points on varieties over $K,$ and Rumely’s existence theorem. As an application we give a birational characterization of arithmetic surfaces $\mathcal X /S$ in terms of the generic points of the closed fibre. The characterization we give implies the existence of finite morphisms to $\mathbb P ^{1}_{S}.$
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Additional Information
  • Barry Green
  • Affiliation: Department of Mathematics, University of Stellenbosch, Stellenbosch 7602, South Africa
  • MR Author ID: 76490
  • Email: bwg@land.sun.ac.za
  • Received by editor(s): December 5, 1995
  • Additional Notes: This paper is part of the author’s Habilitation Thesis, University of Heidelberg, January 1995. The author would like to thank the Deutsche Forschungsgemeinschaft and the University of Heidelberg for supporting this work.
  • © Copyright 1998 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 350 (1998), 1379-1393
  • MSC (1991): Primary 11G30, 11R58, 12J10, 14G25
  • DOI: https://doi.org/10.1090/S0002-9947-98-02094-7
  • MathSciNet review: 1458302