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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Geometric families of constant reductions
and the Skolem property


Author: Barry Green
Journal: Trans. Amer. Math. Soc. 350 (1998), 1379-1393
MSC (1991): Primary 11G30, 11R58, 12J10, 14G25
MathSciNet review: 1458302
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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 ${{\mathchoice {{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\textstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}}{{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\textstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}} {{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\scriptstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}}{{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\scriptscriptstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}}}}_{\!t}.$

The aim of this paper is to study families of constant reductions ${{\mathchoice {{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\textstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}}{{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\textstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}} {{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\scriptstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}}{{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\scriptscriptstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}}}}$ of $F$ prolonging the valuations of $\mathcal V $ and the criterion for them to be principal, that is to be sets of the type ${{\mathchoice {{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\textstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}}{{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\textstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}} {{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\scriptstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}}{{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\scriptscriptstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}}}}_{\!t}.$ 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 ${{\mathchoice {{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\textstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}}{{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\textstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}} {{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\scriptstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}}{{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\scriptscriptstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}}} }$ 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
Email: bwg@land.sun.ac.za

DOI: http://dx.doi.org/10.1090/S0002-9947-98-02094-7
PII: S 0002-9947(98)02094-7
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.
Article copyright: © Copyright 1998 American Mathematical Society