Transactions of the American Mathematical Society

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ISSN 1088-6850(e) ISSN 0002-9947(p)

Geometric families of constant reductions and the Skolem property

Author(s): 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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