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

 

Rumely's local global principle
for algebraic P$\mathcal{S}$C fields over rings


Authors: Moshe Jarden and Aharon Razon
Journal: Trans. Amer. Math. Soc. 350 (1998), 55-85
MSC (1991): Primary 11R23
MathSciNet review: 1355075
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Abstract: Let $ \mathcal{S}$ be a finite set of rational primes. We denote the maximal Galois extension of $ \mathbb{Q}$ in which all $p\in \mathcal{S}$ totally decompose by $N$. We also denote the fixed field in $N$ of $e$ elements $ \sigma _{1},\ldots , \sigma _{e}$ in the absolute Galois group $G( \mathbb{Q})$ of $ \mathbb{Q}$ by $N( {\boldsymbol \sigma })$. We denote the ring of integers of a given algebraic extension $M$ of $ \mathbb{Q}$ by $ \mathbb{Z}_{M}$. We also denote the set of all valuations of $M$ (resp., which lie over $S$) by $ \mathcal{V}_{M}$ (resp., $ \mathcal{S}_{M}$). If $v\in \mathcal{V}_{M}$, then $O_{M,v}$ denotes the ring of integers of a Henselization of $M$ with respect to $v$. We prove that for almost all $ {\boldsymbol \sigma }\in G( \mathbb{Q})^{e}$, the field $M=N( {\boldsymbol \sigma })$ satisfies the following local global principle: Let $V$ be an affine absolutely irreducible variety defined over $M$. Suppose that $V(O_{M,v})\not =\varnothing $ for each $v\in \mathcal{V}_{M}\backslash \mathcal{S}_{M}$ and $V_{\mathrm{sim}}(O_{M,v})\not =\varnothing $ for each $v\in \mathcal{S}_{M}$. Then $V(O_{M})\not =\varnothing $. We also prove two approximation theorems for $M$.


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Additional Information

Moshe Jarden
Affiliation: School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel
Email: jarden@math.tau.ac.il

Aharon Razon
Affiliation: School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel
Email: razon@math.tau.ac.il

DOI: http://dx.doi.org/10.1090/S0002-9947-98-01630-4
PII: S 0002-9947(98)01630-4
Keywords: PAC field over rings, P$\mathcal{S}$C fields over rings, local global principle, global fields, absolute Galois group, Haar measure, valuations, Henselian fields, field of totally $\mathcal{S}$-adic numbers
Received by editor(s): June 14, 1994
Received by editor(s) in revised form: August 1, 1995
Additional Notes: This research was supported by The Israel Science Foundation administered by The Israel Academy of Sciences and Humanities.
The authors thank Joachim Schmid for useful remarks.
Dedicated: To Peter Roquette with gratitude
Article copyright: © Copyright 1998 American Mathematical Society