Strong approximation theorem for absolutely irreducible varieties over the compositum of all symmetric extensions of a global field

Moshe Jarden, Aharon Razon

Research output: Contribution to journalArticlepeer-review

Abstract

Let K be a global field, a proper subset of the set of all primes of K, a finite subset of , and (resp. Ksep) a fixed algebraic (resp. separable algebraic) closure of K with. Let Gal(K) = Gal(Ksep/K) be the absolute Galois group of K. For each , we choose a Henselian (respectively, a real or algebraic) closure of K at in if is non-Archimedean (respectively, archimedean). Then, is the maximal Galois extension of K in Ksep in which each totally splits. For each , we choose a-Adic absolute value of and extend it in the unique possible way to. Finally, we denote the compositum of all symmetric extensions of K by Ksymm. We consider an affine absolutely integral variety V in. Suppose that for each there exists a simple-rational point of V and for each there exists such that in both cases if is non-Archimedean and <![CDATA[\mathbf{z}-\mathfrak{p}|-\mathfrak{p} if is archimedean. Then, there exists such that for all and for all τ â Gal(K), we have if is archimedean and <![CDATA[\mathbf{z}^\tau|-\mathfrak{p} if is non-Archimedean. For , we get as a corollary that the ring of integers of Ksymm is Hilbertian and Bezout.

Original languageEnglish
Pages (from-to)373-380
Number of pages8
JournalGlasgow Mathematical Journal
Volume61
Issue number2
DOIs
StatePublished - 1 May 2019

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