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STRONG APPROXIMATION THEOREM FOR ABSOLUTELY IRREDUCIBLE VARIETIES OVER THE COMPOSITUM OF ALL SYMMETRIC EXTENSIONS OF A GLOBAL FIELD

Part of: General field theory

Published online by Cambridge University Press:  13 July 2018

MOSHE JARDEN  and
AHARON RAZON
MOSHE JARDEN
Affiliation:
Tel Aviv University, Ramat Aviv, Tel Aviv, Israel e-mail: [email protected]
AHARON RAZON
Affiliation:
Elta, Ashdod, Israel e-mail: [email protected]
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Abstract

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Let K be a global field, $\mathcal{V}$ a proper subset of the set of all primes of K, $\mathcal{S}$ a finite subset of $\mathcal{V}$, and ${\tilde K}$ (resp. Ksep) a fixed algebraic (resp. separable algebraic) closure of K with $K_\mathrm{sep}\{\subseteq}{\tilde K}$. Let Gal(K) = Gal(Ksep/K) be the absolute Galois group of K. For each $\mathfrak{p}\in\mathcal{V}$, we choose a Henselian (respectively, a real or algebraic) closure $K_\mathfrak{p}$ of K at $\mathfrak{p}$ in ${\tilde K}$ if $\mathfrak{p}$ is non-archimedean (respectively, archimedean). Then, $K_{\mathrm{tot},\mathcal{S}}=\bigcap_{\mathfrak{p}\in\mathcal{S}}\bigcap_{\tau\in{\rm Gal}(K)}K_\mathfrak{p}^\tau$ is the maximal Galois extension of K in Ksep in which each $\mathfrak{p}\in\mathcal{S}$ totally splits. For each $\mathfrak{p}\in\mathcal{V}$, we choose a $\mathfrak{p}$-adic absolute value $|~|_\mathfrak{p}$ of $K_\mathfrak{p}$ and extend it in the unique possible way to ${\tilde K}$. Finally, we denote the compositum of all symmetric extensions of K by Ksymm. We consider an affine absolutely integral variety V in $\mathbb{A}_K^n$. Suppose that for each $\mathfrak{p}\in\mathcal{S}$ there exists a simple $K_\mathfrak{p}$-rational point $\mathbf{z}_\mathfrak{p}$ of V and for each $\mathfrak{p}\in\mathcal{V}\smallsetminus\mathcal{S}$ there exists $\mathbf{z}_\mathfrak{p}\in V({\tilde K})$ such that in both cases $|\mathbf{z}_\mathfrak{p}|_\mathfrak{p}\le1$ if $\mathfrak{p}$ is non-archimedean and $|\mathbf{z}_\mathfrak{p}|_\mathfrak{p}<1$ if $\mathfrak{p}$ is archimedean. Then, there exists $\mathbf{z}\in V(K_{\mathrm{tot},\mathcal{S}}\cap K_\mathrm{symm})$ such that for all $\mathfrak{p}\in\mathcal{V}$ and for all τ ∈ Gal(K), we have $|\mathbf{z}^\tau|_\mathfrak{p}\le1$ if $\mathfrak{p}$ is archimedean and $|\mathbf{z}^\tau|_\mathfrak{p}<1$ if $\mathfrak{p}$ is non-archimedean. For $\mathcal{S}=\emptyset$, we get as a corollary that the ring of integers of Ksymm is Hilbertian and Bezout.

MSC classification

Primary: 12E30: Field arithmetic
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 61 , Issue 2 , May 2019 , pp. 373 - 380
Copyright
Copyright © Glasgow Mathematical Journal Trust 2018 

References

REFERENCES

1. Bary-Soroker, L., Fehm, A. and Wiese, G., fields, Hilbertian and representations, Galois, J. R. Angew. Math. 712 (2016), 123–139.Google Scholar
2. Cassels, J. W. S. and Fröhlich, A., Algebraic number theory (Academic Press, London, 1967).Google Scholar
3. Fried, M. and Jarden, M., Field arithmetic (3rd edition), Ergebnisse der mathematik. 3, volume 11 (Springer, Heidelberg, 2008).Google Scholar
4. Geyer, W.-D. and Jarden, M., PSC Galois extensions of Hilbertian fields, Math. Nachr. 236 (2002), 119–160.Google Scholar
5. Jarden, M. and Razon, A., appendix by Geyer, W.-D., Skolem density problems over large Galois extensions of global fields, Contemp. Math. 270 (2000), 213–235.Google Scholar
6. Geyer, W.-D., Jarden, M. and Razon, A., Strong approximation theorem for absolutely integral varieties over PSC Galois extensions of global fields, NY J. Math. 23 (2017), 1447–1529.Google Scholar
7. Jarden, M. and Razon, A., Pseudo algebraically closed fields over rings, Israel J. Math. 86 (1994), 25–59.Google Scholar
8. Jarden, M. and Razon, A., Rumely's local global principle for weakly P $\mathcal{S}$C fields over holomorphy domains, Funct. Approx. Comment. Math. 39 (2008), 19–47.Google Scholar
9. Jarden, M., Algebraic patching, Springer monographs in mathematics (Springer, 2011, Heidelberg).Google Scholar
10. Pop, F., Embedding problems over large fields, Ann. Math. 144 (1996), 1–34.Google Scholar
11. Razon, A., Abundance of Hilbertian domains, Manuscr. Math. 94 (1997), 531–542.Google Scholar
12. Razon, A., The elementary theory of e-free PAC domains, Ann. Pure Appl. Logic, 103 (2000), 55–95.Google Scholar