New York Journal of Mathematics Volume 23 (2017) 1447-1529
 Volume 23 Volume Index

Strong approximation theorem for absolutely integral varieties over PSC Galois extensions of global fields

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 Published: October 19, 2017 Keywords: PAC field, strong approximation theorem, stabilizing element, Picard group Subject: 12E30

Abstract
Let K be a global field, V a proper subset of the set of all primes of K, S a finite subset of V, and \tilde K (resp. Ksep) a fixed algebraic (resp. separable algebraic) closure of K. Let Gal(K)=Gal(Ksep/K) be the absolute Galois group of K. For each p∈V we choose a Henselian (respectively, a real or algebraic) closure Kp of K at p in \tilde K if p is nonarchimedean (respectively, archimedean). Then,
Ktot,S=\capp∈S\capτ∈Gal(K)Kpτ
is the maximal Galois extension of K in Ksep in which each p∈S totally splits. For each p∈V we choose a p-adic absolute value | |p of Kp and extend it in the unique possible way to \tilde K.

For σ=(σ1,...,σe)∈Gal(K)e let Ktot,S[σ] be the maximal Galois extension of K in Ktot,S fixed by σ1,...,σe. Then, for almost all σ∈Gal(K)e (with respect to the Haar measure), the field Ktot,S[σ] satisfies the following local-global principle:

Let V be an absolutely integral affine variety in AKn. Suppose that for each p∈S there exists zp∈ Vsimp(Kp) and for each p∈V\S there exists zp∈ V(\tilde K) such that in both cases |zp|p≦1 if p is nonarchimedean and |zp|p<1 if p is archimedean. Then, there exists z∈ V(Ktot,S[σ]) such that for all p∈V and for all τ∈Gal(K) we have: |zτ|p≦1 if p is archimedean and |zτ|p<1 if p is nonarchimedean.

Author information

Wulf-Dieter Geyer:
Department of Mathematics, Universität Erlangen-Nürnberg, Erlangen, Germany
geyer@mi.uni-erlangen.de

Moshe Jarden:
School of Mathematics, Tel Aviv University, Tel Aviv, Israel
jarden@post.tau.ac.il

Aharon Razon:
Elta, Ashdod, Israel
razona@elta.co.il