 

WulfDieter Geyer, Moshe Jarden, and Aharon Razon
Strong approximation theorem for absolutely integral varieties over PSC Galois extensions of global fields view print


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. K_{sep}) a fixed algebraic
(resp. separable algebraic) closure of K.
Let Gal(K)=Gal(K_{sep}/K) be the absolute Galois group of
K.
For each p∈V we choose a Henselian
(respectively, a real or algebraic) closure
K_{p} of K at p
in \tilde K if
p is nonarchimedean (respectively, archimedean).
Then,
K_{tot,S}=\cap_{p∈S}\cap_{τ∈Gal(K)}K_{p}^{τ}
is the maximal Galois extension of K in K_{sep} in which
each
p∈S totally splits.
For each p∈V we choose a padic absolute
value  _{p} of K_{p} and extend it in the unique
possible way to \tilde K.
For σ=(σ_{1},...,σ_{e})∈Gal(K)^{e} let
K_{tot,S}[σ] be the maximal Galois extension of
K in K_{tot,S} fixed by σ_{1},...,σ_{e}.
Then, for almost all σ∈Gal(K)^{e} (with respect to
the Haar
measure), the field K_{tot,S}[σ]
satisfies the following localglobal principle:
Let V
be an absolutely integral affine variety in A_{K}^{n}.
Suppose that for each p∈S there exists
z_{p}∈ V_{simp}(K_{p})
and for each p∈V\S there exists
z_{p}∈ V(\tilde K)
such that in both cases z_{p}_{p}≦1
if p is nonarchimedean and z_{p}_{p}<1 if
p is archimedean.
Then, there exists z∈ V(K_{tot,S}[σ]) such that
for all
p∈V and for all
τ∈Gal(K) we have:
z^{τ}_{p}≦1 if p is
archimedean and z^{τ}_{p}<1 if p
is nonarchimedean.


Author information
WulfDieter Geyer:
Department of Mathematics, Universität ErlangenNürnberg, Erlangen, Germany
geyer@mi.unierlangen.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

