Strong Approximation for Zariski-dense Subgroups of Semi-Simple Algebraic Groups

1984 ◽  
Vol 120 (2) ◽  
pp. 271 ◽  
Author(s):  
Boris Weisfeiler
Keyword(s):  
Strong Approximation ◽  
Algebraic Groups ◽  
Dense Subgroups

scholarly journals Arithmetic purity of strong approximation for semi-simple simply connected groups

2020 ◽  
Vol 156 (12) ◽  
pp. 2628-2649
Author(s):  
Yang Cao ◽  
Zhizhong Huang
Keyword(s):  
Quadratic Form ◽  
Strong Approximation ◽  
Homogeneous Spaces ◽  
Algebraic Groups ◽  
Integral Points ◽  
Spin Group ◽  
Linear Algebraic Groups ◽  
Simply Connected ◽  
Quadratic Hypersurfaces ◽  
Almost Prime

In this article we establish the arithmetic purity of strong approximation for certain semisimple simply connected linear algebraic groups and their homogeneous spaces over a number field $k$. For instance, for any such group $G$ and for any open subset $U$ of $G$ with ${\mathrm {codim}}(G\setminus U, G)\geqslant 2$, we prove that (i) if $G$ is $k$-simple and $k$-isotropic, then $U$ satisfies strong approximation off any finite number of places; and (ii) if $G$ is the spin group of a non-degenerate quadratic form which is not compact over archimedean places, then $U$ satisfies strong approximation off all archimedean places. As a consequence, we prove that the same property holds for affine quadratic hypersurfaces. Our approach combines a fibration method with subgroup actions developed for induction on the codimension of $G\setminus U$, and an affine linear sieve which allows us to produce integral points with almost-prime polynomial values.

scholarly journals On almost strong approximation for algebraic groups

2002 ◽  
Vol 254 (2) ◽  
pp. 441-461 ◽  
Author(s):  
Wai Kiu Chan ◽  
J.S. Hsia
Keyword(s):  
Strong Approximation ◽  
Algebraic Groups
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