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Arithmetic purity of strong approximation for semi-simple simply connected groups
- Part of
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- Journal:
- Compositio Mathematica / Volume 156 / Issue 12 / December 2020
- Published online by Cambridge University Press:
- 01 February 2021, pp. 2628-2649
- Print publication:
- December 2020
-
- Article
- Export citation
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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.
