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

Abstract

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.