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

TL;DR

This paper proves the arithmetic purity of strong approximation for certain semi-simple simply connected algebraic groups over number fields, extending to their homogeneous spaces and affine quadratic hypersurfaces, using a combination of geometric and sieve methods.

Contribution

It establishes the arithmetic purity of strong approximation for specific algebraic groups and their homogeneous spaces, including new cases involving affine quadratic hypersurfaces.

Findings

Strong approximation holds off one or finitely many places for $k$-isotropic groups.

Strong approximation holds off all archimedean places for certain spin groups.

The methods combine fibration, subgroup actions, and an affine sieve for integral points.

Abstract

In this article we establish the arithmetic purity of strong approximation for certain semi-simple simply connected $k$-simple 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 codim$(G\setminus U, G)\geq 2$, we prove that (i) if $G$ is $k$-isotropic, then $U$ satisfies strong approximation off any one (hence any finitely many) place; (ii) if $G$ is the spin group of a non-degenerate quadratic form which is non-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 combinatorial sieve which allows to produce integral points with almost prime polynomial values.