Arithmetic purity: strong approximation and counting integral points on homogeneous spaces

We report recent progress on a joint project with Yang Cao. If an algebraic variety over a number field verifies strong approximation off a finite set of places, it has been first conjectured by Wittenberg that this property is maintained under the removal of any subvariety of codimension two. We say that this variety satisfies arithmetic purity. A closely related question is the density of integral points whose multivariable polynomial values have no common gcd. We confirm the arithmetic purity for semi-simple simply connected isotropic linear algebraic groups, and for affine quadric hypersurfaces, using different methods. They show how the fibration method for rational points and various sieve methods (e.g. affine almost prime linear sieve, Ekedahl’s geometric sieve, Iwaniec’s half-dimensional sieve) match together.