Arithmetic purity of strong approximation

Determine whether every open subset U of a smooth variety X over a number field k satisfying strong approximation off a finite set of primes S also satisfies strong approximation off S whenever the complement X \ U has codimension at least 2 in X.

Background

The paper studies arithmetic purity for strong approximation, which asks whether strong approximation persists after removing a closed subset of codimension at least 2. Unlike weak approximation, strong approximation is not generally a birational invariant, so removing a sufficiently small subset can potentially change the arithmetic behavior of a variety.

The stated conjecture was proposed by Wittenberg and is supported by results for affine spaces, semi-simple simply connected quasi-split groups, related homogeneous spaces, and certain toric varieties. The paper proves the conjecture in the setting of smooth toric varieties over number fields with constant global sections, but does not resolve it for arbitrary smooth varieties satisfying strong approximation off S.

References

Conjecture 1.1. If X is a smooth variety over number field k satisfying strong approximation off a finite set of primes S of k, whether any open subset U of X over k with codim(X \ U, X) ≥ 2 also satisfies strong approximation off S?

— Arithmetic purity of strong approximation for toric varieties with constant global sections  (2608.28204 - Wei et al., 28 Aug 2026) in Conjecture 1.1, Section 1 (Introduction), p. 1

Wittenberg Has proposed the following conjecture in (, Question 2.11): Assume $X$ satisfies strong approximation off a finite set of primes $S$ of $K$. Let $U$ be an open subset of $X$ with $\mathrm{codim}(X-U,X)\geq2$. Does $U$ also satisfies strong approximation off $S$?

— The linear positive torsor presentation and arithmetic purity of strong approximation  (2610.21762 - Wang, 15 Oct 2026) in Section 1.1, Introduction, subsection “Strong approximation and arithmetic purity”