Strong Approximation for a Toric Variety

Equidistribution of rational points and the geometric sieve for toric varieties

“…Since the removal of a codimension two closed subset does not introduce new cohomological obstruction to strong approximation (as first observed by Minčhev [31]), inspired by a question of Wittenberg (see [43,Question 2.11]) on (arithmetic) purity of strong approximation (APSA) (see §2.2) and its recent progress on linear algebraic groups and their homogeneous spaces (see [42,14,12]), it is therefore natural to extend Principle 1.1 to open subvarieties of any fixed almost-Fano variety. We keep using the same notation as above.…”

“…Subsequent work of Pieropan and Schindler [35] deals with the distribution of Campana points on toric varieties. On the other hand, by [42,Theorem] and [14,Theorem 1.3], smooth projective split toric varieties satisfy purity of strong approximation, i.e., for any such W as in Principle 1.2, W(k) is dense in W(A k ).…”

Arithmetic purity of strong approximation for complete toric varieties

“…Wei [26] gave an affirmative answer to Question 1.2 for smooth toric varieties when S = / 0. In [6], Cao, Liang and Xu extended this result to partial smooth equivariant compactifications of homogeneous spaces.…”

“…However, one can expect that strong approximation is invariant among smooth varieties up to a closed subvariety of codimension ≥ 2. Wei [26,Lemma 2.1], and independently Cao and Xu [7,Proposition 3.6] proved that this is true for affine spaces. Based on this evidence, Wittenberg [27,Question 2.11] proposed the following question.…”

“…In this paper, we will give an affirmative answer to Question 1.2 for complete toric varieties (see Theorem 1.5) when S = / 0 and then we provide some other positive evidence (see Theorem 1.6) for Question 1.1. Following Wei [26], we use the descent theory and the combinatorial description of toric varieties to achieve our goals.…”