Strong Approximation with Brauer–Manin Obstruction for Toric Varieties

Abstract

For smooth open toric varieties, we establish strong approximation off infinity with Brauer-Manin obstruction.which is a closed subset of X(A k ). As discovered by Manin, class field theory implies that X(k) ⊆ X(A k ) B . Let Pr ∞ denote the projection from adelic points to finite adelic points.Definition 1.1. Let X be a scheme of finite type over k, and S a finite subset of Ω k . i) If X(k) is dense in X(A S k ), we say X satisfies strong approximation off S. ii) If X(k) is dense in Pr S (X(A k ) Br (X) ), we say X satisfies strong approximation with Brauer-Manin obstruction off S.In this paper, we will study strong approximation for toric varieties defined as follows.Definition 1.2. Let T be a torus over k and X be an integral normal and separated scheme of finite type over k with an action of Twhere m T is the multiplication of T . We simply write (T ֒→ X) or X for this toric variety if the open immersion is clear.The main result of this paper is the following theorem.Theorem 1.3. Any smooth toric variety over k satisfies strong approximation with Brauer-Manin obstruction off ∞ k .As a corollary, we have:Corollary 1.4. Let S be a subset of Ω k , such that ∞ k ⊆ S. Then any smooth toric variety over k satisfies strong approximation with Brauer-Manin obstruction off S.Chambert-Loir and Tschinkel prove the same result in [4] under certain conditions by using harmonic analysis. More precisely, let (T ֒→ X) be a smooth projective toric variety over k and D a T -invariant divisor of X with U = X \ D. Assuming the line bundle −(K X + D) is big where K X is a canonical bundle of X and Pic(U) is free (see the proof of Lemma 3.5.1 in

Approximation forte pour les variétés avec une action d’un groupe linéaire

“…Préliminaires sur les G-variétés. On rappelle un résultat pour les variétés toriques lisses ( [CX,Prop. 2.10]).…”

“…L'approximation forte raffinée pour l'espace affine. Pour un ouvert U d'un espace affine A n satisfaisant codim(A n \ U, A n ) ≥ 2, l'approximation forte hors d'une place v 0 a été établie par Fei Xu et l'auteur dans [CX,Prop. 3.6], et raffinée lorsque la place v 0 est archimédienne dans [CX1,Prop.…”

Strong approximation with Brauer-Manin obstruction for groupic varieties

“…In our previous paper , strong approximation with Brauer–Manin obstruction has been established for open toric varieties. It is natural to ask whether such a result is still true if the torus is replaced by a connected linear algebraic group.…”

“…It is natural to ask whether such a result is still true if the torus is replaced by a connected linear algebraic group. The basic idea in is to construct the standard toric varieties (see [, Definition 2.12]) by using the complement divisors. The group action provides a crucial relation of local integral points for almost all places (see [, Proposition 4.1]).…”

“…The basic idea in is to construct the standard toric varieties (see [, Definition 2.12]) by using the complement divisors. The group action provides a crucial relation of local integral points for almost all places (see [, Proposition 4.1]). In order to prove strong approximation, one further needs to show that any point outside the torus can be approximated by a point in the torus with the same local invariant for all elements in the Brauer groups of a given toric variety (see [, Proposition 4.2]).…”