Integral points of bounded height on toric varieties

Chambert-Loir, Antoine; Tschinkel, Yuri

doi:10.48550/arxiv.1006.3345

Cited by 11 publications

(28 citation statements)

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“…Even though X is an equivariant compactification of G 3 a , the pairs (X, D i ) are neither partial equivariant compactifications of G 3 a nor toric by the previous lemma. Our result is thus not a special case of [CLT10b] or [CLT12].…”

Section: Lemma 24 There Is No Action Of G 3 a On X With An Open Orbit...mentioning

confidence: 66%

“…Results in this direction include complete intersections of large dimension compared to their degree [Bir62], algebraic groups and homogeneous spaces [DRS93, EMS96, EM93, BR95, Mau07, GOS09, WX16], and partial equivariant compactifications [CLT10b,CLT12,TBT13], that is, equivariant compactifications X together with an invariant divisor D. The first case is an application of the circle method, for the latter cases the group structure is exploited by means such as harmonic analysis.…”

Section: Introductionmentioning

confidence: 99%

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Integral points of bounded height on a log Fano threefold

Wilsch¹

2019

Preprint

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Section: Lemma 24 There Is No Action Of G 3 a On X With An Open Orbit...mentioning

confidence: 66%

Section: Introductionmentioning

confidence: 99%

Integral points of bounded height on a log Fano threefold

Wilsch¹

2019

Preprint

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“…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 [4] and also Remark 2.9), they establish asymptotic formulas for integral points of U, which imply that U satisfies strong approximation with Brauer-Manin obstruction off ∞ k .…”

Section: Introductionmentioning

confidence: 61%

Strong Approximation with Brauer–Manin Obstruction for Toric Varieties

Cao¹,

Xu²

2018

Ann. inst. Fourier

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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

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“…as B → ∞, where ρ(Y ′ F ′ ) = rank Pic Y ′ F ′ . Moreover in [8], Chambert-Loir and Tschinkel proved this asymptotic formula with respect to all choices of adelic metric on the anticanonical bundle, in particular the rational points on T are equidistributed with respect to the associated Tamagawa measure, in the sense defined by Peyre (see [27,Sec. 3]).…”

Section: 4mentioning

confidence: 90%

Rational points of bounded height and the Weil restriction

Loughran

2015

Isr. J. Math.

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Given an extension of number fields E ⊂ F and a projective variety X over F , we compare the problem of counting the number of rational points of bounded height on X with that of its Weil restriction over E. In particular, we consider the compatibility with respect to the Weil restriction of conjectural asymptotic formulae due to Manin and others. Using our methods we prove several new cases of these conjectures. We also construct new counterexamples over every number field.

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Integral points of bounded height on toric varieties

Abstract: We establish asymptotic formulas for the number of integral points of bounded height on toric varieties.Résumé. -Nous établissons un développement asymptotique du nombre de points entiers de hauteur bornée dans les variétés toriques.

Cited by 11 publications

References 21 publications

Integral points of bounded height on a log Fano threefold

Integral points of bounded height on a log Fano threefold

Strong Approximation with Brauer–Manin Obstruction for Toric Varieties

Rational points of bounded height and the Weil restriction

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