Points de hauteur bornée, topologie adélique et mesures de Tamagawa

Peyre, Emmanuel

doi:10.5802/jtnb.405

Cited by 67 publications

(102 citation statements)

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“…Une motivation de ce travail est de démontrer la conjecture de Manin dans le cas de S 3 , autrement dit démontrer unéquivalent asymptotique de N H,S3 (B). Pour une présentation générale et complète de ce domaine, nous renvoyons aux articles de présentation de Peyre [19,20,21].…”

Section: Résuméunclassified

Répartition des points rationnels sur la cubique de Segre

Bretèche

2007

Proceedings of the London Mathematical Society

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We study the distribution of rational points on Segre's cubic defined as the locus of points x ∈ P 5 such thatIn particular, we verify Manin's conjecture concerning the asymptotic behaviour of the number of non-trivial rational points of height at most B. est connu sous le nom de cubique de Segre. Nous la noterons S 3 . Cet ensemble peutêtre vu comme une hypersurface de P 4 = {x ∈ P 5 : x 1 + x 2 + x 3 + x 4 + x 5 + x 6 = 0}. Cette cubique

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Section: Résuméunclassified

Répartition des points rationnels sur la cubique de Segre

Bretèche

2007

Proceedings of the London Mathematical Society

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“…The conjectures of Manin [7] and Peyre [16] give a precise prediction for the asymptotic behaviour of N (U ; B), as B → ∞, for normal del Pezzo surfaces in terms of certain invariants associated to a minimal resolution. The conjecture has now been resolved for several singular cubic surfaces over Q.…”

Section: N (U ; B) = # {T ∈ U (Q) : H (T) B }mentioning

confidence: 95%

Counting rational points on the Cayley ruled cubic

Bretèche

Browning²,

Salberger

2015

European Journal of Mathematics

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“…We construct an adelic metric ( · v ) v∈ΩK on the anticanonical line bundle ω −1 S in the sense of [Pey03] such that the Arakelov height (ω n ) j,n with the k-th and l-th columns removed, and t := k + l if k < i < l, and t := k + l − 1 otherwise.…”

Section: The Leading Constantmentioning

confidence: 99%

O-minimality on twisted universal torsors and Manin's conjecture over number fields

Frei¹,

Pieropan²

2016

Ann. Sci. École Norm. Sup.

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Abstract. Manin's conjecture predicts the distribution of rational points on Fano varieties. Using explicit parameterizations of rational points by integral points on universal torsors and lattice-point-counting techniques, it was proved for several specific varieties over Q, in particular del Pezzo surfaces. We show how this method can be implemented over arbitrary number fields, by proving Manin's conjecture for a singular quartic del Pezzo surface of type A 3 + A 1 . The parameterization step is treated in high generality with the help of twisted integral models of universal torsors. To make the counting step feasible over arbitrary number fields, we deviate from the usual approach over Q by placing higher emphasis on the geometry of numbers in the framework of o-minimal structures.

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Points de hauteur bornée, topologie adélique et mesures de Tamagawa

Cited by 67 publications

References 1 publication

Répartition des points rationnels sur la cubique de Segre

Répartition des points rationnels sur la cubique de Segre

Counting rational points on the Cayley ruled cubic

O-minimality on twisted universal torsors and Manin's conjecture over number fields

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