Points de hauteur bornée sur les variétés de drapeaux en caractéristique finie

Peyre, Emmanuel

doi:10.4064/aa152-2-5

Cited by 6 publications

(4 citation statements)

References 18 publications

(19 reference statements)

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“…Manin's problem on function fields has been studied very scarcely until now: one must nevertheless quote the works of Bourqui [Bou02,Bou03,Bou11], which completely solve the case of toric varieties (employing harmonic analysis but also the universal torsor method), as well as the papers [LY] and [Pey12] which treat independently the case of generalised flag varieties, Peyre's work containing moreover an interpretation of the constant. Peyre's method is analogous to Franke, Manin and Tschinkel's for flag varieties over number fields in [FMT], the role of the results of Langlands being played by those of Morris about Eisenstein series over function fields.…”

Section: Manin's Problem Over Function Fieldsmentioning

confidence: 99%

Motivic Euler products in motivic statistics

Bilu

Howe

2021

Alg. Number Th.

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A motivic height zeta function associated to a family of varieties parametrised by a curve is the generating series of the classes, in the Grothendieck ring of varieties, of moduli spaces of sections of this family with varying degrees. This memoir is devoted to the study of the motivic height zeta function associated to a family of varieties with generic fiber having the structure of an equivariant compactification of a vector group. It is a motivic analogue of Chambert-Loir and Tschinkel's work [CLT12] solving Manin's problem for integral points of bounded height on partial equivariant compactifications of vector groups over number fields, and a generalisation of Chambert-Loir and Loeser's paper [ChL]. Our main theorem, stated in the introduction and proved in the final chapter of the text, describes the convergence of this motivic height zeta function with respect to a topology on the Grothendieck ring of varieties coming from the theory of weights in cohomology. We deduce from it the asymptotic behaviour, as the degree goes to infinity, of a positive proportion of the coefficients of the Hodge-Deligne polynomial of the above moduli spaces: in particular, we get an estimate for their dimension and the number of components of maximal dimension.Our method, based on the Poisson summation formula, is inspired by the one used in [CLT12], but the motivic setting requires several major adaptations besides the ones already present in [ChL]. The main contribution of this text is the introduction of a notion of motivic Euler product for series with coefficients in the Grothendieck ring of varieties, analogous to the classical Euler products encountered in number theory. It relies on the construction of geometric objects called symmetric products, which generalise the notion of symmetric power of a variety. These Euler products and symmetric products allow moreover a suitable extension of Hrushovski and Kazhdan's motivic Poisson formula, which is used to rewrite the motivic height zeta function as a sum, over characters, of series with coefficients in a Grothendieck ring of varieties with exponentials. The aforementioned weight topology on this ring is constructed using Saito's theory of mixed Hodge modules, as well as Denef and Loeser's motivic vanishing cycles and Thom-Sebastiani theorem.

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Section: Manin's Problem Over Function Fieldsmentioning

confidence: 99%

Motivic Euler products in motivic statistics

Bilu

Howe

2021

Alg. Number Th.

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“…Sous les seules hypothèses que nous avons données, la convergence du produit eulérien dans (2.0.1) n'est a priori pas assurée. Nous renvoyons à [Pey03] pour des précisions sur les hypothèses permettant de montrer la convergence en utilisant Weil-Deligne. Pour la classe de variétés étudiée dans cet article, ces hypothèses sont vérifiées et la convergence du produit peut d'ailleurs se voir de manière élémentaire.…”

Section: Conjectures De Manin Géométriquesunclassified

“…(2.0.1) L'interprétation conceptuelle de γ(X) se fait en termes du volume de l'espace adélique associé à la variété X × k k(C ) pour une certaine mesure de Tamagawa, cf. [Pey95,Pey03]. Sous les seules hypothèses que nous avons données, la convergence du produit eulérien dans (2.0.1) n'est a priori pas assurée.…”

Section: Conjectures De Manin Géométriquesunclassified

Exemples de comptages de courbes sur les surfaces

Bourqui

2013

Math. Ann.

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Soit X une surface dont l'anneau de Cox a une seule relation, laquelle vérifie en outre une certaine propriété de linéarité. Nous montrons, sous une hypothèse simple, que les conjectures de Manin géométriques valent pour certains degrés du cône effectif dual de X (notamment pour ces degrés l'espace de modules de morphismes a la dimension attendue). Le résultat s'applique à une classe de surfaces de del Pezzo généralisées qui a été intensément étudiée dans le cadre de la conjecture de Manin arithmétique. Some examples of curves countings on surfaces.Let X be a surface whose Cox ring has a single relation satisfying moreover a kind of linearity property. Under a simple assumption, we show that the geometric Manin's conjectures hold for some degrees lying in the dual of the effective cone of X (in particular, for those degrees the moduli space of morphisms has the expected dimension). The result applies to a class of generalized del Pezzo surfaces which has been intensively studied in the context of the arithmetic Manin's conjecture.

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“…Now remark that, disregarding convergence issues, the expression (2.6.20) makes sense for any variety X satisfying hypotheses 1.1, not only the toric ones. Under suitable extra hypotheses on X, Peyre showed that the Euler product in (2.6.20) is indeed convergent and predicted that (2.6.20) should coincide with the constant c appearing in question 1.8 (in fact Peyre's construction applies to a far more general context, including the case of nonconstant families; (2.6.20) is interpreted as the volume of an adelic space associated to X, with respect to a certain Tamagawa measure; see [Pey03a] for more details). Thus we will have checked that Peyre's prediction holds when X is toric.…”

mentioning

confidence: 99%

Asymptotic behaviour of rational curves

Bourqui

2011

Preprint

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We investigate the asympotic behaviour of the moduli space of morphisms from the rational curve to a given variety when the degree becomes large. One of the crucial tools is the homogeneous coordinate ring of the variey. First we explain in details what happens in the toric case. Then we examine the general case. This is a revised and slightly expanded version of notes for a course delivered during the summer school on rational curves held in June 2010 at Institut Fourier, Grenoble.

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Points de hauteur bornée sur les variétés de drapeaux en caractéristique finie

Cited by 6 publications

References 18 publications

Motivic Euler products in motivic statistics

Motivic Euler products in motivic statistics

Exemples de comptages de courbes sur les surfaces

Asymptotic behaviour of rational curves

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