Height Zeta Functions of Equivariant Compactifications of Unipotent Groups

Shalika, Joseph; Tschinkel, Yuri

doi:10.1002/cpa.21572

Cited by 15 publications

(12 citation statements)

References 18 publications

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“…A similar proof works for smooth equivariant compactifications of other algebraic groups and Manin's conjecture for such varieties has been established in many cases, see e.g. [CLT02], [STBT07] and [ST16].…”

Section: A Conjectural Description Of Exceptional Sets In Manin's Con...supporting

confidence: 53%

Geometric consistency of Manin's Conjecture

Lehmann¹,

Sengupta²,

Tanimoto³

2018

Preprint

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We propose an explicit description of the exceptional set in Manin's Conjecture. Our proposal includes the rational point contributions from any subvariety or cover with larger geometric invariants. We prove that this set is contained in a thin subset of rational points, verifying there is no counterexample to Manin's Conjecture which arises from incompatibility of geometric invariants.the setis contained in a thin subset of X(F ).

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Section: A Conjectural Description Of Exceptional Sets In Manin's Con...supporting

confidence: 53%

Geometric consistency of Manin's Conjecture

Lehmann¹,

Sengupta²,

Tanimoto³

2018

Preprint

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“…Manin's conjecture for rational points, extensively studied now for more than three decades, predicts an asymptotic formula for the counting function of rational points of bounded height on rationally connected algebraic varieties over number fields. The class of equivariant compactifications of homogeneous spaces has proved to be a particularly fertile testing ground for the conjecture [6, 8, 23, 36, 38–40, 61, 62, 66]. The related problem of counting integral points on homogeneous spaces has received much attention as well, both classically (see, for example, [33, 35]), and recently, as attested by [10, 25–27, 64, 65].…”

Section: Introductionmentioning

confidence: 99%

Campana points of bounded height on vector group compactifications

Pieropan

Smeets

Tanimoto

et al. 2020

Proc. London Math. Soc.

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We initiate a systematic quantitative study of subsets of rational points that are integral with respect to a weighted boundary divisor on Fano orbifolds. We call the points in these sets Campana points. Earlier work of Campana and subsequently Abramovich shows that there are several reasonable competing definitions for Campana points. We use a version that delineates well different types of behavior of points as the weights on the boundary divisor vary. This prompts a Manin-type conjecture on Fano orbifolds for sets of Campana points that satisfy a klt (Kawamata log terminal) condition. By importing work of Chambert-Loir and Tschinkel to our setup, we prove a log version of Manin's conjecture for klt Campana points on equivariant compactifications of vector groups. Contents

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“…certain equivariant compactifications of algebraic groups. An equivariant compactification of an algebraic group G is a (projective and smooth) variety X which has an open subset isomorphic to G and which is endowed with an action G × X → X of G extending the group law G × G → G. Apart from toric varieties which are exactly the equivariant compactifications of algebraic tori, Manin's problem has been solved for equivariant compactifications of vector groups ( [CLT02], [CLT12]), as well as for compactifications of certain non-commutative algebraic groups ( [ShTBT03], [ShTBT07], [ShT04], [ShT16], [TBT], [TT]).…”

Section: Manin's Problem Via Harmonic Analysismentioning

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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Height Zeta Functions of Equivariant Compactifications of Unipotent Groups

Abstract: Abstract. We prove Manin's conjecture for bi-equivariant compactifications of unipotent groups.

Cited by 15 publications

References 18 publications

Geometric consistency of Manin's Conjecture

Geometric consistency of Manin's Conjecture

Campana points of bounded height on vector group compactifications

Motivic Euler products in motivic statistics

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