# Fonctions ZÊta Des Hauteurs Des Espaces Fibrés

@inproceedings{ChambertLoir2000FonctionsZD, title={Fonctions Z{\^E}ta Des Hauteurs Des Espaces Fibr{\'e}s}, author={Antoine Chambert-Loir and Yuri Tschinkel}, year={2000} }

In this paper we study the compatibility of Manin’s conjectures concerning asymptotics of rational points on algebraic varieties with certain natural geometric constructions. More precisely, we consider locally trivial fibrations constructed from torsors under linear algebraic groups. The main problem is to understand the behaviour of the height function as one passes from fiber to fiber - a difficult problem, even though all fibers are isomorphic. We will be mostly interested in fibrations… Expand

#### 25 Citations

Rational Points on Homogeneous Varieties and Equidistribution of Adelic Periods

- Mathematics
- 2008

Let U := L\G be a homogeneous variety defined over a number field K, where G is a connected semisimple K-group and L is a connected maximal semisimple K-subgroup of G with finite index in its… Expand

Rational points on compactifications of semi-simple groups

- Mathematics
- 2007

We prove Manin's conjecture concerning the distribution of rational points of bounded height, and its refinement by Peyre, for wonderful compactifications of semi-simple algebraic groups over number… Expand

Distribution of orders in number fields

- Mathematics
- 2014

AbstractIn this paper, we study the distribution of orders of bounded discriminants in number fields. We use the zeta functions introduced by Grunewald, Segal, and Smith. In order to carry out our… Expand

Inhomogeneous cubic congruences and rational points on del Pezzo surfaces

- Mathematics
- 2010

Abstract For given non-zero integers a, b, q we investigate the density of solutions (x, y) ∈ ℤ2 to the binary cubic congruence ax2 + by3 ≡ 0 mod q, and use it to establish the Manin conjecture for a… Expand

Height zeta functions of equivariant compactifications of semi-direct products of algebraic groups

- Mathematics
- 2011

We apply the theory of height zeta functions to study the asymptotic distribution of rational points of bounded height on projective equivariant compactifications of semi-direct products.

Proportion of ordinarity in some families of curves over finite fields

- Mathematics
- 2019

A curve over a field of characteristic $p$ is called ordinary if the $p$-torsion of its Jacobian as large as possible, that is, an $\mathbb{F}_p$ vector space of dimension equal to its genus. In this… Expand

Distribution of rational points of bounded height on equivariant compactifications of $\mathrm{PGL}_2$ I

- Mathematics
- 2015

We study the distribution of rational points of bounded height on a one-sided equivariant compactification of $\mathrm{PGL}_2$ using automorphic representation theory of $\mathrm{PGL}_2$.

The Ergodic Theory of Lattice Subgroups

- Mathematics
- 2009

We prove mean and pointwise ergodic theorems for general families of averages on a semisimple algebraic (or S-algebraic) group G, together with an explicit rate of convergence when the action has a… Expand

The Hasse norm principle for abelian extensions

- Mathematics
- 2015

Abstract:We study the distribution of abelian extensions of bounded discriminant of a number field $k$ which fail the Hasse norm principle. For example, we classify those finite abelian groups $G$… Expand

Average bounds for the $$\ell $$ℓ-torsion in class groups of cyclic extensions

- Mathematics
- 2017

For all positive integers $$\ell $$ℓ, we prove non-trivial bounds for the $$\ell $$ℓ-torsion in the class group of K, which hold for almost all number fields K in certain families of cyclic… Expand

#### References

SHOWING 1-6 OF 6 REFERENCES

Rational points of bounded height on Fano varieties

- Mathematics
- 1989

a prime pe7Z. Let V be an algebraic variety defined over F and lI~ a metrized line bundle on V, i.e., a system (L, ]'],) consisting of a line bundle L and a family of Banach v-adic metrics on L | F,,… Expand

Height zeta functions of toric varieties

- Mathematics
- 1996

We investigate analytic properties of height zeta functions of toric varieties. Using the height zeta functions, we prove an asymptotic formula for the number of rational points of bounded height… Expand

Rational Points of Bounded Height on Compactifications of Anisotropic Tori

- Mathematics
- 1994

We investigate the analytic properties of the zeta-function associated with heights on equivariant compactifications of anisotropic tori over number fields. This allows to verify conjectures about… Expand

Height zeta functions of toric bundles over flag varieties

- Mathematics
- 1997

We investigate analytic properties of height zeta functions of toric bundles over flag varieties.