HEIGHT ZETA FUNCTIONS OF TWISTED PRODUCTS

MATTHIAS STRAUCH AND YURI TSCHINKEL

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

Contents

1. Introduction 1 2. Geometry 4 3. Metrizations of line bundles 5 4. Height zeta functions 9 References 10

1. Introduction

Let X be a nonsingular algebraic variety over a number field F. We shall denote by X(F) the set of F -rational points of X. Let L = (L, (∥·∥_{v})) be a metrized line bundle on X, i.e., a line bundle L together with a family of v-adic metrics, where v runs over the set Val(F) of normalized valuations of F. A metrized line bundle L defines a height function

H_{L} : X(F) → R_{>0}.

For appropriate subvarieties U ⊂ X and line bundles L we have

N(U, L, B) := #{x ∈ U(F) | H_{L}(x) ≤ B} < ∞

for all B ∈ R_{>0} (e.g., this holds for any U ⊂ X if L is ample). We are interested in the asymptotic behavior of this counting function as B → ∞. It is expected that the behavior of such asymptotics can be described in geometrical terms [7, 1].

Definition 1.1. Let us denote by Pic(X) the Picard group of X, Pic(X)_{R} = Pic(X) ⊗ R. The cone of effective divisors of X is the closed cone Λ_{eff}(X) ⊂ Pic(X)_{R} generated by the classes of effective divisors.

Let [L] ∈ Pic(X) be the class of L. Denote by K_{X} the canonical line bundle on X.

Definition 1.2. Let L be a line bundle on X such that [L] is contained in the interior of the cone Λ_{eff}(X). Define

a(L) := inf{a ∈ R | a[L] + [K_{X}] ∈ Λ_{eff}(X)}.

Assume that Λ_{eff}(X) is a finitely generated polyhedral cone. Define b(L) as the codimension of the minimal face of Λ_{eff}(X) which contains a(L)[L] + [K_{X}].

In this paper we will consider the following class of varieties. Let X be a smooth projective algebraic variety over F with an action of a split algebraic torus T and let G be a semisimple split linear algebraic group over F. Denote by P an F -rational parabolic subgroup of G. Assume that we have a morphism η : P → T, it will define an action of P on X × G by (x, g)p → (xη(p), p^{−1}g). There exists a smooth projective quotient of X × G by P which we denote by Y = Y_{η} := X ×^{P} G. This quotient is a locally trivial bundle over W = P \G with fiber X.

For example, let us choose a Borel subgroup B ⊂ G containing the maximal unipotent subgroup U ⊂ B, set T = B/U and let η be the canonical projection. Applying this construction to some smooth equivariant compactification of T we will obtain an equivariant compactification of the homogeneous space G/U. Such compactifications are called horospherical varieties, see [8, 6] for a classification and a detailed study of their geometric properties over algebraically closed fields.

Definition 1.3. Let U ⊂ X be a locally closed subset. We define the height zeta function by the series

Z(U, L, s) := ∑_{x∈U(F)} H_{L}(x)^{−s}.

For appropriate U and L the series converges for Re(s) ≫ 0.

A tauberian theorem relates its analytic properties with the asymptotic properties of the counting function N(U, L, B). Analytic properties of height zeta functions are understood for flag varieties [7] and for toric varieties [2, 3, 4].

Theorem 1.4. Let X be a smooth projective toric variety over a number field F, Y = X ×^{P} G a twisted product as described above. Denote by Y^{∘} := T ×^{P} G ⊂ Y the Zariski open subset, obtained as a twisted product of T with P \G. Let L be a metrized line bundle on Y (with the metrization introduced below), such that its class [L] is contained in the interior of the cone of effective divisors. Then there exists a δ(L) > 0 such that the height zeta function has the following representation:

Z(Y^{∘}, L, s) = \frac{Θ(L)}{(s − a(L))^{b(L)}} + \frac{h(s)}{(s − a(L))^{b(L)−1}} ,

where a(L) ∈ Q_{>0}, h(s) is a function which is holomorphic for Re(s) ≥ a(L) − δ(L) and Θ(L) ≠ 0.

Theorem 1.5. For rational points on Y^{∘} ⊂ Y we have the following asymptotic formula

N(Y^{∘}, L, B) = \frac{Θ(L)}{a(L)(b(L) − 1)!}B^{a(L)}(log B)^{b(L)−1}(1 + o(1))

for B → ∞, where Θ(L) ≠ 0. Moreover,

Θ(−K_{Y}) = α(Y)τ(K_{Y}),

where α(Y) ∈ Q is a constant related to the geometry of the cone of effective divisors and τ(K_{Y}) is the Tamagawa number defined by Peyre [9].

This proves the conjecture of Batyrev and Manin [1] about the distribution of rational points of bounded height for this class of varieties and the conjecture of Peyre about the asymptotic constant [9] (with respect to a specific metrization). As special cases it includes the results for flag varieties and for toric varieties obtained in [7, 1, 4]. Actually, it is not necessary to assume that T and G are split over F. In this paper we present the relevant constructions and give an outline of the argument. The proofs will appear in [12].

Similar results hold for twisted products of flag varieties with flag varieties. However, the proofs require certain restrictions on the twist η [11]. An example which doesn’t fit the picture given by the conjecture is contained in [5].

Acknowledgements. We are very grateful to V. Batyrev and J. Franke for helpful discussions and collaboration on related questions. The first author was supported by the Graduiertenkolleg of the Mathematics Institute of the University of Bonn. Part of this work was done while the second author was visiting ENS in Paris and MPI in Bonn. He would like to thank both institutions for ideal working conditions.

2. Geometry

Let G be a split semisimple algebraic group defined over a number field F and P_{0} a minimal parabolic subgroup of G. Let P be a standard parabolic subgroup, X^{∗}(P) the group of F -rational characters of P. We denote by W = P \G be the generalized flag variety. There is a monomorphism with finite cokernel X^{∗}(P) → Pic(W). We introduce coordinates in X^{∗}(P)_{C} as follows. Let Δ_{0} be the set of simple roots determined by the maximal torus T_{0} in the center of a Levi component of P_{0}. Let ǎ_{0}^{+} ⊂ X^{∗}(P_{0}) be the closure of the positive Weyl chamber determined by P_{0}. There is an embedding of X^{∗}(P) ↪ X^{∗}(P_{0}). Let ǎ_{P}^{+} be the closed cone ǎ_{0}^{+} ∩ X^{∗}(P)_{R} which is spanned by a canonical basis ω_{1}, ..., ω_{r}. We identify s′ = (s_{1}, ..., s_{r}) ∈ C^{r} with s_{1}ω_{1} + ⋯ + s_{r}ω_{r} ∈ X^{∗}(P)_{C}.

Let X be a variety with an action τ : X ×T → X of a split algebraic torus T and η a morphism P → T. Denote by π_{X} : X × T → X the canonical projection. The twisted product Y := X ×^{P} G has the following properties: We have have a canonical morphism π : Y → W under which Y is a locally trivial fiber bundle over W with fiber X. Therefore, if X is non-singular, so is Y. Let L be a line bundle on X and suppose that ψ : τ^{∗}L → π_{X}^{∗}L is a T -linearization of L. Using this T -linearization one gets a canonical line bundle L^{Y} on Y. Note that this line bundle and even its isomorphism class in Pic(Y) depend on the choice of the T -linearization. In fact, we get a homomorphism

Pic^{T} (X) → Pic(Y)

from the group of isomorphism classes of T -linearized line bundles on X to the Picard group of Y. One can show that if X is non-singular there is an exact sequence

0 → Pic(W) ⟶^{π^{∗}} Pic(Y) → Pic(X) → 0.

Now let us consider the case when X is a toric variety. Let T be a d-dimensional split algebraic torus over F. Let M be the lattice of F -rational characters of T and N the dual lattice. Smooth equivariant compactifications of algebraic tori are described by combinatorial objects - complete regular d-dimensional fans Σ ⊂ N ⊗ R (see [2]). Let Σ(1) := {e_{1}, ..., e_{n}} be the set of 1-dimensional generators of Σ. Denote by P L(Σ) the lattice of piecewise linear integral functions on N. Clearly, we have an embedding M ↪ P L(Σ). An integral piecewise linear function φ ∈ P L(Σ) is determined by the vector s = (s_{1}, ..., s_{n}), where s_{i} ∈ Z is the value of φ on the generator of some 1-dimensional cone in Σ(1), (i = 1, ..., n). It will be convenient for us to work with these coordinates which we can assume to be complex. We write φ_{s} for the complex valued piecewise linear function thus obtained.

Let η : P → T be a morphism and Y = Y_{η} := X ×_{P} G the twisted product. We have a homomorphism η̌ : M → X^{∗}(P).

Proposition 2.1. 1. There is an exact sequence

0 → M → P L(Σ) ⊕ X^{∗}(P) → Pic(Y) → 0.

2. The cone of effective divisors Λ_{eff}(Y) is the image of the simplicial cone P L(Σ)^{+} ⊕ ǎ_{P}^{+}, where P L(Σ)^{+} ⊂ P L(Σ)_{R} is the cone of piecewise linear functions φ_{s} such that φ_{s}(e_{j}) ≥ 0 for all j = 1, ..., n.

3. The anticanonical class is given by the class of (φ_{Σ}, 2ρ_{P} ,), where φ_{Σ}(e_{j}) = 1 for all j = 1, ..., n and 2ρ_{P} is the sum of roots occuring in the unipotent radical of P (counted with multiplicities).

One can find a description of these geometric properties for spherical varieties over algebraically closed fields in [6].

3. Metrizations of line bundles

Let X be a smooth projective variety over F and let L be a line bundle on X.

Definition 3.1. Let v be a place of F. A v-adic metric on L is a family (∥ · ∥_{x})_{x∈X(F_{v})} of v-adic norms

∥ · ∥_{x} : x^{∗}L → R

such that for any open subset U ⊂ X and any s ∈ L(U) the map

U(F_{v}) → R, x ↦ ∥x^{∗}s∥_{x},

is continuous, where U(F_{v}) is given the v-adic topology.

We will write (∥ · ∥_{x}) instead of (∥ · ∥_{x})_{x∈X(F_{v})} if the place v is clear from the context. In the sequel we will use the symbol ∥ · ∥_{v} to denote a v-adic metric on L.

The compactness of X(F_{v}) implies that any two v-adic metrics ∥ · ∥_{v} and ∥ · ∥′_{v} on L are equivalent in the sense that there exist constants c_{2} ≥ c_{1} > 0 such that for all x ∈ X(F_{v}) and all s ∈ x^{∗}L we have [10]

c_{1}∥s∥_{x} ≤ ∥s∥′_{x} ≤ c_{2}∥s∥_{x}.

Given a v-adic metric ∥ · ∥_{v} on L we get a v-adic metric ∥ · ∥_{v}^{∨} on the dual line bundle L^{∨} by setting

∥t∥_{x}^{∨} := \frac{|t(s)|_{v}}{∥s∥_{x}}

for arbitrary s ∈ x^{∗}L − {0}. If we are given a v-adic metric ∥ · ∥′_{v} on another line bundle L′ then we get a canonical v-adic metric ∥ · ∥_{v}∥ · ∥′_{v} on L ⊗ L′.

Suppose that L is generated by global sections, and let (s_{j})_{j=0,...,q} be a family of F -rational global sections which generate the space Γ(X, L). Let v ∈ Val(F), x ∈ X(F_{v}) and s ∈ x^{∗}L − {0}. We put

∥s∥_{x} := ( max_{0≤j≤q} |\frac{s_{j}}{s} (x)|_{v})^{−1}

and ∥0∥_{x} = 0. The family (∥ · ∥_{x})_{x∈X(F_{v})} is then a v-adic metric on L, called the v-adic metric associated with the family (s_{j})_{j=0,...,q} of global sections of L.

Definition 3.2. An adelic metric on L is a family (∥ · ∥_{v})_{v∈Val(F)} of v-adic metrics ∥ · ∥_{v} on L such that the following holds: there are line bundles L_{1}, L_{2} on X which are generated by global sections, families (s_{j}^{(1)})_{j=0,...,q_{1}} (resp. (s_{j}^{(2)})_{j=0,...,q_{2}}) of global sections of L_{1} (resp. L_{2} ) generating Γ(X, L_{1}) (resp. Γ(X, L_{2}) ) and an isomorphism L → L_{1}⊗L_{2}^{∨} such that for almost all v the norm ∥·∥_{v} is the pullback of ∥·∥_{v}^{(1)}·(∥·∥_{v}^{(2)})^{∨} under this isomorphism, where ∥·∥_{v}^{(1)} (resp. ∥·∥_{v}^{(2)}) is the v-adic metric associated to (s_{j}^{(1)})_{j=0,...,q_{1}} (resp. (s_{j}^{(2)})_{j=0,...,q_{2}}).

In the sequel we will write (∥ · ∥_{v}) for an adelic metric (∥ · ∥_{v})_{v∈Val(F)}. A pair L = (L, (∥ · ∥_{v})) consisting of a line bundle L on X and an adelic metric (∥ · ∥_{v}) on L will be called a metrized line bundle or a metrization of L.

Definition 3.3. Let L be a metrized line bundle on X. It defines the height function H_{L} : X(F) → R_{>0} by

H_{L}(x) = ∏_{v∈Val(F)} ∥x_{v}^{∗}s∥_{x_{v}}^{−1},

where x_{v} is the image of x under the embedding X(F) → X(F_{v}) and s is any F -rational local section of L at x with s(x) ≠ 0.

It is easy to show that for a given x ∈ X(F) almost all factors in the product equal 1. By the product formula, the height is independent of the choice of an F -rational section s.

Let Y be a twisted product of a flag variety W with a smooth projective variety X equipped with an action of a split algebraic torus T. Let L a line bundle on X and ψ : τ^{∗}L → π_{X}^{∗}L its T -linearization. Suppose that we are given a v-adic metric ∥ · ∥_{v} on L. Let t ∈ T (F_{v}). For any x ∈ X(F_{v}) we have an induced isomorphism

ψ_{x,t} : x^{∗}L = (xt, t^{−1})^{∗}(τ^{∗}L) ⟶^{∼} (xt, t^{−1})^{∗}(π_{X}^{∗}L) = (xt)^{∗}L.

This allows to define ∥ · ∥_{x,t} : x^{∗}L → R by ∥s∥_{x,t} := ∥ψ_{x,t}(s)∥_{xt}. It is easy to check that ∥ · ∥_{v,t} := (∥ · ∥_{x,t})_{x∈X(F_{v})} is a new v-adic metric on L.

Denote by K_{T,v} ⊂ T (F_{v}) the maximal compact subgroup of T (F_{v}). We shall call the v-adic metric ∥ · ∥_{v} on L admissible if for all t ∈ K_{T,v} we have ∥ · ∥_{v,t} = ∥ · ∥_{v}. This does not depend on ψ. An adelic metric (∥·∥_{v})_{v∈Val(F)} is called admissible if for all v ∈ Val(F) the v-adic metric ∥ · ∥_{v} is admissible. It is clear that if t = (t_{v})_{v} ∈ T (A_{F}) and (∥ · ∥_{v}) is an admissible adelic metric on L then (∥ · ∥_{v,t_{v}})_{v∈Val(F)} is an adelic metric on L. We will denote this metrization of L by L(t).

Let K_{G} = ∏_{v∈Val(F)} K_{G,v} be a maximal compact subgroup of G(A_{F}) such that we have the Iwasawa decomposition G(A_{F}) = P_{0}(A_{F})K_{G}. Let (x, g) ∈ X(F_{v})×G(F_{v}) be an element in the preimage of y ∈ Y (F_{v}) under the projection π_{Y} : X × G → Y. Then, by the construction of L^{Y} via ψ : τ^{∗}L → π_{X}^{∗}L, there is a canonical isomorphism

ζ_{x,g} : y^{∗}L^{Y} → x^{∗}L.

Let g = pk, p ∈ P (F_{v}), k ∈ K_{G,v}. Then we define a v-adic norm

∥ · ∥_{v} : y^{∗}L^{Y} → R

by

∥s∥_{y} := ∥ζ_{x,g}(s)∥_{x,η(p)}.

This is well defined and gives a v-adic metric ∥ · ∥_{v}^{Y} on L^{Y}. If (∥ · ∥_{v}) is an admissible adelic metric on L, then (∥ · ∥_{v}^{Y}) is an adelic metric on L^{Y}. We put L^{Y} := (L^{Y} , (∥ · ∥_{v}^{Y})).

Let (x, g) ∈ π_{Y}^{−1}(y) for y ∈ Y (F), g = pk, p ∈ P (A_{F}), k ∈ K_{G}. By the construction above, we have

H_{L^{Y}} (y) = H_{L(η(p))}(x).

Therefore, we get a formal expression for the height zeta function of the twisted product Y = X ×^{P} G corresponding to a metrized line bundle L^{Y} ⊗ π^{∗}L_{W}, where L is a metrized line bundle on X, consisting of a T -linearized line bundle L and an admissible adelic metric, and L_{W} is a metrized line bundle on W = P \G. Using the relation

H_{L^{Y} ⊗π^{∗}L_{W}} (y) = H_{L^{Y}} (x)H_{L_{W}} (π(y))

we have formally

Z_{Y} (L^{Y} ⊗ L_{W} , s) = ∑_{γ∈P (F)\G(F)} H_{L_{W}} (γ)^{−s} ∑_{y∈π^{−1}(γ)} H_{L^{Y}} (y)^{−s}

= ∑_{γ∈P (F)\G(F)} H_{L_{W}} (γ)^{−s} ∑_{x∈X(F)} H_{L(η(p_{γ}))}(x)^{−s}

= ∑_{γ∈P (F)\G(F)} H_{L_{W}} (γ)^{−s}Z_{X}(L(η(p_{γ})), s),

where γ = p_{γ}k_{γ}, p_{γ} ∈ P (A_{F}), k_{γ} ∈ K_{G}.

Our next goal is to describe explicit metrizations of line bundles on twisted products of toric varieties with generalized flag varieties.

We have the logarithmic map

T (F_{v})/K_{T,v} → N

(resp. N ⊗ R for archimedean v) and we denote by \overline{t}_{v} the image of t_{v} ∈ N. For every g_{v} =∈ G(F_{v}) we choose a representative in P (F_{v}) which we will denote by p(g_{v}). For every g_{v} ∈ G(F_{v}) we define a homomorphism H_{P,v}(g_{v}) : X^{∗}(P)_{C} → C by

H_{P,v}(g_{v}, λ) =< λ, H_{P,v}(g_{v}) >= log(|λ(p(g_{v}))|_{v})

for λ ∈ X^{∗}(P) and extend this by linearity.

Definition 3.4. We define a pairing

H_{Σ,v} : T (F_{v}) × P L(Σ)_{C} → C
H_{Σ,v}(t_{v}, s) = e^{φ_{s}(\overline{t}_{v}) log q_{v}}

where q_{v} equals the number of elements in the residue field of F_{v} for non-archimedean v and log q_{v} = 1 for archimedean v. Denote by H_{Σ} := ∏_{v∈Val(F)} H_{Σ,v}. We define

H_{W,v} : G(F_{v}) × X^{∗}(P)_{C} → C
H_{W,v}(g_{v}, s′) = e^{<s′,H_{P,v}(p(g_{v}))>}

and let H_{W} := ∏_{v∈Val(F)} H_{W,v} and H_{P} := ∑_{v∈Val(F)} H_{P,v}. We define a pairing

H_{Σ,η,v} : (T (F_{v}) × G(F_{v})) × (P L(Σ)_{C} × X^{∗}(P)_{C}) → C
H_{Σ,η,v}((t_{v}, g_{v}), (s, s′)) = H_{Σ,v}(t_{v}η(p(g_{v})), s) · H_{W,v}(g_{v}, s′)

We have the pairing

H_{Σ,η} = ∏_{v∈Val(F)} H_{Σ,η,v} : (T (A_{F})×G(A_{F}))×(P L(Σ)_{C}×X^{∗}(P)_{C}) → C

Proposition 3.5. The pairing H_{Σ,η} descends to a pairing

H_{Σ,η} : Y^{∘}(F) × Pic(Y)_{C} → C.

Let L be a line bundle and [L] its class in Pic(Y). Denote by Y^{∘} = T ×^{P} G ⊂ Y. Then the restriction of the above pairing to Y^{∘}(F) × [L] gives a height function with respect to some metrization of L.

The proof of this proposition proceeds by comparing the metrization of line bundles on Y given by this pairing with the general construction of metrizations of line bundles on twisted products described above.

4. Height zeta functions

For (s, s′) ∈ P L(Σ)_{C} ⊕ X^{∗}(P)_{C} we define a zeta function by the series

Z_{Σ,η}(s, s′) = ∑_{y∈Y^{∘}(F)} H_{Σ,η}(y, (−s, −s′)).

This zeta function is the extension of the one-parameter height zeta function considered above to the whole complexified Picard group.

Proposition 4.1. The series Z_{Σ,η}(s, s′) converges absolutely and uniformly to a holomorphic function for Re(s) ∈ R_{>1}^{n} and Re(s′) ∈ R_{>1}^{r}. Moreover, it descends to a function on Pic(Y)_{C} which is holomorphic in the interior of the cone Λ_{eff}(Y) + [−K_{Y}].

Recall that Y^{∘} → W = P \G is a fiber bundle with fibers isomorphic to T. Using the absolute convergence we can rearrange the terms summing first over the rational points in the base of this fibration and then over the rational points in each fiber. We obtain

Z_{Σ,η}(s, s′) = ∑_{γ∈P (F)\G(F)} H_{W} (γ, −s′) ∑_{t∈T (F)} H_{Σ}(tη(p_{γ}), −s).

Using the Poisson formula on the group T (A_{F}) and the K_{T} -invariance of the pairing H_{Σ} we have [2]:

Proposition 4.2.

Z_{Σ,η}(s, s′) = ∑_{γ∈P (F)\G(F)} e^{<−s′,H_{P} (γ)>} ∫_{A^{∗}} Ĥ_{Σ}(χ, −s)χ^{−1}(η(p_{γ}))dχ

where A^{∗} is the group of topological characters of T (A_{F}) which are trivial on the closed subgroup T (F)K_{T}, dχ is an appropriate Haar measure and Ĥ_{Σ}(χ, −s) is the Fourier transform of the function H_{Σ}(x, −s) on T (A_{F}).

The embedding M ↪ P L(Σ) induces a map of tori T_{P L} → T. We have T_{P L} = G_{m}^{n}. A character χ ∈ A^{∗} defines characters (χ_{1}, ..., χ_{n}) where χ_{j} are characters G_{m}(A_{F}) invariant under the closed subgroup G_{m}(F)K, where K ⊂ G_{m}(A_{F}) is the maximal compact subgroup. Using the explicit computation of Ĥ_{Σ}(χ, −s) [2], and interchanging summation and integration we have

Z_{Σ,η}(s, s′) =

= ∫_{A^{∗}} ∏_{j=1}^{n} L(s_{j}, χ_{j})ζ_{Σ,∞}(s, χ)( ∑_{γ∈P (F)\G(F)} e^{<−s′,H_{P} (γ)>}χ^{−1}(η(p_{γ})))dχ =

= ∫_{A^{∗}} ∏_{j=1}^{n} L(s_{j}, χ_{j})E_{P}^{G}(s′, χ ∘ η)ζ_{Σ,∞}(s, χ)dχ.

Here we denoted by

E_{P}^{G}(s′, χ ∘ η) := ∑_{γ∈P (F)\G(F)} e^{<−s′,H_{P} (γ)>}χ^{−1}(η(p_{γ}))

the Eisenstein series and by L(s, χ) the Hecke L-function.

The analytic continuation of this integral and the description of its poles uses uniform estimates of Hecke L-functions, Eisenstein series and ζ_{Σ,∞}(s, χ). There is a canonical map y : A^{∗} → ∑_{v|∞} M_{R} with finite kernel. Let ∥ ∥_{∞} be a norm on ∑_{v|∞} M_{R} and ∥ ∥ a norm on X^{∗}(P)_{R}.

Proposition 4.3. Let E_{P}^{G}(s′, χ) be the Eisenstein series with unramified character χ ∈ (P (A_{F})/P (F))^{∗}. For all ε > 0 there exists a δ > 0 such that for all s′ with Re(s′) contained in any compact K in the domain 0 < |Re (s_{j}) − 1| < δ for j = 1, ..., r and all χ ∈ (P (A_{F})/P (F))^{∗} there exists a constant c(K, ε) depending only on K and ε such that

|E_{P}^{G}(s′, χ)| ≤ c(K, ε)(1 + ∥Im(s′)∥ + ∥y(χ)∥_{∞})^{ε}.

The proof of this proposition uses the Phragmen-Lindelöf principle and the theory of the constant term of Eisenstein series. The proof of our theorem 1.4 is similar to the proof of the main technical theorem in [4]. The proof of the theorem 1.5 combines the proofs of corresponding statements for toric varieties [3] and for generalized flag varieties [9].

References

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[10] J. Silverman, The theory of height functions, in Arithmetic geometry, ed. G. Cornell, J. Silverman, Springer, New York (1986).

[11] M. Strauch, Rational points of bounded height on twisted products of generalized flag varieties, in preparation.

[12] M. Strauch, Yu. Tschinkel, Height zeta functions of toric bundles over flag varieties, in preparation.

Mathematisches Institut, Universität Bonn, Beringstr. 1, 53117 Bonn, Germany

E-mail address: mats@rhein.iam.uni-bonn.de

Dept. of Mathematics, U.I.C., Chicago, (IL) 60607-7045, U.S.A.

E-mail address: yuri@math.uic.edu