Tamagawa numbers of polarized algebraic varieties

Dedicated to Professor Yu.I. Manin on his 60th birthday

Victor V. Batyrev and Yuri Tschinkel

Abstract

Let L = (L, ∥ · ∥_{v}) be an ample metrized invertible sheaf on a smooth quasi-projective algebraic variety V over a number field F. Denote by N(V, L, B) the number of rational points in V having Lheight ≤ B. In this paper we consider the problem of a geometric and arithmetic interpretation of the asymptotic for N(V, L, B) as B → ∞ in connection with recent conjectures of Fujita concerning the Minimal Model Program for polarized algebraic varieties.

We introduce the notions of L-primitive varieties and L-primitive fibrations. For L-primitive varieties V over F we propose a method to define an adelic Tamagawa number τ_{L}(V) which is a generalization of the Tamagawa number τ(V) introduced by Peyre for smooth Fano varieties. Our method allows us to construct Tamagawa numbers for Q-Fano varieties with at worst canonical singularities.

In a series of examples of smooth polarized varieties and singular Fano varieties we show that our Tamagawa numbers express the dependence of the asymptotic of N(V, L, B) on the choice of v-adic metrics on L.

Contents

1 Introduction 3

2 Geometry of L-polarized varieties 8 2.1 L-closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Kodaira energy and α_{L}(V) . . . . . . . . . . . . . . . . . . . . 10 2.3 L-primitive varieties . . . . . . . . . . . . . . . . . . . . . . . 13 2.4 L-primitive fibrations and descent of metrics . . . . . . . . . . 17

3 Heights and asymptotic formulas 20 3.1 Basic terminology and notations . . . . . . . . . . . . . . . . . 20 3.2 Weakly and strongly L-saturated varieties . . . . . . . . . . . 21 3.3 Adelic L-measure and τ_{L}(V) . . . . . . . . . . . . . . . . . . . 26 3.4 Main strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.5 L-primitive fibrations and τ_{L}(V) . . . . . . . . . . . . . . . . . 33

4 Height zeta-functions 35 4.1 Tauberian theorem . . . . . . . . . . . . . . . . . . . . . . . . 35 4.2 Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.3 Symmetric product of a curve . . . . . . . . . . . . . . . . . . 36 4.4 Homogeneous spaces G/P . . . . . . . . . . . . . . . . . . . . 39 4.5 Toric varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5 Singular Fano varieties 45 5.1 Weighted projective spaces . . . . . . . . . . . . . . . . . . . . 45 5.2 Vaughan-Wooley cubic . . . . . . . . . . . . . . . . . . . . . . 47 5.3 Cubic xyz = u^{3} . . . . . . . . . . . . . . . . . . . . . . . . . . 49

1 Introduction

Let F be a number field (a finite extension of Q), Val(F) the set of all valuations of F, F_{v} the v-adic completion of F with respect to v ∈ Val(F), and | · |_{v} : F_{v} → R the v-adic norm on F_{v} normalized by the conditions |x|_{v} = |N_{F_{v}/Q_{p}}(x)|_{p} for p-adic valuations v ∈ Val(F).

Consider a projective space P^{m} with standard homogeneous coordinates (z_{0}, ..., z_{m}) and a locally closed quasi-projective subvariety V ⊂ P^{m} defined over F (we want to stress that V is not assumed to be projective). Let V (F) be the set of points in V with coordinates in F. A standard height function H : P^{m}(F) → R_{>0} is defined as follows

H(x) := ∏_{v∈Val(F)} max_{j=0,...,m}{|z_{j}(x)|_{v}}.

A basic fact about the standard height function H claims that the set

{x ∈ P^{m}(F) : H(x) ≤ B}

is finite for any real number B [29]. We set

N(V, B) = #{x ∈ V (F) : H(x) ≤ B}.

It is an experimental fact that whenever one succeeds in proving an asymptotic formula for the function N(V, B) as B → ∞, one obtains the asymptotic

N(V, B) = c(V)B^{a(V)}(log B)^{b(V)−1}(1 + o(1))  (1)

with some constants a(V) ∈ Q, b(V) ∈ \frac{1}{2}Z, and c(V) ∈ R_{>0}. We want to use this observation as our starting point. It seems natural to ask the following:

Question A. For which quasi-projective subvarieties V ⊂ P^{m} defined over F do there exist constants a(V) ∈ Q, b(V) ∈ \frac{1}{2}Z and c(V) ∈ R_{>0} such that the asymptotic formula (1) holds?

Question B. Does there exist a quasi-projective variety V over F with an asymptotic which is different from (1)?

In this paper we will be interested not in Questions A and B themselves but in a related to them another natural question:

Question C. Assume that V is an irreducible quasi-projective variety over a number field F such that the asymptotic formula (1) holds. How to compute the constants a(V), b(V) and c(V) in this formula via some arithmetical properties of V over F and geometrical properties of V over C?

To simplify our terminology, it will be convenient for us to postulate:

Assumption. For all quasi-projective V′, V with V′ ⊂ V ⊂ P^{m} and |V (F)| = ∞ there exists the limit

lim_{B→∞} \frac{N(V′, B)}{N(V, B)}.

The following definitions have been useful to us:

Definition S_{1}. A smooth irreducible quasi-projective subvariety V ⊂ P^{m} over a number field F is called weakly saturated, if |V (F)| = ∞ and if for any locally closed subvariety W ⊂ V with dim W < dim V one has

lim_{B→∞} \frac{N(W, B)}{N(V, B)} < 1.

It is important to remark that Question C really makes sense only for weakly saturated varieties. Indeed, if there were a locally closed subvariety W ⊂ V with dim W < dim V and

lim_{B→∞} \frac{N(W, B)}{N(V, B)} = 1,

then it would be enough to answer Question C for each irreducible component of W and for all possible intersections of these components (i.e., one could forget about the existence of V and reduce the situation to a lowerdimensional case). In general, it is not easy to decide whether or not a given locally closed subvariety V ⊂ P^{m} is weakly saturated. We expect (and our assumption implies this) that the orbits of connected subgroups G ⊂ P GL(m + 1) are examples of weakly saturated varieties V ⊂ P^{m} (see 3.2.8).

Definition S_{2}. A smooth irreducible quasi-projective subvariety V ⊂ P^{m} with |N(V, B)| = ∞ is called strongly saturated, if for all dense Zariski open subsets U ⊂ V, one has

lim_{B→∞} \frac{N(U, B)}{N(V, B)} = 1.

First of all, if V ⊂ P^{m} is a strongly saturated subvariety, then for any locally closed subvariety W ⊂ V with dim W < dim V, one has

lim_{B→∞} \frac{N(W, B)}{N(V, B)} = 0,

i.e., V is weakly saturated.

On the other hand, if V ⊂ P^{m} is weakly saturated, but not strongly saturated, then there must be an infinite sequence W_{1}, W_{2}, … of pairwise different locally closed irreducible subvarieties W_{i} ⊂ V with dim W_{i} < dim V and |W_{i}(F)| = ∞ such that for an arbitrary positive integer k one has

0 < lim_{B→∞} \frac{N(W_{1} ∪ ⋯ ∪ W_{k}, B)}{N(V, B)} < 1.

Moreover, in this situation one can always choose the varieties W_{i} to be strongly saturated (otherwise one could find W′_{i} ⊂ W_{i} with dim W′_{i} < dim W_{i} with the same properties as W_{i} etc.). The strong saturatedness of each W_{i} implies that

lim_{B→∞} \frac{N(W_{i_{1}} ∩ ⋯ ∩ W_{i_{l}}, B)}{N(V, B)} = 0

for all pairwise different i_{1}, …, i_{l} and l ≥ 2. In particular, one has

∑_{i=1}^{k} lim_{B→∞} \frac{N(W_{i}, B)}{N(V, B)} = lim_{B→∞} \frac{N(W_{1} ∪ ⋯ ∪ W_{k}, B)}{N(V, B)} < 1 ∀k > 0.

Definition F. Let V be a weakly saturated quasi-projective variety in P^{m} and W_{1}, W_{2}, … an infinite sequence of strongly saturated irreducible subvarieties W_{i} having the property

0 < θ_{i} := lim_{B→∞} \frac{N(W_{i}, B)}{N(V, B)} < 1 ∀i > 0.

We say that the set {W_{1}, W_{2}, …} forms an asymptotic arithmetic fibration on V, if the following equality holds

∑_{i=1}^{∞} θ_{i} = 1.

The main purpose of this paper is to explain some geometric and arithmetic ideas concerning weakly saturated varieties and their asymptotic arithmetic fibrations by strongly saturated subvarieties. It seems that the cubic bundles considered in [9] are examples of such a fibration. We want to remark that most of the above terminology grew out of our attempts to restore a conjectural picture of the interplay between the geometry of algebraic varieties and the arithmetic of the distribution of rational points on them after we have found in [9] an example which contradicted general expectations formulated in [4].

In section 2 we consider smooth quasi-projective varieties V over C together with a polarization L = (L, ∥ · ∥_{h}) consisting of an ample line bundle L on V equipped with a positive hermitian metric ∥·∥_{h}. Our main interest in this section is a discussion of geometric properties of V in connection with the Minimal Model Program [27] and its version for polarized algebraic varieties suggested by Fujita [20, 23, 24]. We introduce our main geometric invariants α_{L}(V), β_{L}(V), and δ_{L}(V) for an arbitrary L-polarized variety V. It is important to remark that we will be only interested in the case α_{L}(V) > 0. The number α_{L}(V) was first introduced in [4, 5], it equals to the opposite of the so called Kodaira energy (investigated by Fujita in [20, 23, 24]). Our basic geometric notion in the study of L-polarized varieties V with α_{L}(V) > 0 is the notion of an L-primitive variety. In Fujita’s program for polarized varieties with negative Kodaira energy L-primitive varieties play the same role as Q-Fano varieties in Mori’s program for algebraic varieties with negative Kodaira dimension. In particular, one expects the existence of so called L-primitive fibrations, which are analogous to Q-Fano fibrations in Mori’s program. We show that on L-primitive varieties there exists a canonical volume measure. Moreover, this measure allows us to construct a descent of hermitian metrics to the base of L-primitive fibrations. Many geometric ideas of this section are inspired by [4, 5].

In section 3 we introduce our main arithmetic notions of weakly and strongly L-saturated varieties. Our first main diophantine conjecture claims that if an adelic L-polarized quasi-projective algebraic variety V over a number field F is strongly L-saturated, then the corresponding L-polarized complex algebraic variety V (C) is L-primitive. Moreover, we conjecture that if an adelic L-polarized quasi-projective algebraic variety V over a number field F is weakly L-saturated, then the corresponding L-polarized complex algebraic variety V (C) admits an L-primitive fibration having infinitely many fibers W defined over F which form an asymptotic arithmetic fibration. These conjectures allow us to establish a connection between the geometry of V (C) and the arithmetic of V. Following this idea, we explain a construction of an adelic measure on an arbitrary L-primitive variety V with α_{L}(V) > 0 and of the corresponding Tamagawa number τ_{L}(V) as a regularized adelic integral of this measure. Our construction generalizes the definition of Tamagawa measures associated with a metrization of the canonical line bundle due to Peyre [30]. We expect that for strongly L-saturated varieties V the number τ_{L}(V) reflects the dependence of the constant c(V) in the asymptotic formula (1) on the adelic metrization of the ample line bundle L. We discuss the natural question about the behavior of the adelic constant τ_{L}(W) for fibers W in L-primitive fibrations on weakly L-saturated varieties.

In section 4 we show that our diophantine conjectures agree with already known examples of asymptotic formulas established for polarized algebraic varieties through the study of analytic properties of height zeta functions.

In Section 5 we illustrate our expectations for the constants a(V), b(V) and c(V) in the asymptotics of N(V, B) on some examples of smooth Zariski dense subsets V in Fano varieties with singularities.

We would like to thank J.-L. Colliot-Thélène for his patience and encouragement. We are very grateful to B. Mazur, Yu. I. Manin, L. Ein and A. Chambert-Loir for their comments and suggestions. We thank the referee for several useful remarks.

2 Geometry of L-polarized varieties

2.1 L-closure

Let V be a smooth irreducible quasi-projective algebraic variety over C, V (C) the set of closed points of V, L an ample invertible sheaf on V, i.e., L^{⊗k} = i^{∗}O_{P^{m}}(1) for some k > 0 and some embedding i : V ↪ P^{m}. Since we don’t assume V to be compact, the invertible sheaf L on V itself contains too little information about the embedding i : V ↪ P^{m}. For instance, let V be an affine variety of positive dimension. Then the space of global sections of L is infinite dimensional and we don’t know anything about the projective closure of V in P^{m} even though we know that the invertible sheaf L^{⊗k} is isomorphic to i^{∗}O_{P^{m}}(1). This situation changes if one considers L together with a positive hermitian metric, i.e., an ample metrized invertible sheaf L associated with L. Let us choose a positive hermitian metric h on O_{P^{m}}(1) (e.g. Fubini-Study metric) and denote by ∥ · ∥_{h} the induced metric on L^{⊗k}. Thus we obtain a metric ∥ · ∥ on L by putting ∥s(x)∥ := ∥s^{k}(x)∥_{h}^{1/k} for any x ∈ V (C) and any section s ∈ H^{0}(U, L) over an open subset U ⊂ V.

Definition 2.1.1 We call a pair L = (L, ∥·∥) an ample metrized invertible sheaf associated with L. We denote by L^{⊗ν} the pair (L^{⊗ν}, ∥ · ∥^{ν}).

Our next goal is to show that an ample metrized invertible sheaf contains almost complete information about the projective closure of V in P^{m}.

Definition 2.1.2 Let L = (L, ∥ · ∥) be an ample metrized invertible sheaf on a complex irreducible quasi-projective variety V. We denote by

H_{bd}^{0}(V, L)

the subspace of H^{0}(V, L) consisting of those global sections s of L over V such that the corresponding continuous function x ↦ ∥s(x)∥ (x ∈ V (C)) is globally bounded on V (C) from above by a positive constant C(s) depending only on s. We call H_{bd}^{0}(V, L) the space of globally bounded sections of L.

Proposition 2.1.3 Let \overline{V} be the normalization of the projective closure of V with respect to the embedding i : V ↪ P^{m} with L^{⊗k} = i^{∗}O_{P^{m}}(1). Denote by e : \overline{V} → P^{m} the corresponding finite projective morphism. Then one has a natural isomorphism

H_{bd}^{0}(V, L^{⊗k}) ≅ H^{0}(\overline{V} , e^{∗}O_{P^{m}}(1)).

Proof. Since \overline{V} (C) is compact, the continuous function x ↦ ∥s(x)∥ is globally bounded on \overline{V} (C) for any s ∈ H^{0}(\overline{V} , e^{∗}O_{P^{m}}(1)). Therefore, we obtain that H^{0}(\overline{V} , e^{∗}O_{P^{m}}(1)) is a subspace of H_{bd}^{0}(V, L^{⊗k}).

Now let f ∈ H_{bd}^{0}(V, L^{⊗k}) be a globally bounded on V (C) section of i^{∗}O_{P^{m}}(1). Since H_{bd}^{0}(V, L^{⊗k}) is a subspace of H^{0}(V, L^{⊗k}), the section f uniquely extends to a global meromorphic section \overline{f} ∈ H^{0}(\overline{V} , e^{∗}O_{P^{m}}(1)). Since a bounded meromorphic function is holomorphic, \overline{f} is a global regular section of e^{∗}O_{P^{m}}(1) (we apply the theorem of Riemann to some resolution of singularities ρ : X → \overline{V} and use the fact that ρ_{∗}O_{X} = O\overline{V}). Thus we have

H_{bd}^{0}(V, L^{⊗k}) ⊂ H^{0}(\overline{V} , e^{∗}O(1)).
□

Definition 2.1.4 We define a the graded C-algebra

A(V, L) = ⨁_{ν≥0} H_{bd}^{0}(V, L^{⊗ν}).

Using 2.1.3, one immediately obtains:

Corollary 2.1.5 The graded algebra A(V, L) is finitely generated.

Definition 2.1.6 We call the normal projective variety

\overline{V}^{L} = Proj A(V, L).

the L-closure of V with respect to an ample metrized invertible sheaf L.

Remark 2.1.7 By 2.1.3, \overline{V}^{L} is isomorphic to \overline{V}. Therefore, we have obtained a way to define the normalization of the projective closure of V with respect to an L^{⊗k}-embedding via a notion of an ample metrized invertible sheaf L on V.

2.2 Kodaira energy and α_{L}(V)

Let X be a normal irreducible algebraic variety of dimension n. We denote by Div(X) (resp. by Z_{n−1}(X)) the group of Cartier divisors (resp. Weil divisors) on X. An element of Div(X) ⊗ Q (resp.Z_{n−1}(X) ⊗ Q) is called a Q-Cartier divisor (resp. a Q-divisor). By K_{X} we denote a divisor of a meromorphic differential n-form on X, where K_{X} is considered as an element of Z_{n−1}(X).

Definition 2.2.1 Let X be a projective variety and L be an invertible sheaf on X. The Iitaka-dimension κ(L) is defined as

κ(L) = \{ \begin{matrix}−∞ & if H^{0}(X, L^{⊗ν}) = 0 for all ν > 0 \\ Max dim ϕ_{L^{⊗ν}}(X) : & H^{0}(X, L^{⊗ν}) ≠ 0 \end{matrix}

where ϕ_{L^{⊗ν}}(X) is the closure of the image of X under the rational map

ϕ_{L^{⊗ν}} : X → P(H^{0}(X, L^{⊗ν})).

A Cartier divisor L is called semi-ample (resp. effective), if L^{⊗ν} is generated by global sections for some ν > 0 (resp. κ(L) ≥ 0).

Remark 2.2.2 The notions of Iitaka-dimension, ampleness and semi-ample-ness obviously extend to Q-Cartier divisors. Let L be a Cartier divisor. Then for all κ_{1}, k_{2} ∈ N we set

κ(L^{⊗k_{1}/k_{2}}) := κ(L),
L^{⊗k_{1}/k_{2}} is ample ⇔ L is ample,

and

L^{⊗k_{1}/k_{2}} is semi-ample ⇔ L is semi-ample.

Definition 2.2.3 Let X be a smooth projective variety. We denote by NS(X) the group of divisors on X modulo numerical equivalence and set NS(X)_{R} = NS(X) ⊗ R. By [L] we denote the class of a divisor L in NS(X). The cone of effective divisors Λ_{eff}(X) ⊂ NS(X)_{R} is defined as the closure of the subset

⋃_{κ(L)≥0} R_{≥0}[L] ⊂ NS(X)_{R}.

Definition 2.2.4 Let V be a smooth quasi-projective algebraic variety with an ample metrized invertible sheaf L, \overline{V}^{L} the L-closure of V and ρ some resolution of singularities

ρ : X → \overline{V}^{L}.

We define the number

α_{L}(V) = inf{t ∈ Q : t[ρ^{∗}L] + [K_{X}] ∈ Λ_{eff}(X)}.

and call it the L-index of V.

Remark 2.2.5 It is easy to see that the L-index does not depend on the choice of ρ.

Remark 2.2.6 The L-index α_{L}(V) for smooth projective varieties V was first introduced in [4] and [5]. We remark that the opposite number −α_{L}(V) coincides with the notion of Kodaira energy introduced and investigated by Fujita in [20, 23, 24]:

κϵ(V, L) = −α_{L}(V) = − inf{t ∈ Q : κ((L)^{⊗t} ⊗ K_{V}) ≥ 0}.

From the viewpoint of our diophantine applications it is much more natural to consider α_{L}(V) instead of its opposite −α_{L}(V). The only reason that we could see for introducing the number −α_{L}(V) instead of α_{L}(V) is some kind of compatibility between the notions of Kodaira energy and Kodaira dimension, e.g. Kodaira energy must be positive (resp. negative) iff the Kodaira dimension is positive (resp. negative).

The following statement was conjectured in [4] (see also [5, 21, 22]):

Conjecture 2.2.7 (Rationality) Assume that α_{L}(V) > 0. Then α_{L}(V) is rational.

Remark 2.2.8 It was shown in [5] that this conjecture follows from the Minimal Model Program. In particular, it holds for dim V ≤ 3. If dim V = 1, then the only possible values of α_{L}(V) are numbers 2/k with (k ∈ N). If dim V = 2, then α_{L}(V) ∈ {2/k, 3/l} with (k, l ∈ N).

Definition 2.2.9 A normal irreducible algebraic variety W is said to have at worst canonical (resp. terminal) singularities if K_{W} is a Q-Cartier divisor and if for some (or every) resolution of singularities

ρ : X → W

one has

K_{X} = ρ^{∗}(K_{W}) ⊗ O(D)

where D is an effective Q-Cartier divisor (resp. the support of the effective divisor D coincides with the exceptional locus of ρ). Irreducible components of the exceptional locus of ρ which are not contained in the support of D are called crepant divisors of the resolution ρ.

Definition 2.2.10 A normal irreducible algebraic variety W is called a canonical Q-Fano variety, if W has at worst canonical singularities and K_{W}^{−1} is an ample Q-Cartier divisor. A maximal positive rational number r(W) such that K_{W}^{−1} = L^{⊗r(W)} for some Cartier divisor L is called the index of a canonical Q-Fano variety W (obviously, one has r(W) = α_{L}(W) for some positive metric on L).

The following conjecture is due to Fujita [20]:

Conjecture 2.2.11 (Spectrum Conjecture) Let S(n) be the set all possible values of α_{L}(V) for smooth quasi-projective algebraic varieties V of dimension ≤ n with an ample metrized invertible sheaf L. Then for any ε > 0 the set

{α_{L}(V) ∈ S(n) : α_{L}(V) > ε}

is finite.

This conjecture follows from the Minimal Model Program [27] and from the following conjecture on the boundedness of index for Fano varieties with canonical singularities:

Conjecture 2.2.12 (Boundedness of Index) The set of possible values of index r(W) for canonical Q-Fano varieties W of dimension n is finite.

In particular, both conjectures are true for Q-Fano varieties of dimension n ≤ 3 [2, 20, 23, 24, 28, 32].

2.3 L-primitive varieties

Definition 2.3.1 Let X be a projective algebraic variety. We call an effective Q-divisor D rigid, if κ(D) = 0.

Proposition 2.3.2 An effective Q-divisor D on X is rigid if and only if there exist finitely many irreducible subvarieties D_{1}, …, D_{l} ⊂ X (l ≥ 0) of codimension 1 such that D = r_{1}D_{1} + … + r_{l}D_{l} with r_{1}, …, r_{l} ∈ Q_{>0} and

dim H^{0}(X, O(n_{1}D_{1} + … + n_{l}D_{l})) = 1 ∀ (n_{1}, …, n_{l}) ∈ Z_{≥0}^{l}.

Proof. Let D be rigid. Take a positive integer m_{0} such that m_{0}D is a Cartier divisor and dim H^{0}(X, O(m_{0}D)) = 1. Denote by D_{1}, …, D_{l} the irreducible components of the divisor (s) of a non-zero section s ∈ H^{0}(X, O(mD)). One has

(s) = m_{1}D_{1} + ⋯ + m_{l}D_{l} m_{1}, …, m_{l} ∈ N.

Since O(D_{i}) admits at least one global non-zero section we obtain that

dim H^{0}(X, O(n′_{1}D_{1} + … + n′_{l}D_{l})) ≥ dim H^{0}(X, O(n_{1}D_{1} + … + n_{l}D_{l})),

whenever n_{1} ≥ n′_{1}, …, n_{l} ≥ n′_{l} for (n′_{1}, …, n′_{l}), (n_{1}, …, n_{l}) ∈ Z_{≥0}^{l}. This implies that

dim H^{0}(X, O(n_{1}D_{1} + … + n_{l}D_{l})) ≥ 1 ∀ (n_{1}, …, n_{l}) ∈ Z_{≥0}^{l}.

On the other hand, for any (n_{1}, …, n_{l}) ∈ Z_{≥0}^{l} there exists a positive integer n_{0} such that n_{0}m_{1} ≥ n_{1}, …, n_{0}m_{l} ≥ n_{l}. Therefore,

dim H^{0}(X, O(n_{1}D_{1} + … + n_{l}D_{l})) ≤ dim H^{0}(X, O(n_{0}m_{0}D)) = 1,

since κ(n_{0}m_{0}D) = κ(D) = 0.

□

Corollary 2.3.3 Let D_{1}, …, D_{l} ⊂ X be all irreducible components of the support of a rigid Q-Cartier divisor D. Then a linear combination

n_{1}D_{1} + … + n_{l}D_{l}, n_{1}, …, n_{l} ∈ Z

is a principal divisor, iff n_{1} = ⋯ = n_{l} = 0.

Proof. Assume that n_{1}D_{1} + … + n_{l}D_{l} is linearly equivalent to 0. Then the effective Cartier divisor D_{0} = ∑_{n_{i}≥0} n_{i}D_{i} is linearly equivalent to the effective Cartier divisor D′_{0} = ∑_{n_{j}<0}(−n_{j})D_{j}. Since D_{0} and D′_{0} have different supports we have dim H^{0}(X, O(D_{0})) ≥ 2. Contradiction to 2.3.2.

□

Definition 2.3.4 Let V be a smooth quasi-projective algebraic variety with an ample metrized invertible sheaf L and \overline{V}^{L} the projective L-closure of V. The variety V is called L-primitive, if the number α_{L}(V) is rational and if for some resolution of singularities

ρ : X → \overline{V}^{L}

one has ρ^{∗}(L)^{⊗α_{L}(V)} ⊗ K_{X} = O(D), where D is a rigid effective Q-Cartier divisor on X.

Remark 2.3.5 It is easy to see that the notion of an L-primitive variety doesn’t depend on the choice of a resolution of singularities ρ. Since V is smooth, we can always assume that the natural mapping

ρ : ρ^{−1}(V) → V

is an isomorphism.

Example 2.3.6 Let V_{1} and V_{2} be two smooth quasi-projective varieties with ample metrized invertible sheaves L_{1} and L_{2} (resp. on V_{1} and V_{2}). Assume that V_{1} (resp. V_{2} ) is L_{1}-primitive (resp. L_{1}-primitive). Then the product V = V_{1} × V_{2} is L-primitive, where L = π_{1}^{∗}L_{1} ⊗ π_{2}^{∗}L_{2}.

Our main list of examples of L-primitive varieties is obtained from canonical Q-Fano varieties:

Example 2.3.7 Let V be the set of nonsingular points of a canonical Q-Fano variety W with an ample metrized invertible sheaf L = (L, ∥ · ∥) such that K_{W}^{−1} = L^{⊗r(W)} ( W = \overline{V}^{L}). Then V is an L-primitive variety with L-index r(W). Indeed, let ρ : X → W be a resolution of singularities. By 2.2.10, we have ρ^{∗}(L)^{⊗r(W)} ⊗ K_{X} = O(D), where D is an effective Q-Cartier divisor. Since the support of D consists of exceptional divisors with respect to ρ, D is rigid (see 2.3.2).

We expect that the above examples cover all L-primitive varieties:

Conjecture 2.3.8 (Canonical Q-Fano contraction) Let V be an Lprimitive variety with α_{L}(V) > 0. Then there exists a resolution of singularities ρ : X → \overline{V}^{L} and a birational projective morphism π : X → W to a canonical Q-Fano variety W such that π^{∗}K_{W}^{−1} ≅ ρ^{∗}(L)^{α_{L}(V)} (i.e., α_{L}(V) = r(W)) and the support of D (ρ^{∗}(L)^{⊗r(W)} ⊗ K_{X} = O(D)) is contained in the exceptional locus of π.

The above conjecture is expected to follow from the Minimal Model Program using the existence and termination of flips (in particular, it holds for toric varieties).

The following statement will be important in our construction of Tamagawa numbers for L-primitive varieties defined over a number field:

Conjecture 2.3.9 (Vanishing) For V an L-primitive variety such that α_{L}(V) > 0 we have

h^{i}(X, O_{X}) = 0 ∀ i > 0

for any resolution of singularities

ρ : X → \overline{V}^{L}

such that the support of the Q-Cartier divisor ρ^{∗}(L)^{⊗α_{L}(V)} ⊗ K_{X} is a Q-Cartier divisor with normal crossings. In particular, Pic(X) is a finitely generated abelian group and one has a canonical isomorphism

Pic(X) ⊗ Q ≅ NS(X) ⊗ Q.

Remark 2.3.10 Theorem 1-2-5 in [27] implies the vanishing for the structure sheaf for Q-Fano varieties with canonical singularities (even with logterminal singularities). All canonical (and log-terminal singularities) are rational and it follows that the higher cohomology of the structure sheaf on any desingularization of a canonical or a log-terminal Fano variety must also vanish (by Leray spectral sequence). Therefore, we would obtain the vanishing 2.3.9 for all L-primitive varieties which are birationally equivalent to a Fano variety with at worst log-terminal singularities. The existence of a canonical Q-contraction 2.3.8 would insure this.

Definition 2.3.11 Let V be an L-primitive variety with α_{L}(V) > 0, ρ : X → \overline{V}^{L} any resolution of singularities, D_{1}, …, D_{l} irreducible components of the support of the rigid effective Q-Cartier divisor D with O(D) = ρ^{∗}(L)^{⊗α_{L}(V)} ⊗ K_{X}. We shall call

Pic(V, L) := Pic(X \ ⋃_{i=1}^{l} D_{i})

the L-Picard group of V. The number

β_{L}(V) := rk Pic(V, L)

will be called the L-rank of V. We define the L-cone of effective divisors Λ_{eff}(V, L) ⊂ Pic(V, L) ⊗ R as the image of Λ_{eff}(X) ⊂ NS(X)_{R} = Pic(X) ⊗ R under the natural surjective R-linear mapping

ρ̃ : Pic(X) ⊗ R → Pic(V, L) ⊗ R.

Remark 2.3.12 By 2.3.3, one obtains the exact sequence

0 → Z[D_{1}] ⊕ ⋯ ⊕ Z[D_{l}] → Pic(X) →^{ρ̃} Pic(V, L)→0  (2)

and therefore

β_{L}(V) = rk Pic(X) − l.

Using these facts, it is easy to show that the group Pic(V, L) and the cone Λ_{eff}(V, L) do not depend on the choice of a resolution of singularities ρ : X → \overline{V}^{L}.

The above conjecture holds in dimension n ≤ 3 as a consequence of the Minimal Model Program. More precisely, it is a consequence of Conjecture 2.3.8 and the following weaker statement:

Conjecture 2.3.13 (Polyhedrality) Let V be an L-primitive variety with α_{L}(V) > 0. Then Λ_{eff}(V, L) is a rational finitely generated polyhedral cone.

Definition 2.3.14 Let (A, A_{R}, Λ) be a triple consisting of a finitely generated abelian group A of rank k, a k-dimensional real vector space A_{R} = A⊗R and a convex k-dimensional finitely generated polyhedral cone Λ ∈ A_{R} such that Λ ∩ −Λ = 0 ∈ A_{R}. For Re(s) contained in the interior of the cone Λ we define the X -function of Λ by the integral

X_{Λ}(s) := ∫_{Λ^{∗}} e^{−<s,y>}dy

where Λ^{∗} ∈ A_{R}^{∗} is the dual cone to Λ and dy is the Lebesgue measure on A_{R}^{∗} normalized by the dual lattice A^{∗} ⊂ A_{R}^{∗} where A^{∗} := Hom(A, Z).

Remark 2.3.15 If Λ is a finitely generated rational polyhedral cone the function X_{Λ}(s) is a rational function in s. However, the explicit determination of this function might pose serious computational problems.

Definition 2.3.16 Let V be an L-primitive smooth quasi-projective algebraic variety with a metrized invertible sheaf L and α_{L}(V) > 0. Let X be any resolution of singularities ρ : X → \overline{V}^{L}. We consider the triple

(Pic(V, L), Pic(V, L)_{R}, Λ_{eff}(V, L))

and the corresponding X -function. Assuming that Pic(V, L) is a finitely generated abelian group (cf. 2.3.9) and that Λ_{eff}(V, L) is a polyhedral cone (cf. 2.3.13), we define the constant γ_{L}(V) ∈ Q by

γ_{L}(V) := X_{Λ_{eff}(V,L)}(ρ̃(−[K_{X}])).

2.4 L-primitive fibrations and descent of metrics

Let V be an L-primitive variety of dimension n. We show that there exists a canonical measure on V (C) which is uniquely defined up to a positive constant.

In order to construct this measure we choose a resolution of singularities ρ : X → \overline{V}^{L} and a positive integer k_{2} such that k_{2}D is a Cartier divisor, where O(D) ≅ (ρ^{∗}L)^{⊗α_{L}(V)} ⊗ K_{X}. Then k_{1} = k_{2}α_{L}(V) is a positive integer. Let g ∈ H^{0}(X, O(k_{2}D)) be a non-zero global section (by 2.3.2, it is uniquely defined up to a non-zero constant). We define a measure ω_{L}(g) on X(C) as follows. Choose local complex analytic coordinates z_{1}, …, z_{n} in some open neighborhood U_{x} ⊂ X(C) of a point x ∈ V (C). We write the restriction of the global section g to U_{x} as

g = s^{k_{2}α_{L}(V)}(dz_{1} ∧ ⋯ ∧ dz_{n})^{⊗k_{2}} = s^{k_{1}}(dz_{1} ∧ ⋯ ∧ dz_{n})^{⊗k_{2}},

where s is a local section of L . Then we set

ω_{L}(g) := (\frac{\sqrt{−1}}{2})^{n} ∥s∥^{α_{L}(V)}(dz_{1} ∧ d\overline{z}_{1}) ∧ ⋯ ∧ (dz_{n} ∧ d\overline{z}_{n}).

By a standard argument, one obtains that ω_{L}(g) doesn’t depend on the choice of local coordinates in U_{x} and that it extends to the whole complex space X(C). It remains to notice that the restriction of the measure ω_{L}(g) to V (C) ⊂ X(C) does not depend on the choice of ρ. So we obtain a welldefined measure on V (C).

Remark 2.4.1 We note that the measure ω_{L}(g) depends on the choice of g ∈ H^{0}(X, O(k_{2}D)). More precisely, it multiplies by |c|^{1/k_{2}} if we multiply g by some non-zero complex number c. Thus we obtain that the mapping

∥ · ∥_{L} : H^{0}(X, O(k_{2}D)) → R_{≥0}
g ↦ ∫_{X(C)} ω_{L}(g) = ∫_{V (C)} ω_{L}(g)

satisfies the property

∥cg∥_{L} = |c|^{1/k_{2}}∥g∥_{L} ∀c ∈ C^{∗}.

Definition 2.4.2 Let V be a smooth quasi-projective variety with an ample metrized invertible sheaf L, ρ : X → \overline{V}^{L} a resolution of singularities. A regular projective morphism π : X → Y to a projective variety Y (dim Y < dim X) is called an L-primitive fibration on V if there exists a Zariski dense open subset U ⊂ Y such that the following conditions are satisfied:

(i) for any point y ∈ U(C) the fiber V_{y} = π^{−1}(y) ∩ V is a smooth quasiprojective L-primitive subvariety;

(ii) α_{L}(V) = α_{L}(V_{y}) > 0 for all y ∈ U(C); (iii) for any k ∈ N such that kα_{L}(V) ∈ Z,

L_{k} := R^{0}π_{∗} (ρ^{∗}(L)^{⊗α_{L}(V)} ⊗ K_{X})^{⊗k}

is an ample invertible sheaf on Y.

We propose the following version of the Fibration Conjecture of Fujita (see [20]):

Conjecture 2.4.3 (Existence of Fibrations) Let V be an arbitrary smooth quasi-projective variety with an ample metrized invertible sheaf L and α_{L}(V) > 0. Then there exists a resolution of singularities ρ : X → \overline{V}^{L} such that X admits an L-primitive fibration π : X → Y on some dense Zariski open subset V′ ⊂ V.

Remark 2.4.4 From the viewpoint of the Minimal Model Program, Conjecture 2.4.3 is equivalent to the statement about the conjectured existence of Q-Fano fibrations for algebraic varieties of negative Kodaira-dimension (cf. [27]). The existence of an L-primitive fibration is equivalent to the fact that the graded algebra

R(V, L) = ⊕_{ν≥0}H^{0}(X, M^{kν}), M := ρ^{∗}(L)^{⊗α_{L}(V)} ⊗ K_{X}

is finitely generated (cf. 2.4 in [4]). One can define Y as Proj R(V, L) and X as a common resolution of singularities of \overline{V}^{L} and of the indeterminacy locus and generic fiber of the natural rational map \overline{V}^{L} → Y (cf. [26]).

It is important to observe that a metric ∥ · ∥ on L induces natural metrics on all ample invertible sheaves L_{k} on Y:

Definition 2.4.5 Let L_{k} be an ample invertible sheaf on Y as above. We define a metric ∥ · ∥_{L,k} on L_{k} as follows. Let y ∈ Y (C) be a closed point, U ⊂ Y be a Zariski open subset containing y, and s ∈ H^{0}(U, L_{k}) is a section with s(y) ≠ 0. Then we set

∥s(y)∥_{L,k} := (∫_{V_{y}(C)} ω_{L}(π^{∗}s))^{k} ,

where V_{y}(C) is the fiber over y of the L-primitive fibration π^{∗}, π^{∗}s the πpullback of s restricted to L-primitive variety V_{y}(C), and ω_{L}(π^{∗}s) the corresponding to π^{∗}s volume measure on V_{y}(C). We call ∥ · ∥_{L,k} a k-adjoint descent to Y of a metric ∥ · ∥ on L.

3 Heights and asymptotic formulas

3.1 Basic terminology and notations

Let F be a number field, O_{F} ⊂ F the ring of integers in F, Val(F) the set of all valuations of F, F_{v} the completion of F with respect to a valuation v ∈ Val(F), Val(F)_{∞} = {v_{1}, …, v_{r}} the set of all archimedean valuations of F. For any algebraic variety X over a field F we denote by X(F) the set of its K-rational points.

Definition 3.1.1 Let E be a vector space of dimension m + 1 over F, O_{E} ⊂ E a projective O_{F} -module of rank m+1 and ∥·∥_{v_{1}}, …, ∥·∥_{v_{r}} the set of Banach norms on the real or complex vector spaces E_{v_{i}} = E ×_{F} F_{v_{i}} corresponding to elements of Val(F)_{∞} = {v_{1}, …, v_{r}}. It is well-known that the above data for E define a family {∥ · ∥_{v}, v ∈ Val(F)} of v-adic metrics for a standard invertible sheaf O(1) on P(E). If x ∈ X(F) is a point and s ∈ H^{0}(U, O(1)) is a section over an open subset U ⊂ P(E) containing x, then we denote by ∥s(x)∥_{v} the corresponding v-adic norm of s at x. We set Õ(1) = (O(1), ∥·∥_{v}) to be the standard invertible sheaf O(1) on P(E) together with a family vadic metrics ∥ · ∥_{v} defined by the above data and we call Õ(1) the standard ample metrized invertible sheaf on P(E).

Definition 3.1.2 Let X be an algebraic variety over F. For any point x ∈ X(F) and any regular function f ∈ H^{0}(U, O_{X}) on an open subset U ⊂ P(E) containing x, we define the v-adic norm ∥f(x)∥_{v} := |f(x)|_{v}. We call this family of v-adic metrics on O_{X} the canonical metrization of the structure sheaf O_{X}.

Definition 3.1.3 Let X be a quasi-projective algebraic variety, L a very ample invertible sheaf on X, i : X ↪ P(E) an embedding with L = i^{∗}O(1). We denote by L = (L, ∥·∥_{v}) the sheaf L together with v-adic metrics induced from a family of v-adic metrics on the standard ample metrized invertible sheaf Õ(1) on P(E). In this situation we call L a very ample metrized sheaf on X and write L = i^{∗}Õ(1).

Definition 3.1.4 Let L = (L, ∥ · ∥_{v}) be a very ample metrized invertible sheaf on X. Then for any point x ∈ X(F), the L-height of x is defined as

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

where s ∈ Γ(U, L) is a nonvanishing at x section of L over some open subset U ⊂ X.

Remark 3.1.5 Using a canonical metrization of the structure sheaf O_{X}, the linear mapping

S^{k}(Γ(U, L)) → Γ(U, L^{⊗k}) (k > 0),

and the F -bilinear mapping

Γ(U, L^{⊗k}) × Γ(U, L^{⊗−k}) → Γ(U, O_{X}),

one immediately sees that a family of v-adic metrics on an invertible sheaf L allows to define a family of v-adic metrics on L^{⊗k} and on any invertible sheaf M such that there exist integers k_{1}, k_{2} (k_{2} ≠ 0) with L^{⊗k_{1}} = M^{⊗k_{2}}. In this situation we write

M = (M, ∥ · ∥_{v}) := (L^{⊗k_{1}/k_{2}}, ∥ · ∥_{v}^{k_{1}/k_{2}}),

or simply M = L^{k_{1}/k_{2}}. Obviously, one obtains

H_{M}(x) = (H_{L}(x))^{k_{1}/k_{2}} ∀ x ∈ X(F).

Definition 3.1.6 Let L be an ample invertible sheaf on a quasi-projective variety X and k a positive integer such that L^{⊗k} is very ample. We define an ample metrized invertible sheaf L = (L, ∥ · ∥_{v}) on X associated with L by considering L^{⊗k} := (L^{⊗k}, ∥ · ∥_{v}^{k}) as a very ample metrized invertible sheaf on X.

3.2 Weakly and strongly L-saturated varieties

Let V be an arbitrary quasi-projective algebraic variety over F with an ample metrized invertible sheaf L. We always assume that V (F) is infinite and set

N(V, L, B) := #{x ∈ V (F) : H_{L}(x) ≤ B}.

Here and in 3.4 we will work under the following Assumption 3.2.1 For all quasi-projective V′, V with V′ ⊂ V and |V (F)| = ∞ there exists the limit

lim_{B→∞} \frac{N(V′, L, B)}{N(V, L, B)}.

Definition 3.2.2 We call an irreducible quasi-projective algebraic variety V with an ample metrized invertible sheaf L weakly L-saturated if for any Zariski locally closed subset W ⊂ V with dim W < dim V, one has

lim_{B→∞} \frac{N(W, L, B)}{N(V, L, B)} < 1.

Definition 3.2.3 We call an irreducible quasi-projective algebraic variety V with an ample metrized invertible sheaf L strongly L-saturated if for any dense Zariski open subset U ⊂ V, one has

lim_{B→∞} \frac{N(U, L, B)}{N(V, L, B)} = 1.

Definition 3.2.4 Let V be a weakly L-saturated variety, W ⊂ V a locally closed strongly saturated subvariety of smaller dimension. Then we call W an L-target of V, if

0 < lim_{B→∞} \frac{N(W, L, B)}{N(V, L, B)} < 1.

Theorem 3.2.5 Let V be an arbitrary quasi-projective algebraic variety with an ample metrized invertible sheaf L. Assume that |V (F)| = ∞ and that 3.2.1 holds. Then we have:

(i) if V is strongly L-saturated then V is weakly L-saturated;

(ii) V contains finitely many weakly L-saturated subvarieties W_{1}, …, W_{k} with

lim_{B→∞} \frac{N(W_{1} ∪ ⋯ ∪ W_{k}, L, B)}{N(V, L, B)} = 1.

(iii) V contains a strongly L-saturated subvariety W having the property

0 < lim_{B→∞} \frac{N(W, L, B)}{N(V, L, B)} < 1.

(iv) if V is weakly saturated and if it doesn’t contain a dense Zariski open subset U ⊂ V which is strongly saturated then V contains infinitely many L-targets.

Proof. (i) Let W ⊂ V be a Zariski closed subset with dim W < dim V and U = V \ W. Then

lim_{B→∞} \frac{N(W, L, B)}{N(V, L, B)} + lim_{B→∞} \frac{N(U, L, B)}{N(V, L, B)} = lim_{B→∞} \frac{N(V, L, B)}{N(V, L, B)} = 1.

Since V is strongly L-saturated, we have

lim_{B→∞} \frac{N(U, L, B)}{N(V, L, B)} = 1

and therefore

lim_{B→∞} \frac{N(W, L, B)}{N(V, L, B)} = 0 < 1.

(ii) Let W ⊂ V be a minimal Zariski closed subset such that

lim_{B→∞} \frac{N(W, L, B)}{N(V, L, B)} = 1

and W_{1}, …, W_{k} irreducible components of W. It immediately follows from the minimality of W that each W_{i} is weakly saturated.

(iii) Let W ⊂ V be an irreducible Zariski closed subset of minimal dimension such that

lim_{B→∞} \frac{N(W, L, B)}{N(V, L, B)} = 1.

The minimality of W implies that W is weakly saturated.

(iv) By (iii) the set of L-targets is nonempty. Assume that the set of all L-targets is finite: {W_{1}, …, W_{k}}. The strong saturatedness of each W_{i} implies that

lim_{B→∞} \frac{N(W_{i_{1}} ∩ ⋯ ∩ W_{i_{l}}, L, B)}{N(V, L, B)} = 0

for all pairwise different i_{1}, …, i_{l} ∈ {1, …, k} and l ≥ 2. In particular, one has

∑_{i=1}^{k} lim_{B→∞} \frac{N(W_{i}, L, B)}{N(V, L, B)} = lim_{B→∞} \frac{N(W_{1} ∪ ⋯ ∪ W_{k}, L, B)}{N(V, L, B)} < 1.

We set U := V \ (W_{1} ∪ ⋯ ∪ W_{k}). Then

lim_{B→∞} \frac{N(U, L, B)}{N(V, L, B)} > 0.

Since U is not strongly saturated, there exists an irreducible Zariski closed subset W_{0} ⊂ V of minimal dimension < dim V such that

lim_{B→∞} \frac{N(W_{0} ∩ U, L, B)}{N(U, L, B)} > 0.

It follows from the minimality of W_{0} that W_{0} is strongly saturated. On the other hand, one has

lim_{B→∞} \frac{N(W_{0}, L, B)}{N(V, L, B)} ≥ lim_{B→∞} \frac{N(W_{0} ∩ U, L, B)}{N(V, L, B)} > 0,

i.e., W_{0} is an L-target and W_{0} ∉ {W_{1}, …, W_{k}}. Contradiction. □

Definition 3.2.6 Let V be a weakly L-saturated variety and W_{1}, W_{2}, … an infinite sequence of strongly saturated irreducible subvarieties W_{i} having the property

0 < θ_{i} := lim_{B→∞} \frac{N(W_{i}, L, B)}{N(V, L, B)} < 1 ∀i > 0.

We say that the set {W_{1}, W_{2}, …} forms an asymptotic arithmetic Lfibration on V, if the following equality holds

∑_{i=1}^{∞} θ_{i} = 1.

We expect that the main source of examples of weakly and strongly saturated varieties should come from the following situation:

Proposition 3.2.7 Assume 3.2.1 and let G ⊂ P GL(n + 1) be a connected linear algebraic group acting on P^{n} and V := Gx ⊂ P^{n} a G-orbit of a point x ∈ P^{n}(F). Then V is weakly Õ(1)-saturated.

Proof. Let W ⊂ V be an arbitrary locally closed subset with dim W < dim V, \overline{W} ⊂ V its Zariski closure in V and U := V \\overline{W} ⊂ V the corresponding dense Zariski open subset of V. Then V is covered by the open subsets gU, where g runs over all elements in G(F) (this follows from the fact that G is unirational and that G(F) is Zariski dense in G [10]). Therefore, the orbit of x ∈ V (F) under G(F) is Zariski dense in V. Since the Zariski topology is noetherian we can choose a finite subcovering: V = ⋃_{i=1}^{k} g_{i}U (g_{i} in G(F)). Considering g_{i} ∈ G(F) as matrices in P GL(n+1) and using standard properties of heights [29], one obtains positive constants c_{i} such that

H_{L}(g_{i}(x)) ≤ c_{i}H_{L}(x)

for all x ∈ P^{n}(F). It is clear that for c_{0} := ∑_{i=1}^{k} c_{i} we have

N(U, L, B) ≤ N(V, L, B) ≤ c_{0}N(U, L, B).

It follows that

\frac{1}{c_{0}} ≤ lim_{B→∞} \frac{N(U, L, B)}{N(V, L, B)} ≤ 1.

Hence

lim_{B→∞} \frac{N(W, L, B)}{N(V, L, B)} ≤ lim_{B→∞} \frac{N(\overline{W} , L, B)}{N(V, L, B)} ≤ 1 − \frac{1}{c_{0}} < 1.

We can reformulate the statement of 3.2.7 as follows:

Proposition 3.2.8 Let G be a connected linear algebraic group, H ⊂ G a closed subgroup and V := G/H. If 3.2.1 holds then V is weakly saturated with respect to any G-equivariant projective embedding of V.

It is easy to see that V = G/H is not necessarily weakly saturated with respect to projective embeddings which are not G-equivariant:

Example 3.2.9 Let S ⊂ P^{8} be the anticanonically embedded Del Pezzo surface which is a blow up of a rational point in P^{2}. Denote by L the metrized anticanonical sheaf on S. The unique exceptional curve C ⊂ S is contained in the union of two open subsets U_{0}, U_{1} ⊂ S where U_{0} ≅ U_{1} ≅ A^{2}. Therefore, S can be considered as a projective compactification of the algebraic group G_{a}^{2} (after an identification of G_{a}^{2} with U_{0} or U_{1}). This compactification is not G_{a}^{2}-equivariant. One has

a_{L}(U_{0}) = a_{L}(U_{1}) = a_{L}(C) = 2,

but

a_{L}(U_{0} \ C) = a_{L}(U_{1} \ C) = 1.

Hence, U_{0} and U_{1} are not weakly L-saturated.

It is easy to show that an equivariant compactification of G/H is not necessarily strongly L-saturated:

Example 3.2.10 Let V = P^{1} × P^{1}. Then V is a G-homogeneous variety with G = GL(2) × GL(2). However, V is not strongly L-saturated for L := π_{1}^{∗}O(k_{1}) ⊗ π_{2}^{∗}O(k_{2}) (k_{1}, k_{2} ∈ N), if k_{1} ≠ k_{2}.

3.3 Adelic L-measure and τ_{L}(V)

Now we define an adelic measure ω_{L} corresponding to an ample metrized invertible sheaf L on an L-primitive variety V with α_{L}(V) > 0 which satisfies the assumption 2.3.9. This is a generalization of a construction due to Peyre ([30]) for V being a smooth projective variety and L the metrized canonical line bundle, which in its turn is a generalization of the classical construction of Tamagawa measures on the adelic points of algebraic groups.

Let V be an L-primitive variety of dimension n, ρ : X → \overline{V}^{L} a resolution of singularities, k_{2} a positive integer such that k_{1} = k_{2}α_{L}(V) ∈ Z and

(ρ^{∗}(L)^{α_{L}(V)} ⊗ K_{X})^{⊗k_{2}} ≅ O(D),

where D is a rigid effective Cartier divisor on X.

Definition 3.3.1 Let g be a non-zero element of the 1-dimensional F -vector space H^{0}(X, O(D)) (g is defined uniquely up to an element of F^{∗}). Let v ∈ Val(F). We define a measure ω_{L,v}(g) on V (F_{v}) as follows. Choose local v-analytic coordinates x_{1,v}, …, x_{n,v} in some open neighborhood U_{x} ⊂ X(F_{v}) of a point x ∈ X(F_{v}). We write the restriction of the global section g to U_{x} as

g = s^{k_{1}}(dx_{1,v} ∧ ⋯ ∧ dx_{n,v})^{k_{2}}

where s is a local section of L. Define a v-adic measure on U_{x} as

ω_{L,v}(g) := ∥s∥_{v}^{k_{1}/k_{2}}dx_{1,v} ⋯ dx_{n,v} = ∥s∥_{v}^{α_{L}(V)}dx_{1,v} ⋯ dx_{n,v},

where dx_{1,v} ⋯ dx_{n,v} is the usual normalized Haar measure on F_{v}^{n}. By a standard argument, one obtains that ω_{L,v}(g) doesn’t depend on the choice of local coordinates in U_{x} and that it extends to the whole v-adic space X(F_{v}). The restriction of ω_{L,v}(g) doesn’t depend on the choice of ρ. So we obtain a well-defined v-adic measure on V (F_{v}).

Remark 3.3.2 We remark that ω_{L,v}(g) depends on the choice of a global section g ∈ H^{0}(X, O(D)): if g′ = cg (c ∈ F^{∗}) is another global section, then

ω_{L,v}(g′) = |c|_{v}^{1/k}ω_{L,v}(g).

Our next goal is to obtain an explicit formula for the integral

d_{v}(V) := ∫_{V (F_{v})} ω_{L,v}(g) = ∫_{X(F_{v})} ω_{L,v}(g)

for almost all v ∈ Val(F). We use the p-adic integral formula of Denef ([16] Th. 3.1) in the same way as the p-adic formula of A. Weil ([35] Th. 2.2.3) was used by Peyre in [30].

Remark 3.3.3 By [1, 26], we can choose ρ in such a way that ρ is defined over F and all irreducible components D_{1}, …, D_{l} (l ≥ 0) of the support of D = ∑_{i=1}^{l} m_{i}D_{i} are smooth divisors with normal crossings over the algebraic closure \overline{F}.

Definition 3.3.4 Let G be the image of Gal(\overline{F} /F) in the symmetric group S_{l} that acts by permutations on D_{1}, …, D_{l}. We set I := {1, …, l} and denote by I/G the set of all G-orbits in I. For any J ⊂ I we set

D_{J} := \{ \begin{matrix}⋂_{j∈J} D_{j}, & if J ≠ ∅ \\ X \ ⋃_{j∈I} D_{j}, & if J = ∅ \end{matrix} ,

D_{J}^{∘} = D_{J} \ ⋃_{j∉J} D_{j}.

(D_{J} is defined over F if and only if J is a union of some of G-orbits in I/G.).

We can extend X to a projective scheme X of finite type over O_{F} and divisors D_{1}, …, D_{l} to codimension-1 subschemes D_{1}, …, D_{l} in X such that for almost all non-archimedean v ∈ Val(F) the reductions of X and D_{1}, …, D_{l} modulo ℘_{v} ⊂ O_{F} are smooth projective varieties X_{v} and D_{v,1}, …, D_{v,l} over the algebraic closure \overline{k_{v}} of the residue field k_{v} with D_{v,i} ≠ D_{v,j} for i ≠ j. Moreover, we can assume that

D_{v,J} := \{ \begin{matrix}⋂_{j∈J} D_{v,j}, & if J ≠ ∅ \\ X \ ⋃_{j∈I} D_{v,j}, & if J = ∅ \end{matrix}

are also smooth over \overline{k_{v}}.

Definition 3.3.5 A non-archimedean valuation v ∈ Val(F) which satisfies all the above assumptions will be called a good valuation for the pair (X , {D_{i}}_{i∈I}).

Definition 3.3.6 Let G_{v} ⊂ G be a cyclic subgroup generated by a representative of the Frobenius element in Gal(\overline{k_{v}}/k_{v}). We denote by I/G_{v} the set of all G_{v}-orbits in I. If j ∈ I/G_{v}, then we set b_{j} to be the length of the corresponding G_{v}-orbit and put r_{j} = m_{j}/k_{2}, where m_{j} is the multiplicity of irreducible components of D corresponding to the G_{v}-orbit j.

The following theorem is a slightly generalized version of Th. 3.1 in [16]:

Theorem 3.3.7 Let v ∈ Val(F) be a good non-archimedean valuation for (X , {D_{i}}_{i∈I}). Then

d_{v}(V) = ∫_{X(F_{v})} ω_{L,v}(g) = \frac{c_{∅}}{q_{v}^{n}} + \frac{1}{q_{v}^{n}} ∑_{∅≠J⊂I/G_{v}} c_{J} ∏_{j∈J} (\frac{q_{v}^{b_{j}} − 1}{q_{v}^{b_{j}(r_{j}+1)} − 1}) ,

where q_{v} is the cardinality of k_{v}, and c_{J} is the cardinality of the set of k_{v}- rational points in D_{J}^{∘}.

Let us consider an exact sequence of Galois Gal(\overline{F} /F)-modules:

0 → Z[D_{1}] ⊕ ⋯ ⊕ Z[D_{l}] → Pic(X) →^{ρ̃} Pic(X \ ⋃_{i=1}^{l} D_{i})→0  (3)

Theorem 3.3.8 Assume that X has the property h^{1}(X, O_{X}) = 0. Then

d_{v}(V) = 1 + \frac{1}{q_{v}} Tr(Φ_{v}|Pic(X \ ⋃_{i=1}^{l} D_{i}) ⊗ Q_{l}) + O (\frac{1}{q_{v}^{1+ε}} ) ,

where

ε = min{1/2, r_{1}, …, r_{l}}

and Φ_{v} is the Frobenius morphism.

Proof. By conjectures of Weil proved by Deligne [15], one has

\frac{c_{J}}{q_{v}^{n}} = O (\frac{1}{q_{v}} ) J ≠ ∅

(since dim D_{J} ≤ n − 1 for J ≠ ∅) and

\frac{c_{∅}}{q_{v}^{n}} = ∑_{k=0}^{2n}(−1)^{k}Tr(Φ_{v}|H_{c}^{k}(X \ ⋃_{i=1}^{l} D_{i}, Q_{l})),

where H_{c}^{k}(·, Q_{l}) denotes the étale cohomology group with compact supports. Using long cohomology sequence of the pair (X , ⋃_{i} D_{i}), one obtains isomorphisms

H_{c}^{k}(X \ ⋃_{i=1}^{l} D_{i}, Q_{l}) = H_{c}^{k}(X , Q_{l}) for k = 2n, 2n − 1

and the short exact sequence

0 → H_{c}^{2n−2}(X \ ⋃_{i=1}^{l} D_{i}, Q_{l}) → H_{c}^{2n−2}(X , Q_{l}) → ⨁_{i=1}^{l} H_{c}^{2n−2}(D_{i}, Q_{l}) → 0.

Using isomorphisms

H_{c}^{2n−2}(D_{i}, Q_{l}(n − 1)) ≅ Q_{l}, H_{c}^{2n}(X , Q_{l}(n)) ≅ Q_{l},

Poincaré duality

H_{c}^{2n−2}(X , Q_{l}(n − 1)) × Pic(X) →^{∼} Q_{l}

and the vanishing property h^{1}(X, O_{X}) = 0, we obtain

H_{c}^{2n−1}(X \ ⋃_{i=1}^{l} D_{i}, Q_{l}) = 0

and

\frac{c_{∅}}{q_{v}^{n}} = 1 + \frac{1}{q_{v}} Tr(Φ_{v}|Pic(X \ ⋃_{i=1}^{l} D_{i}) ⊗ Q_{l})

+ ∑_{k=0}^{2n−3} \frac{(−1)^{k}}{q_{v}^{n}} Tr(Φ_{v}|H_{c}^{k}(X \ ⋃_{i=1}^{l} D_{i}, Q_{l}))

= 1 + \frac{1}{q_{v}} Tr(Φ_{v}|Pic(X \ ⋃_{i=1}^{l} D_{i}) ⊗ Q_{l}) + O (\frac{1}{q_{v}^{3/2}} ) .

On the other hand, for J ≠ ∅ one has

∏_{j∈J} (\frac{q_{v}^{b_{j}} − 1}{q_{v}^{b_{j}(r_{j}+1)} − 1}) = O (\frac{1}{q_{v}^{1+ε_{0}}} ) ,

where ε_{0} = min{r_{1}, …, r_{l}}.

□

Definition 3.3.9 We define the convergency factors

λ_{v} := \{ \begin{matrix}L_{v}(1, Pic(X \ ⋃_{i=1}^{l} D_{i})), & if v is good \\ 1 & otherwise \end{matrix}

where L_{v} is the local factor of the Artin L-function corresponding to the G-module Pic(X \ ⋃_{i=1}^{l} D_{i}) and we set

ω_{L,S} := \sqrt{|disc(F)|}^{−n} ∏_{v∈Val(F)} λ_{v}^{−1}ω_{L,v}(g),

where disc(F) is the absolute discriminant of O_{F} and S is the set of bad valuations.

By the product formula, ω_{L,S} doesn’t depend on the choice of g.

Definition 3.3.10 Denote by A_{F} the adele ring of F. Let \overline{X(F)} be the closure of X(F) in X(A_{F}) (in direct product topology). Under the vanishing assumption h^{1}(X, O_{X}) = 0, we define the constant

τ_{L}(V) = lim_{s→1}(s − 1)^{β_{L}(V)}L_{S}(s, Pic(X \ ⋃_{i=1}^{l} D_{i})) ∫\overline{X(F)} ω_{L,S},

where β_{L}(V) is the rank of the submodule of Gal(\overline{F} /F)-invariants of the module Pic(X \ ⋃_{i=1}^{l} D_{i}).

3.4 Main strategy

Now we proceed to discuss our main strategy in understanding the asymptotic for the number N(V, L, B) as B → ∞ for an arbitrary L-polarized quasi-projective variety. Again, we shall make the assumption 3.2.1. Our approach consists in 4 steps including 3 subsequent simplifications of the situation:

Step 1 (reduction to weakly L-saturated varieties): By 3.2.5 (ii), every quasi-projective L-polarized variety V contains a finite number of weakly Lsaturated varieties W_{1}, … W_{k} such that

lim_{B→∞} \frac{N(W, L, B)}{N(V, L, B)} = 1.

Therefore, it would be enough to understand separately the asymptotics of N(W_{i}, L, B), i ∈ {1, …, k} modulo the asymptotics N(W_{i_{1}}∩⋯∩W_{i_{l}}, L, B) for low-dimensional subvarieties W_{i_{1}}∩⋯∩W_{i_{l}}, where {i_{1}, …, i_{l}} ⊂ {1, …, k} are subsets of pairwise different elements with l ≥ 2.

For our next reduction step, we need:

Conjecture 3.4.1 Let V be a weakly L-saturated variety which doesn’t contain an open Zariski dense and strongly L-saturated subset U ⊂ V. Then the set of L-targets of V forms an asymptotic arithmetic L-fibration.

Step 2 (reduction to strongly L-saturated varieties): Let V be an arbitrary weakly L-saturated variety. Then either V contains a strongly Lsaturated Zariski open subset or, according to 3.4.1, we obtain an asymptotic arithmetic L-fibration of V by L-targets. In the first situation, it is enough to understand the asymptotic of N(U, L, B) for the strongly L-saturated variety U (we note that the complement V \ U consists of low-dimensional irreducible components). In the second situation, it is enough to understand the asymptotic of N(W_{i}, L, B) for each of the L-targets W_{i} ⊂ V.

For our next reduction step, we need:

Conjecture 3.4.2 Let V be a smooth strongly L-saturated quasi-projective variety. Then the complex analytic variety V (C) is L-primitive.

Step 3 (reduction to L-primitive varieties): Every quasi-projective algebraic variety V is a disjoint union of finitely many locally closed smooth subvarieties V_{i}. Therefore, if one knows the asymptotic for each N(V_{i}, L, B) then one immediately obtains the asymptotic for N(V, L, B).

Definition 3.4.3 Let V be an L-primitive algebraic variety over a number field F, ρ : X → \overline{V}^{L} a desingularization over F of the closure of V with the exceptional locus consisting of smooth irreducible divisors D_{1}, …, D_{l}. We consider Pic(X) and Pic(V, L) as Gal(\overline{F} /F)-modules and we denote by β_{L}(V) the rank of Gal(\overline{F} /F)-invariants in Pic(V, L) and by δ_{L}(V) the cardinality of the cohomology group

H^{1}(Gal(\overline{F} /F), Pic(V, L)).

Remark 3.4.4 Using the long exact Galois-cohomology sequence associated with (3), one immediately obtains that β_{L}(V) and δ_{L}(V) do not depend on the choice of the resolution ρ.

Step 4 (expected asymptotic formula): Let V be a strongly Lsaturated (and L-primitive) smooth quasi-projective variety. Assume that a_{L}(V) > 0. Then we expect that the following asymptotic formula holds:

N(V, L, B) = c_{L}(V)B^{α_{L}(V)}(log B)^{β_{L}(V)−1} (1 + o(1)) ,

where

c_{L}(V) := \frac{γ_{L}(V)}{α_{L}(V)(β_{L}(V) − 1)!}δ_{L}(V)τ_{L}(V),

γ_{L}(V) is an invariant of the triple (Pic(V, L), Pic(V, L)_{R}, Λ_{eff}(V, L)) (2.3.16), δ_{L}(V) is a cohomological invariant of the Gal(\overline{F} /F)-module Pic(V, L) (3.4.3) and τ_{L}(V) is an adelic invariant of a family of v-adic metrics {∥ · ∥_{v}} on L (3.3.10).

In sections 4 and 5 we discuss some examples which show how the constants

α_{L}(V), β_{L}(V), δ_{L}(V), γ_{L}(V), τ_{L}(V)

appear in asymptotic formulas for the number of rational points of bounded L-height on algebraic varieties. Naturally, we expect that the exhibited behavior is typical. However, we also feel that one should collect more examples which could help to clarify the general situation.

3.5 L-primitive fibrations and τ_{L}(V)

We proceed to discuss our observations concerning the arithmetic conjecture 3.4.1 at its relation to the geometric conjecture 2.4.3.

Let V be a weakly L-saturated smooth quasi-projective variety with a_{L}(V) > 0 which is not strongly saturated and which doesn’t contain Zariski open dense strongly saturated subvarieties. We distinguish the following two cases: Case 1. V is not L-primitive. In this case we expect that some Zariski open dense subset U ⊂ V admits an L-primitive fibration which is defined by a projective regular morphism π : X → Y over F to a low-dimensional normal irreducible projective variety Y satisfying the conditions (i)-(iii) in 2.4.2, for an appropriate smooth projective compactification X of U (see 2.4.3). It seems natural to expect that all fibers satisfy the vanishing assumption 2.3.9. Thus we see that for any y ∈ Y (F) such that V_{y} = π^{−1}(y) ∩ V is L-primitive we can define the adelic number τ_{L}(V_{y}). Furthermore, we expect that every L-target W is contained in an appropriate L-primitive subvariety V_{y} which is a fiber of the L-primitive fibration π : V → U on V. In particular, Step 4 of our main strategy implies that if every L-target W coincides with a suitable L-primitive fiber V_{y} then one should expect the asymptotic

N(V, L, B) = c_{L}(V)B^{α_{L}(V)}(log B)^{β_{L}(V)−1} (1 + o(1)) ,

where the numbers α_{L}(V) (resp. β_{L}(V)) coincide with the numbers α_{L}(V_{y}) (resp. β_{L}(V_{y})) for the corresponding L-targets V_{y} and the constant c_{L}(V) is equal to the sum

∑_{y} c_{L}(V_{y}) = ∑_{y} \frac{γ_{L}(V_{y})}{α_{L}(V_{y})(β_{L}(V_{y}) − 1)!}δ_{L}(V_{y})τ_{L}(V_{y}),

where y runs over all points in Y (F) such that V_{y} is an L-target of V. It is natural to try to understand the dependence of τ_{L}(V_{y}) on the choice of a point y ∈ Y (F). We expect that the number τ_{L}(V_{y}) can be interpreted as a “height” of y. More precisely, the examples we considered suggest the following:

Conjecture 3.5.1 There exist a family of v-adic metrics on K_{Y} and two positive constants c_{2} > c_{1} > 0 such that

c_{1}H_{F}(y) ≤ τ_{L}(V_{y}) ≤ c_{2}H_{F}(y) ∀y ∈ Y (F) ∩ U,

where U ⊂ Y is some dense Zariski open subset and F is a metrized Qinvertible sheaf associated with the Q-Cartier divisor L_{1}^{−1} ⊗ K_{Y} (recall that L_{1} is the tautological ample Q-Cartier divisor on Y defined by the graded ring R(V, L) (see 2.4.4)).

Case 2. V is L-primitive (but not strongly saturated!). We don’t know examples of a precise asymptotic formula in this situation.

Example 3.5.2 Let V be a Fano diagonal cubic bundle over P^{3} with the homogeneous coordinates (X_{0} : X_{1} : X_{2} : X_{3}) defined as a hypersurface in P^{3} × P^{3} by the equation

X_{0}Y_{0}^{3} + X_{1}Y_{1}^{3} + X_{2}Y_{2}^{3} + X_{3}Y_{3}^{3} = 0

in P^{3} ×P^{3} (see [9]). We expect that V is not strongly saturated with respect to a metrized anticanonical sheaf L := O(3, 1) and that the corresponding L-targets are the splitting diagonal cubics in fibers of the natural projection π : V → P^{3} (this leads to the failure of the expected asymptotic formula in Step 4 for this example).

The next example was suggested to us by Colliot-Thélène:

Example 3.5.3 Let V be an analogous diagonal quadric bundle over P^{3} defined as a hypersurface in P^{3} × P^{3} by the equation

X_{0}Y_{0}^{2} + X_{1}Y_{1}^{2} + X_{2}Y_{2}^{2} + X_{3}Y_{3}^{2} = 0.

For infinitely many fibers V_{x} = π_{1}^{−1}(x) (x ∈ P^{3}(F)) we have rk Pic(V_{x}) = 2. At the same time, we have also rk Pic(V) = 2. We consider the height function associated to some metrization of the line bundle L := O(3, 2). On the one hand, we think that the asymptotic on the whole variety is c(V)B log B(1+o(1)) for B → ∞ with some c(V) > 0. On the other hand, if X_{0}X_{1}X_{2}X_{3} is a square in F we get already about B log B solutions. Another important observation is the expected convergency of the series

∑_{x∈P^{3}(F):V_{x}≅P^{1}×P^{1}} c(V_{x}).

The latter would be a consequence of the following two facts. First, the condition V_{x} ≅ P^{1} × P^{1} (x = (X_{0} : X_{1} : X_{2} : X_{3})) is equivalent to the conditions that V_{x} contains an F -rational point and that the product X_{0}X_{1}X_{2}X_{3} is a square in F. The number of F -rational points x′ = (X_{0} : X_{1} : X_{2} : X_{3} : Z) with H_{O(1)}(x′) ≤ B lying on the hypersurface with the equation X_{0}X_{1}X_{2}X_{3} = Z^{2} in the weighted projective space P^{4}(1, 1, 1, 1, 2) can be estimated from above by B^{2}(log B)^{3}(1 + o(1)). Secondly, we expect

c(V_{x}) = H_{O(4)}^{−1}(x)(1 + o(1)),

uniformly over the base P^{3}(F). This would imply the claimed convergency.

4 Height zeta-functions

4.1 Tauberian theorem

One of the main techniques in the proofs of asymptotic formulas for the counting function

N(V, L, B) := #{x ∈ V (F) : H_{L}(x) ≤ B }

has been the use of height zeta functions. Let L be an ample metrized invertible sheaf on a smooth quasi-projective algebraic variety X. We define the height zeta function by the series

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

which converges absolutely for Re(s) ≫ 0. After establishing the analytic properties of Z(X, L, s) one uses the following version of a Tauberian theorem:

Theorem 4.1.1 ([14]) Suppose that there exist an ε > 0 and a real number Θ(L) > 0 such that

Z(X, L, s) = \frac{Θ(L)}{(s − a)^{b}} + \frac{f(s)}{(s − a)^{b−1}}

for some a > 0, b ∈ N and some function f(s) which is holomorphic for Re(s) > a − ε. Then we have the following asymptotic formula

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

for B → ∞.

4.2 Products

Let X_{1} and X_{2} be two smooth quasi-projective varieties with ample metrized invertible sheaves L_{1} and L_{2} (resp. on X_{1} and X_{2}). Denote by X = X_{1} × X_{2} the product and by L the product of L_{1} and L_{2} (with the obvious metrization). Clearly,

Z(X, L, s) = Z(X_{1}, L_{1}, s) · Z(X_{2}, L_{2}, s).

Assume that for i = 1, 2 we have

Z(X_{i}, L_{i}, s) = \frac{Θ(L_{i})}{(s − α_{L_{i}}(X_{i}))^{β_{L_{i}}(X_{i})}} + \frac{f_{i}(s)}{(s − α_{L_{i}}(X_{i}))^{β_{L_{i}}(X_{i})}}

with some functions f_{i}(s) which are holomorphic in the domains Re(s_{i}) > α_{L_{i}}(X_{i}) − ε for some ε > 0. There are two possibilities:

Case 1: α_{L_{1}}(X) = α_{L_{2}}(X). In this situation the constant Θ(L) at the pole of highest order β_{L_{1}}(X_{1}) + β_{L_{2}}(X_{2}) is given by Θ(L) = Θ(L_{1})Θ(L_{2}).

Case 2: α_{L_{1}}(X) < α_{L_{2}}(X). In this situation the constant is a sum

Θ(L) = ∑_{x∈X_{1}(F)} H_{L_{1}}^{−α_{L_{2}}}(x)Θ(L_{2}).

Consider the projection X → X_{1} and denote by V_{x} the fiber over x ∈ X_{1}(F). We notice

τ_{L}(V_{x}) = H_{L_{1}}^{−α_{L_{2}}}(x)τ_{L_{2}}(X_{2}).

We denote by M the Q-Cartier divisor π_{1}^{∗}L_{1}^{α_{L_{2}}} ⊗ K_{V_{1}}. We obtain that π_{1}^{∗}L_{1}^{α_{L_{2}}} = M ⊗ K_{V_{1}}^{−1}. So we have τ_{L}(V_{x}) ∼ H_{M⊗K_{V_{1}}^{−1}}^{−1}. We observe that

Tamagawa numbers of fibers depend on the height of the points on the base.

4.3 Symmetric product of a curve

Let C be a smooth irreducible curve of genus g ≥ 2 over F. We denote by X = C^{(m)} the m-th symmetric product of C and by Y := Jac(C) the Jacobian of C. We fix an m > 2g − 2. We have a fibration

π : C^{(m)} → Y,

with P^{m−g} as fibers. We denote by V_{y} a fiber over y ∈ Y (F).

Let C̃ → Spec(O_{F}) be a smooth model of C over the integers and L an ample hermitian line bundle on C̃. It defines a height function

H_{L} : C(F) → R_{>0}

which extends to X(F). Observe that α_{L}(X) = (m + 1 − g)/d, where d := deg_{Q}(L). Consider the height zeta function

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

This function was introduced by Arakelov in [3].

Theorem 4.3.1 [17] Let L be an ample hermitian line bundle on C̃. There exist an ε(L) > 0 and a real number Θ(L) ≠ 0 such that the height zeta function has the following representation

Z(X, L, s) = \frac{Θ(L)}{(s − α_{L}(X))} + f(s)

with some function f(s) which is holomorphic for Re(s) > α_{L}(X) − ε(L).

This is Theorem 8 in ([17], p. 422). Arakelov gives an explicit expression for the constant Θ(L) ([3]). We are very grateful to J.-B. Bost for pointing out to us that Arakelov’s formula is not correct and for allowing us to use his notes on the Arakelov zeta function.

Theorem 4.3.2 With the notations above we have

Θ(L) = ∑_{y∈Y (F)} τ_{L}(V_{y}).

Proof. We outline the proof for F = Q. For Re(s) ≫ 0 one can rearrange the order of summation and one obtains

Z(X, L, s) := ∑_{y∈Y (F)} ∑_{x∈V_{y}(F)} H_{L}(x)^{−s}.

It is proved in ([17], p. 420-422) that the sums

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

have simple poles at s = α_{L}(X) with non-zero residues and that one can “sum” these expressions to obtain a function with a simple pole at s = α_{L}(X) and meromorphic continuation to Re(s) > α_{L}(X) − ε(L) for some ε(L) > 0. Moreover, the residue at this pole is obtained as a sum over y ∈ Y (F) of the residues of Z(V_{y}, L, s).

Choosing an element z(y) in the class of y ∈ Jac(C)(F), and denoting by E(y) := Γ(C, O(z(y))) one can identify the fiber as V_{y} = P(E(y)), where f ∈ E(y) = Γ(C, O(z(y))) is mapped to z(y) + div(f). The height is given by the formula

H_{L}(z(y) + div(f)) := H_{L}(z(y)) exp(∫_{C(C)} log |f|c_{1}(L)).

This defines a metrization of the anticanonical line bundle on V_{y} = P(E(y)). Assuming the Tamagawa number conjecture for P^{n} (for suitable metrizations of the line bundle O(1) on P^{n}) we obtain

lim_{s→α_{L}(X)}(s − α_{L}(X))Z(V_{y}, L, s) = τ_{L}(V_{y}).

□

One can write down an explicit formula for τ_{L}(V_{y}). For f ∈ E(y)_{C}\{0} we define

Φ(f) := exp(\frac{1}{d} ∫_{C(C)} log |f|c_{1}(L))

and we put Φ(0) = 0. It follows that

τ_{L}(V_{y}) = \frac{1}{2}α_{L}(X)H_{L}(y)^{−α_{L}(X)} · \frac{vol(\{f ∈ E(y)_{R} | Φ(f) ≤ 1\})}{vol(E(y)_{R}/E(y))},

where the volumes are calculated with respect to some Lebesgue measure on E(z(y))_{R}. Arakelov relates this last expression to the Neron-Tate height of y ∈ Jac(C). A detailed calculation due to Bost indicates that Arakelov’s formula is correct only up-to O(1).

4.4 Homogeneous spaces G/P

Let G be a split semisimple linear algebraic group defined over a number field F. It contains a Borel subgroup P_{0} defined over F and a maximal torus which is split over F. Let P be a standard parabolic. Denote by Y_{P} = P \G (resp. X = P_{0}\G) the corresponding flag variety.

A choice of a maximal compact subgroup K such that G(A_{F}) = P_{0}(A_{F})K defines a metrization on every line bundle L on the flag varieties Y_{P} ([19], p. 426). We will denote by

H_{L} : P (F)\G(F) → R_{>0}

the associated height. We consider the height zeta function

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

Theorem 4.4.1 Let L be an ample metrized line bundle on X. There exist an ε(L) > 0 and a real number Θ(L) ≠ 0 such that the height zeta function has the following representation

Z(X, L, s) = \frac{Θ(L)}{(s − α_{L}(X))^{β_{L}(X)}} + \frac{f(s)}{(s − α_{L}(X))^{β_{L}(X)−1}}

with some function f(s) which is holomorphic for Re(s) > α_{L}(X) − ε(L).

This theorem follows from the identification of the height zeta function with an Eisenstein series and from the work of Langlands. The formula (2.10) in ([19], p. 431) provides an expression for Θ(L) which we will now analyze.

There is a canonical way to identify the faces of the closed cone of effective divisors Λ_{eff}(X) ⊂ Pic(X)_{R} with Λ_{eff}(Y_{P}) as P runs through the set of standard parabolics. A line bundle L such that its class is contained in the interior of the cone ∈ Λ_{eff}(X) defines a line bundle

[L_{Y}] := α_{L}(X)[L] + [K_{X}]

which is contained in the interior of the face Λ_{eff}(Y) ⊂ Λ_{eff}(X) for some Y = Y_{P}. We have a fibration π_{L} : X → Y with fibers isomorphic to the flag variety V := P_{0}\P. A fiber over y ∈ Y (F) will be denoted by V_{y}. Denote by K_{Y} the canonical line bundle on Y with the metrization defined above.

Theorem 4.4.2 We have

Θ(L) = ∑_{y∈Y (F)} γ_{L}(X)τ_{L}(V_{y}).

Proof. In the domains of absolute and uniform convergence we can rearrange the order of summation and we obtain

Z(X, L, s) = ∑_{y∈Y (F)} ∑_{x∈P_{0}(F)\P (F)} H_{L}(yx)^{−s}.

One can check that the sums

Z(V_{y}, L, s) := ∑_{x∈V_{y}(F)} H_{L}(x)^{−s} = ∑_{x∈P_{0}(F)\P (F)} H_{L}(yx)^{−s}

have poles at s = α_{L}(X) of order β_{L}(X) with non-zero residues, and that they admit meromorphic continuation to α_{L}(X) − ε(L) for some ε(L) > 0. Moreover, the constant Θ(L) is obtained as a sum over y ∈ Y (F) of the residues of Z(V_{y}, L, s). From the Tamagawa number conjecture for P_{0}\P (with varying metrizations of the anticanonical line bundle) we obtain

lim_{s→α_{L}(X)}(s − α_{L}(X))^{β_{L}(X)}Z(V_{y}, L, s) = γ_{L}(X)τ_{L}(V_{y}).

The sum

∑_{y∈Y (F)} γ_{L}(X)τ_{L}(V_{y})

converges for [L_{Y}] contained in the interior of Λ_{eff}(Y). □

Let us recall the explicit formula for τ_{L}(V_{y}) (see (2.9) in [19], p. 431):

lim_{s→α_{L}(X)}(s − α_{L}(X))^{β_{L}(X)} ∑_{x∈P_{0}(F)\P (F)} H_{L}(xy)^{−s} = γ_{1}c_{P} H_{L_{Y}^{−1}⊗K_{Y}} (y)

where γ_{1} ∈ Q is an explicit constant and the constant c_{P} is defined in ([19], p. 430). It follows that

Θ(L) = γ_{1}c_{P} ∑_{y∈Y (F)} H_{L_{Y}^{−1}⊗K_{Y}} (y).

Next we observe that there is an explicit constant γ_{2} ∈ Q such that we have

γ_{2}τ_{L}(V_{y}) = c_{P} · H_{L_{Y}^{−1}⊗K_{Y}} (y)

for all y ∈ Y (F). To see this, we first identify c_{P} = γ_{2}τ_{L}(V), this is done by a computation of local factors of intertwining operators ([30], p.160-161). The next step involves the comparison of Tamagawa measures on V_{y} for varying y ∈ Y (F). Finally, we have γ_{L}(X) = γ_{1}γ_{2}.

4.5 Toric varieties

There are many equivalent ways to describe a toric variety X over a number field F together with some projective embedding (see, for example, [12, 25, 13]). For us, it will be useful to view X = X_{Σ} as a collection of the following data ([6]):

1. A splitting field E of the algebraic torus T and the Galois group G = Gal(E/F).

2. The lattice of E-rational characters of T, which we denote by M and its dual lattice N.

3. A G-invariant complete fan Σ in N_{R} = N ⊗ R.

There is an isomorphism between the group of G-invariant integral piecewise linear functions φ ∈ P L(Σ)^{G} and classes of T -linearized line bundles on X_{Σ}. For φ ∈ P L(Σ)^{G} we denote the corresponding line bundle by L(φ).

We define metrizations of line bundles as follows. Let G_{v} ⊂ G be the decomposition group at v. We put N_{v} = N^{G_{v}} for the lattice of G_{v}-invariants of N for non-archimedean valuations v and N_{v} = N_{R}^{G_{v}} for archimedean v. We have the logarithmic map

T (F_{v})/T (O_{v}) → N_{v}

which is an embedding of finite index for all non-archimedean v, an isomorphism of lattices for almost all non-archimedean valuations and an isomorphism of real vector spaces for archimedean valuations. We denote by \overline{t}_{v} the image of t_{v} ∈ T (F_{v}) under this map.

Definition 4.5.1 ([6], p. 607) For every φ ∈ P L(Σ)^{G} and t_{v} ∈ T (F_{v}) we define the local height function

H_{Σ,v}(t_{v}, φ) := e^{φ(\overline{t}_{v}) log q_{v}}

where q_{v} is the cardinality of the residue field of F_{v} for non-archimedean valuations and log q_{v} = 1 for archimedean valuations. For t ∈ T (A_{F}) we define the global height function as

H_{Σ}(t, φ) := ∏_{v∈Val(F)} H_{Σ,v}(t_{v}, φ).

We proved in ([6], p. 608) that this pairing can be extended to a pairing

H_{Σ} : T (A_{F}) × P L(Σ)_{C}^{G} → C

and that it defines a simultaneous metrization of T -linearized line bundles on X. We will denote such metrized line bundles by L = L(φ). We consider the height zeta function

Z(T, L, s) = ∑_{t∈T (F)} H_{L}(t)^{−s}.

Theorem 4.5.2 ([8]) Let L be an invertible sheaf on X (with the metrization introduced above) such that its class [L] is contained in the interior of Λ_{eff}(X). There exist an ε(L) > 0 and a Θ(L) > 0 such that the height zeta function has the following representation

Z(L, T, S) = \frac{Θ(L)}{(s − α_{L}(X))^{β_{L}(X)}} + \frac{f(s)}{(s − α_{L}(X))^{β_{L}(X)−1}}

where f(s) is a function which is holomorphic for Re(s) > α_{L}(X) − ε(L).

Remark 4.5.3 The computation of the constants α_{L}(X) and β_{L}(X) in specific examples is a problem in linear programming. For the anticanonical line bundle on a smooth toric variety X we have α_{K^{−1}}(X) = 1 and β_{K^{−1}}(X) = dim P L(Σ)_{R}^{G} − dim M_{R}^{G}.

Our goal is to identify the constant Θ(L). Let us recall some properties of toric varieties and introduce more notations (see [6]). The cone of effective divisors Λ_{eff}(X) is generated by the classes of irreducible components of X\T which we denote by [D_{1}], ..., [D_{r}]. These divisors correspond to Galois orbits Σ_{1}(1), ..., Σ_{r}(1) on the set of 1-dimensional cones in Σ. The line bundle L defines a face Λ(L) of Λ_{eff}(X). We denote by J = J(L) the maximal set of indices J ⊂ [1, ..., r] such that we have

α_{L}(X)[L] + [K_{X}] = ∑_{j∈J} r_{j}[D_{j}]

with [D_{j}] ∈ Λ(L) and some r_{j} ∈ Q_{>0}. We denote by I = I(L) the set of indices i ∉ J(L). We denote by M_{J} the lattice given by

M_{J} := {m ∈ M | < e, m >= 0 R_{≥0}e ∈ ∪_{i∈I}Σ_{i}(1)},

We denote by M_{I} := M/M_{J} and by N_{∗} the corresponding dual lattices. We have an exact sequence of algebraic tori

1 → T_{I} → T → T_{J} → 1

which induces a map π_{L} : T (F) → T_{J}(F) with finite cokernel and an exact sequence of lattices

0 → N_{I} → N → N_{J} → 0.

The restriction of the fan Σ ⊂ N_{R} to N_{I,R} will be denoted by Σ_{I}. It is again a G-invariant fan and it will define an equivariant compactification X_{I} of T_{I}. The class of the piecewise linear function φ_{I} ∈ P L(Σ_{I})^{G} with φ_{I}(e) = 1 for e ∈ ∪_{i∈I}Σ_{i} corresponds to the class of the anticanonical line bundle [−K_{I}] ∈ Pic(X_{I}).

The line bundle L defines a fibration of varieties π_{L} : X_{Σ} → Y with fibers isomorphic to X_{I}, which, when restricted to T, gives rise to the exact sequence of tori above. We denote the fiber over y ∈ T_{J}(F) by X_{I,y}.

Theorem 4.5.4 We have

Θ(L) = γ_{L}(X)δ_{L}(X) ∑_{y∈π_{L}(T (F))} τ_{L}(X_{I,y}).

Proof. Let L = L(φ) be an invertible sheaf on T with the metrization introduced above. In the domain of absolute and uniform convergence we can rearrange the order of summation and we obtain

Z(T, L, s) = ∑_{y∈π_{L}(T (F))} ∑_{x∈T_{I,y}(F)} H_{Σ}(yx, φ)^{−s}.

From the proof of our main theorem in [8] it follows that the sums

Z(T_{I,y}, L, s) = ∑_{x∈T_{I}(F)} H_{Σ}(yx, φ)^{−s}

have a pole at α_{L}(X) of order β_{L}(X). Moreover, the constant Θ(L) is obtained as a sum over y ∈ π_{L}(T (F)) of residues of Z(T_{I,y}, L, s). Now we want to use the Tamagawa number conjecture for the anticanonical line bundle (with varying metrizations) on the toric variety X_{I} to conclude the proof.

In [7] we proved this conjecture for a specific metrization and under the assumption that the fan Σ is regular. We want to demonstrate that our proof goes through in the general case needed above.

Our main idea was to use the Poisson summation formula on the adelic group T (A_{F}) and to obtain an integral representations for the height zeta function. We denote by A_{I} = (T_{I}(A_{F})/KT_{I}(F))^{∗} the group of unitary characters of T_{I}(A_{F}) which are trivial on T_{I}(F) and on the maximal compact subgroup K ⊂ T_{I}(A_{F}). Using the the adelic definition of the height function we obtain

Z(T_{I,y}, L, s) = ∑_{t∈T_{I}(F)} H_{L}(yt)^{−s} = ∫_{A_{I}} dχ ∫_{T_{I}(A_{F})} H_{L}(yt)^{−s}χ(t)dμ,

where dμ is a Haar measure on T (A_{F}) and dχ is the orthogonal Haar measure on A_{I}. To apply our technical theorem in [7] about the analytic continuation and the residues of such integrals we need to know that

∫_{T_{I}(A_{F})} H_{L}(yt)^{−s}χ(t)dμ = ∏_{i∈I} L_{i}(χ_{i}, s) · ζ_{Σ_{I}}(χ, s) · ζ_{∞}(χ, s)

where

ζ_{Σ_{I}}(χ, s) = ∏_{v∈Val(F)} ζ_{Σ_{I},v}(χ, s)

is an absolutely convergent Euler product for Re(s) > α_{L(X)} − ε(L), L_{i}(χ_{i}, s) are Artin L-functions (with some induced characters χ_{i}) and ζ_{∞}(χ, s) satisfies certain growth conditions.

First we observe that it is unnecessary to assume that the fan Σ_{I} is regular. Using our definition of the height function we see that the calculation of the Fourier transform (see [6]) reduces to summations of the function q_{v}^{φ(\overline{t_{v}})+im_{v}} (m_{v} ∈ M_{I,R}) over the lattice N_{I,v} (resp. to integrations in cones for archimedean valuation). A piecewise linear function φ induces a piecewise linear function on any subdivision of the fan. Clearly, the result of such summations and integrations does not depend on any subdivisions.

Next we see that for a fixed y ∈ T_{J}(F) we have H_{L,v}(yt) = H_{L,v}(t) for almost all v and all t ∈ T_{I}(A_{F}). Now we can refer to lemma 5.10 in [8] which proves the required statement. The local integrals for the remaining finitely many non-archimedean valuations will be absorbed into ζ_{Σ_{I}}(χ, s). And finally, we need to check that the estimates of the Fourier transform of H_{L,v}(yt) at archimedean valuations are still satisfied for any y ∈ T_{J}(F). This is straightforward.

We can now apply the main technical theorem of [7] and obtain

lim_{s→α_{L}}(s − α_{L}(X))^{β_{L}(X)}Z(T_{I,y}, L, s) = γ_{L}(X)δ_{L}(X)τ_{L}(X_{I,y}).

□

Remark 4.5.5 It is possible to compute τ_{L}(X_{I,y}) and to observe that it is related to the height of the point y ∈ T_{J}(F).

Remark 4.5.6 Similar statements hold for equivariant compactifications of homogeneous spaces G/U where G is a split reductive group and U is its maximal unipotent subgroup [33]. We hope that these results can be extended to equivariant compactifications of other homogeneous spaces, in particular, to equivariant compactifications of reductive and non-reductive groups.

5 Singular Fano varieties

5.1 Weighted projective spaces

Let W := P(w) = P(w_{0}, …, w_{n}) be a weighted projective space of dimension n with weights w = (w_{0}, …, w_{n}). We remark that W is a rational variety over Q with Pic(W) ≅ Z. Moreover, the anticanonical class K_{W}^{−1} is an ample Q-Cartier divisor. So W is a (singular) Fano variety of index

r = \frac{w_{0} + ⋯ + w_{n}}{l.c.m.\{w_{0}, …, w_{n}\}}.

One could try to generalize the method of Schanuel [31] for counting Qrational points of bounded height on usual projective spaces to the case of weighted projective spaces. Let z_{0}, z_{1}, …, z_{n} be homogeneous coordinates on Y. Then a first approximation to counting points of bounded height would be a counting of all (n + 1)-tuples (x_{0}, x_{1}, …, x_{n}) ∈ Z^{n} \ {0} satisfying the conditions

|x_{i}| ≤ B^{\frac{w_{i}}{w_{0}+w_{1}+···+wn}} i = 0, …, n.

Since the volume of the domain restricted by these inequalities is

B = ∏_{i=0}^{n} B^{\frac{w_{i}}{w_{0}+w_{1}+···+wn}} ,

one could expect that the asymptotic number of solutions of these inequalities agrees with “expected” linear growth for the anticanonical height. However, this “intuition” turns out to be wrong, in general, because the singularities of Y could be even worse than canonical. A typical class of singularities that appear on Y are so called log-terminal singularities introduced by Kawamata [28]. We give below a simple example of a Del Pezzo surface with a logterminal singularity and we show that for every dense Zariski open subsets U ⊂ W the number N(U, B) of F -rational points of anticanonical height ≤ B in U has more than linear growth:

N(U, B) = c(U)B^{2− \frac{4}{m+2}} (1 + o(B)).

Moreover, there are no dense Zariski open subsets U′ ⊂ X such that the adelic term in the constant c(U) in the asymptotic formula for N(U, B) would be independent of U for U ⊂ U′.

Example 5.1.1 (Del-Pezzo surface with a log-terminal singularity) Let W = P(1, 1, m) be a singular weighted projective plane with weights (1, 1, m), m ≥ 2. Then the anticanonical class of W is an ample Q-Cartier divisor (i.e., W is Del Pezzo surface) and p = (0 : 0 : 1) is the unique singularity of W. Let X → W be the minimal resolution of the singularity at p ∈ W. Then X is isomorphic to a ruled surface F_{m} = P(O_{P^{1}} ⊕ O_{P^{1}}(m)) and the exceptional divisor E = f^{−1}(p) is a smooth rational curve which is a section of the P^{1}-bundle over P^{1} and ⟨E, E⟩ = −m. Then we have

K_{X} = f^{∗}K_{W} + \frac{2 − m}{m}E.

Therefore, p is canonical ⇔ m = 2 and p is log-terminal ⇔ m ≥ 3.

The group Pic(X) is isomorphic to Z^{2} where {[E], [C]} is a Z-basis. Moreover, [E], [C] are generators for the cone Λ_{eff}(X) of effective divisors in Pic(X)_{R}. We have

K_{X} = −2E − (m + 2)C,

L := f^{∗}(−K_{X}) = \frac{m + 2}{m}E + (m + 2)C

and

α_{L}(W) = inf\{t ∈ Q : t[L] + [K_{X}] ∈ Λ_{eff}(X)\} = \frac{2m}{m + 2}.

Since X is a smooth toric variety, we can apply our main result in [8] and obtain the following:

Theorem 5.1.2 Let π : W \ p → P^{1} be the natural projection, C_{x} the fiber of π over x ∈ P^{1}(Q). Then for any dense Zariski open subset in W \ p, one has

N(U, B) = c(U)B^{2− \frac{4}{m+2}} (1 + o(B))

Moreover,

c(U) = ∑_{x∈P^{1}(Q)∩π(U)} c(C_{x}).

5.2 Vaughan-Wooley cubic

Example 5.2.1 Let Y ⊂ P^{5} be a singular cubic defined by the equation z_{0}z_{1}z_{2} − z_{3}z_{4}z_{5} = 0, X the intersection of Y with the linear subspace in P^{5} with the equation:

z_{0} + z_{1} + z_{2} − z_{3} − z_{4} − z_{5} = 0.

Vaughan and Wooley proved [34]:

Theorem 5.2.2 Let U ⊂ X be the complement in X to the following 15 divisors D_{i_{1}i_{2}i_{3}}, D_{ij} ⊂ X ({i_{1}, i_{2}, i_{3}} = {0, 1, 2}, i ∈ {0, 1, 2}, j ∈ {3, 4, 5}), where

D_{i_{1}i_{2}i_{3}} = {(z_{0} : …: z_{5}) ∈ P^{5} : z_{i_{1}} = z_{3}, z_{i_{2}} = z_{4}, z_{i_{3}} = z_{5}}

D_{ij} = {(z_{0} : …: z_{5}) ∈ X : z_{i} = z_{j} = 0}.

If N(U, B) is the number of Q-rational points in U of the anticanonical height ≤ B, then there exist some constants c_{1} > c_{2} > 0 such that

c_{2}B^{2}(log B)^{5} ≤ N(U, B) ≤ c_{1}B^{2}(log B)^{5}.

We want to show that this result is compatible with predictions in [4]. First of all we note that Y is a 4-dimensional toric Fano variety: an equivariant compactification of a 4-dimensional algebraic torus T with respect to a 4-dimensional polyhedron Δ with 6 lattice vertices

v_{0} = (0, 0, 0, 0), v_{1} = (1, 0, 0, 0), v_{2} = (0, 1, 0, 0),
v_{3} = (0, 0, 1, 0), v_{4} = (0, 0, 0, 1), v_{5} = (1, 1, −1, −1)

(Δ is the support of global sections of a very ample divisor Y corresponding to the embedding Y ↪ P^{4}). The polyhedron Δ has 9 faces Θ_{ij} of codimension 1:

Θ_{ij} = Conv({v_{0}, v_{1}, v_{2}, v_{3}, v_{4}, v_{5}} \ {v_{i}, v_{j}}).

Each face Θ_{ij} defines an torus invariant divisor Y_{ij} ⊂ Y \ T such that D_{ij} = Y_{ij} ∩ X. It is easy to check that all singularities of Y are at worst terminal and that the hypersurface X ⊂ Y intersects all strata Y_{ij} transversally. From these facts we obtain that the only exceptional divisors with the discrepancy 0 that appear in a resolution of singularities of X come from singularities in X ∩ T. We write down the affine equation of X ∩ T as

1 + x + y − z − t − \frac{xy}{zt} ,

where T = Spec Q[x^{±1}, y^{±1}, z^{±1}, t^{±1}]. From this equation one immediately sees that the only singularities in X ∩ T are the A_{1}-double points lying on the curve C : x = y = z = t. Therefore, we obtain rk Cl(X) = rk Cl(Y) = 9 − dim T = 5. Moreover, there exists exactly one crepant divisor (over C) in a resolution of singularities of X. So the predicted power of log B in the asymptotic formula for N(U, B) is (rk Cl(X) − 1) + 1 = 5.

5.3 Cubic xyz = u^{3}

We consider the singular cubic surface X ⊂ P^{3} over Q given by the homogeneous equation xyz = u^{3}. This is a toric variety, an equivariant compactification of the torus

T = SpecQ[x, y, z]/(xyz − 1)

given by the condition u ≠ 0. We can fix an isomorphism T ≅ G_{m}^{2} by choosing {x, y} as a basis of the group of algebraic characters of T. Consider the problem of the computation of the asymptotic of

N(T, B) = Card{(x, y) ∈ (Q^{∗})^{2} : H(x, y) ≤ B }

for B → ∞, where

H(x, y) = ∏_{v∈Val(Q)} max{∥x∥_{v}, ∥y∥_{v}, ∥(xy)^{−1}∥_{v}, ∥1∥_{v}}

This problem is addressed in [18]. We would like to use this problem as a down-to-earth illustration of our general theory of height zeta functions of toric varieties. First of all we note that the relation ∥x∥_{v}∥y∥_{v}∥(xy)^{−1}∥_{v} = ∥1∥_{v} = 1 implies that

H_{v}(x, y) := max{∥x∥_{v}, ∥y∥_{v}, ∥(xy)^{−1}∥_{v}}.

Since l_{1} = log ∥x∥_{v}, l_{2} = log ∥y∥_{v}, l_{3} = log ∥(xy)^{−1}∥_{v} are linear functions on the logarithmic space N_{R,v} ≅ R^{2}:

N_{R,v} = \{ \begin{matrix}T (Q_{v})/T (O_{v}) ⊗_{Z} R, & if v = p ∈ Spec Z \\ T (Q_{v})/T (O_{v}), & if v = ∞, \end{matrix}

we can consider log h_{v}(x, y) as a piecewise linear function on N_{R,v}. Let e_{1} = (−2, 1), e_{2} = (1, −2) and e_{3} = (1, 1) be lattice vectors in Z^{2} ⊂ R^{2}. We define the following 3 convex cones in R^{2}:

σ_{1} = R_{≥0}e_{2} + R_{≥0}e_{3},
σ_{2} = R_{≥0}e_{1} + R_{≥0}e_{3},
σ_{3} = R_{≥0}e_{1} + R_{≥0}e_{2}.

Then N_{R,v} = ⋃_{i=1}^{3} σ_{i} and the restriction of log H_{v}(x, y) to σ_{i} coincides with the linear function l_{i}. Let

A := ⨁_{v∈Val(Q} T (Q_{v})/T (O_{v})

be the logarithmic adelic group. In order to compute the height zeta function

Z(s) = ∑_{(x,y)∈(Q^{∗})^{2}=T (Q)} H(x, y)^{−s}

we use the natural homomorphism Log of T (Q) to A. Denote by B the subgroup Log(T (Q)) ⊂ A. We remark that the kernel of Log consists of 4 elements of finite order in (Q^{∗})^{2} and the quotient A/B is isomorphic to R^{2}. Moreover the functions H_{v}(x, y)^{−s} on each T (Q_{v})/T (O_{v}) define a natural extension of h(x, y)^{−s} to a function on A. So we obtain:

Z(s) = 4 ∑_{b∈B} ∏_{v∈Val(Q)} H_{v}(b_{v})^{−s}  (4)

The main idea of our proof in [6] is to apply the Poisson summation formula on the group A and to express the height zeta function Z(s) as an integral

Z(s) = \frac{4}{(2π)^{2}} ∫_{R^{2}} (∏_{p} Q_{p}(s, im) · Q_{∞}(s, im)) dm,

where m = (m_{1}, m_{2}) ∈ R^{2}, dm = dm_{1}dm_{2},

Q_{p}(s, im) = ∑_{(b_{1,p},b_{2,p})∈Z^{2}} h_{p}(b_{p})^{−s}p^{i<b,m>},

and

Q_{∞}(s, im) = ∫_{R^{2}} H_{∞}(b)^{−s} exp(i < b, m >).

An exact computation of Q_{p}(s, im) and Q_{∞}(s, im) can be obtained by a subdivision of each of the cones σ_{1}, σ_{2}, σ_{3} into a union of 3 subcones generated by a basis of the lattice Z^{2} ⊂ R^{2}. (From the viewpoint of toric geometry this means that we reduce the counting problem for rational points in a torus with respect to a singular compactification to a counting problem for rational points in a torus with respect to the minimal resolution of singularities of this compactification). This calculation is done in [6], Section 2 for arbitrary smooth toric varieties. What remains is the analytic continuation of the integral and the identification of the constant at the leading pole. For this it is necessary to work on the whole complexified space P L(Σ)_{C} and to invoke the technical theorems in Section 6 in [7]. Applying the main theorem of [7], we obtain

N(T, B) = \frac{γ_{K^{−1}}(X)δ_{K^{−1}}(X)τ_{K^{−1}}(X)}{6!}B(log B)^{6}(1 + o(1))

for B → ∞. The constants are as follows: γ_{K^{−1}}(X) = 1/36, δ(X) = 1 and τ_{K^{−1}}(X) = τ_{K^{−1}}(X)_{∞} ∏_{p} τ_{K^{−1}}(X)_{p} where

τ_{K^{−1}}(X)_{p} = (1 + \frac{7}{p} + \frac{1}{p^{2}} ) · (1 − 1/p)^{7}

for all primes p and τ_{K^{−1}}(X)_{∞} = 9 · 4. Similar statements hold over any number field. One can compute the constant γ_{K^{−1}}(X) by observing that Λ_{eff}^{∗}(X) (the dual cone to the cone of effective divisors) is a union of two simplicial cones.

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Mathematisches Institut, Universität Tübingen Auf der Morgenstelle 10, 72076 Tübingen, Germany e-mail: batyrev@bastau.mathematik.uni-tuebingen.de and Dept. of Mathematics, U.I.C. Chicago, (IL) 60607-7045, U.S.A. e-mail: yuri@math.uic.edu