POINTS OF BOUNDED HEIGHT ON EQUIVARIANT COMPACTIFICATIONS

OF VECTOR GROUPS, I

by

Antoine Chambert-Loir & Yuri Tschinkel

Abstract. — We prove asymptotic formulas for the number of rational points of bounded height on certain equivariant compactifications of the affine plane. Résumé. — Nous établissons un développement asymptotique du nombre de points rationnels de hauteur bornée sur certaines compactifications équivariantes du plan affine.

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1. Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2. Heights, Poisson formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3. Projective space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4. Blow-ups of P^{2} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Introduction

Let X be a smooth projective algebraic variety defined over a number field F and X(F) the set of rational points of X. Let ℒ be a metrized ample line bundle and

H_{ℒ} : X(F) → R_{>0}

the associated exponential height (cf. [17], [18]). We are interested in the asymptotic behavior of the counting function

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

as B → ∞, where U ⊂ X is some Zariski open subset. There is a conjectural framework describing such asymptotics for varieties with ample (or, more generally, effective) anticanonical class (cf. [7], [1], [14], [17], [2] and references therein). In particular, it is expected that

N(U, K_{X}^{−1}, B) = \frac{Θ(X)}{r!}B(log B)^{r}(1 + o(1)),

as B → ∞, for some appropriate U ⊂ X. Here K_{X}^{−1} is the metrized anticanonical line bundle on X, r = rk Pic(X)−1 and Θ(X) is a product of a Tamagawa type number τ(K_{X}) (which depends on the metrization), a rational number α(X) defined in terms of the cone of effective divisors Λ_{eff}(X) and the order of the non-trivial part of the Brauer group | Br(X)/ Br(F)|. There is a similar description for arbitrary polarizations ℒ (cf. [2]).

These conjectures have been proved flag varieties ([7]), toric varieties ([2]), and toric bundles induced from torsors ([21], [5], [4]). The proofs use a precise combinatorial description of all geometric and arithmetic invariants of the varieties: line bundles, metrizations of the line bundles etc. (for example, in terms of lattices, cones and fans). Such a description is possible because representations of reductive groups are rigid (don’t admit deformations). Consequently, the corresponding varieties don’t have moduli.

The only other known approach to asymptotics of rational points on algebraic varieties is the classical circle method in analytic number theory. Varieties which can be treated by this method do admit moduli. However, one of the drawbacks is that so far it works only for varieties which are complete intersections of small degree d in projective spaces P^{n} of large dimension (very roughly, n ≫ 2^{d}) (mild singularities are allowed). In particular, these complete intersections have Picard group Z. There is a promising generalization of the circle method to complete intersections in other varieties (for example, toric varieties) due to E. Peyre (cf. [15]), which should provide examples of asymptotics for varieties with moduli and with Pic(X) of higher ranks, once the necessary estimates are established. As a reference to the circle method let us mention the papers by H. Davenport, R. Heath-Brown, Ch. Hooley on smooth cubic hypersurfaces (cf. [6], [10], [11]), by B. Birch and by W. Schmidt on general complete intersections (cf. [3], [20]).

In this paper we prove asymptotics of rational points of bounded height on varieties which admit moduli and which at the same time are closely relatated to (non-reductive) linear algebraic groups. More precisely, we consider equivariant compactifications of the additive group G_{a}^{n}. For n = 2 it can be shown that all such compactifications are obtained as blow-ups of P^{2}, or Hirzebruch surfaces in points which are fixed under the action of G_{a}^{2}. Notice that a variety (even P^{2}) may admit non-isomorphic structures as an equivariant compactification of G_{a}^{n}. A similar “minimal model program” of equivariant compactifications of G_{a}^{n} is a non-trivial problem already for n = 3 (work in progress [9]).

In this paper we study in detail the example of a blow-up of P^{2} in r Qrational points which are all contained in the line at infinity P^{1} (with the equation x_{0} = 0). The moduli space of such surfaces X is ℳ_{0,r}. It is easy to see that X is a smooth projective equivariant compactification of G_{a}^{2} with Pic(X) = Z^{r+1}, trivial Brauer group and a simplicial cone of effective divisors Λ_{eff}(X). Denote by U ≃ G_{a}^{2} ⊂ P^{2} the complement to x_{0} = 0. Then the series

Z(U, K_{X}^{−1}, s) = ∑_{x∈U(Q)} H_{K_{X}^{−1}}(x)^{−s}

is absolutely and uniformly convergent to a holomorphic function for Re(s) ≫ 1. One of the main results of this paper is the following:

Theorem 1. —

Z(U, K_{X}^{−1}, s) = \frac{h(s)}{(s − 1)^{r+1}} ,

where h(s) is holomorphic for Re(s) > 1 − δ (for some δ > 0) and

h(1) = α(X)τ(K_{X}) ≠ 0.

Corollary 2. — We have the following asymptotic formula:

N(U, K_{X}^{−1}, B) = \frac{α(X)τ(K_{X})}{r!}B(log B)^{r}(1 + o(1)),

as B → ∞.

In fact, we will prove asymptotics for every ℒ on X such that its class is contained in the interior of Λ_{eff}(X). We will also give estimates for the growth of the function h(s) in vertical strips in the neighborhood of Re(s) = 1. This implies more precise asymptotic expansions for the counting function N(U, ℒ , B).

We will address the compactifications of G_{a}^{n} (with n > 2) in subsequent papers.

Acknowledgements. The work of the second author was partially supported by the NSA.

1. Geometry

1.1. Generalities. — Let G be an algebraic group and X a smooth projective variety with an action of G. We denote by Pic^{G}(X) the (abelian) group of isomorphy classes of G-linearized line bundles on X (cf. [13], Chap. 1, § 3, Def. 1.6). We shall say that the variety X is an equivariant compactification of G if X has an open dense subset U which is equivariantly isomorphic to G. Well known examples are given by toric varieties which are equivariant compactifications of tori (algebraic groups isomorphic to G_{m}^{n} over the algebraic closure of F, where G_{m} = Spec(F [x, x^{−1}]) is the multiplicative group scheme).

In this paper, we are interested in equivariant compactifications of G_{a}^{n}, where G_{a} = Spec(F [x]) is the additive group scheme (we may call them addic varieties^{(∗)}).

Notice that a variety can be an equivariant compactification of a group G in many non-isomorphic ways, as the following example shows.

Example 1.2. — The projective plane P^{2} is an equivariant compactification of G_{a}^{2} in (essentially) two non-isomorphic ways. One of the possible actions is the standard translation action, fixing a line P^{1} at infinity. All 1-parameter subgroups are lines. The other action has exactly one fixed point. Generic 1-parameter subgroups are conics (cf. [9] for more details, esp. Prop. 3.2).

We quote from [9] the following general geometrical facts about equivariant compactifications of additive groups.

Proposition 1.3. — Let X be a smooth projective equivariant compactification of G_{a}^{n} and D = X\G_{a}^{n} the boundary.

1. The boundary D is a pure codimension 1 subvariety.

2. The Picard group Pic(X) is freely generated by the irreducible components D_{0}, ..., D_{r} of D.

3. The closed cone of effective divisors Λ_{eff}(X) ⊂ Pic(X)_{R} is given by

Λ_{eff}(X) = ⊕_{k=0}^{r}R_{+}[D_{k}].

(∗)Scherzhafter Vorschlag von Yu. I. Manin, Weihnachten 1998

4. There exist integers n_{k} > 1 such that the anticanonical class is given by

−[K_{X}] = ∑_{k=0}^{r} n_{k}[D_{k}].

1.4. Blow-ups. — The basic example of an equivariant compactification of G_{a}^{n} is the projective space P^{n}, with G_{a}^{n} acting on P^{n} = Proj(F [x_{0}, …, x_{n}]) by translation:

((t_{1}, …, t_{n}), (x_{0} : ⋯: x_{n})) ↦ (x_{0} : x_{1} + t_{1}x_{0} : ⋯: x_{n} + t_{n}x_{0})

which stabilizes the “hyperplane at infinity” given by the equation x_{0} = 0. In this paper we consider blow-ups of the projective space π: X → P^{n} in a subscheme Z ⊂ P^{n} of pure codimension ⩾ 2, which is contained in this hyperplane. We denote by ℐ_{Z} the ideal sheaf of Z in P^{n} so that the blow-up is defined by the formula X = Proj(⨁_{j} ℐ_{Z}^{j}). As Z ⊂ Z_{0} is fixed by G_{a}^{n}, the universal property of the blow-up implies that the action of G_{a}^{n} on P^{n} lifts uniquely to an action on X.

The geometry of blow-ups of arbitrary subschemes can be very complicated. We shall assume that ℐ_{Z} is the product of a finite number of ideals ℐ_{Z_{k}}, where the Z_{k} are integral subschemes of the hyperplane at infinity in P^{n}. The universal property of blow-ups says that X is the “minimal” scheme mapping to P^{n} on which ℐ_{Z} becomes invertible. An easy lemma in commutative algebra implies that on X, the ℐ_{Z_{k}} are themselves invertible. (The blow-up of a product of ideals is the universal way to make these ideals invertible; it is the same as blowing up successively Z_{1}, then the strict transform of Z_{2}, etc.) In particular, π factors as X → X_{k} → P^{n}, X_{k} being the blow-up of Z_{k} in P^{n}.

On X, we now have Cartier divisors D_{k} (which are the inverse images of the Z_{k}) and line bundles 𝒪_{X}(D_{k}) equipped with a canonical section s_{D_{k}} ∈ Γ(X, 𝒪_{X}(D_{k})). Moreover, s_{D_{k}} and 𝒪_{X}(D_{k}) are pullbacks of similar objects on X_{k} (which we will denote by the same letters). Note also that 𝒪_{X}(−D_{k}) = ℐ_{Z_{k}} · 𝒪_{X} ⊂ 𝒪_{X} and that by definition, s_{D_{k}} is obtained by dualizing the pullback of the canonical inclusion ℐ_{Z_{k}} → 𝒪_{P^{n}}.

Let D_{0} be the strict transform of Z_{0} in X. We have a canonical isomorphism:

π^{∗}𝒪_{X}(Z_{0}) ≃ 𝒪_{X}(D_{0} + ∑_{k=1}^{r} D_{k}). (1.5)

Denoting by s_{D_{0}} the canonical section of 𝒪_{X}(D_{0}), the tensor product ⨂_{k=0}^{r} s_{D_{k}} equals the pull back on X of the canonical section of 𝒪(Z_{0}).

The D_{k}’s (0 ⩽ k ⩽ r) form a basis of the Picard group of X. We identify the anticanonical sheaf in these coordinates:

Proposition 1.6. — If Z is smooth, then X is a smooth variety and its anticanonical sheaf is given by

K_{X}^{−1} = π^{∗}((n + 1)Z_{0}) ⊗ ⨂_{k=1}^{r−1} 𝒪(−(codim(Z_{k}) − 1)D_{k})

= ⨂_{k=0}^{r−1} 𝒪((2 + dim(Z_{k}))D_{k}).

Proof. — See Hartshorne [8], Ex. 8.5, p. 188.

1.7. Metrizations on blow-ups. — Let S be the spectrum of a Dedekind ring (which will be the ring of integers in F, or a localization of it, or a completion) or the spectrum of a field which is equipped with a valuation. Let X be a projective scheme over S. For a locally free sheaf ℰ of finite rank on X, there are several notions of metrizations corresponding to these various cases. We recall briefly the definitions.

– If S = Spec(F), the spectrum of a field endowed with a valuation, a metric on a ℰ is a family of norms on the fibres ℰ_{x} for x ∈ X(F̄), which vary continuously with x.

– If F̄ = C, one may ask that the dependence is 𝒞^{∞}, and—independently— that the metrics are hermitian in the fibers.

– If S = Spec(F), where F is the fraction field of a discrete valuation ring R, any flat and projective model (𝒳 , ℰ) over Spec(R) determines a metric according to which a section is of norm ⩽ 1 at a point iff it is integral.

– If S = Spec(F), F being a number field, an adelic metric on ℰ is a collection of metrics for all X_{v}/F_{v}, v being the different places of F. Moreover, one assumes that there exists a model over Spec(𝔬_{F}) which gives the same metrics except at a finite number of places. At these exceptional places the ratios of the two metrics are assumed to be bounded functions on X.

The usual definitions of metrics on subsheafs, quotients, hom’s, etc. are compatible with these notions.

Let X be a quasi-projective flat scheme over S, ℐ a sheaf of ideals on X and Z = V (ℐ). Let π : Y → X be the blow-up of V (ℐ), Y = Proj(⨁_{n} ℐ^{n}). On Y, the inverse image of Z becomes a Cartier divisor D and the line bundle 𝒪(D) is equipped with a canonical section s_{D}. We want to endow 𝒪_{Y} (D) with a metric and to give a formula for the norm of s_{D} at any point of Y \|D|. Note that 𝒪(−D) = ℐ · 𝒪_{Y} ⊂ 𝒪_{Y} and that s_{D} is the pull-back of the canonical inclusion ℐ → 𝒪_{X}.

Choose a locally free sheaf ℰ of finite rank on X with a section σ_{Z} ∈ Γ(X, ℰ) whose scheme of zeroes is Z (existence follows from the quasi-projectivity of X). This induces a surjective homomorphism φ : ℰ^{∨} → ℐ and a closed immersion Y ↪ P(ℰ^{∨}) such that 𝒪(−D) = ℐ ·𝒪_{Y} = 𝒪_{P}(1) and the universal quotient map π^{∗}ℰ^{∨} → 𝒪_{P}(1) on Y is the pullback of φ. Hence, to metrize 𝒪_{Y} (D) it is sufficient to endow ℰ^{∨} with a metric. The quotient metric on 𝒪_{P}(1) is defined as follows: for any local section s of 𝒪_{P}(1), we pose

∥s∥ (y) = inf_{t} ∥t∥

where the infimum is on the local sections t of ℰ^{∨} mapping to s under the canonical surjection π^{∗}φ : π^{∗}ℰ^{∨} → 𝒪_{P}(1).

Restrict this to Y. This gives a norm on 𝒪_{P}(1)|_{Y} = 𝒪_{Y} (−D). The dual norm on 𝒪_{Y} (D) of the canonical section s_{D} is given by the formula

∥s_{D}∥ = sup_{s≠0} \frac{|⟨s_{D}, s⟩|}{∥s∥} = sup_{t} \frac{|⟨s_{D}, π^{∗}φ(t)⟩|}{∥t∥}

the last supremum being over the non-zero local sections t of π^{∗}ℰ^{∨}. But, away from D on the blow-up, t comes from a local section of ℰ^{∨} and ⟨s_{D}, π^{∗}φ(t)⟩ is exactly the image of t under the surjection φ : ℰ^{∨} → ℐ. Hence, ∥s_{D}∥ is equal to the norm of φ, viewed as a homomorphism ℰ^{∨} → 𝒪_{X}, which by the definition of the dual norm on ℰ^{∨} is exactly the norm of the original section σ_{Z} ∈ Γ(X, ℰ).

(This can be simplified if one regards the blow-up as the closure of the graph of the map X \ Z → P(ℰ^{∨}) induced by σ_{Z}.)

Note the precise meaning of these calculations:

– they are valid if S is any field with a valuation;

– if S is a discrete valuation ring, arithmetic intersection on the integral model gives a result which is compatible with the metrized theory on the generic fibre if the metric on ℰ comes from the model;

– if S is the ring of integers of a number field, they show that we have an adelic metric in the sense of Arakelov geometry provided ℰ is equipped with an adelic metric.

Hence, we have the following theorem:

Theorem 1.8. — Let X be an algebraic variety over a field F, ℐ ⊂ 𝒪_{X} a sheaf of ideals on X and π : Y → X the blow-up of 𝒴 . Let ℰ be a locally free sheaf of finite rank on X with a section σ_{Z} ∈ Γ(X, ℰ) such that V (ℐ) = div(σ_{Z}) as schemes.

Assume ℰ is given a metric. Then the line sheaf 𝒪_{Y} (D) corresponding to the exceptional divisor D on Y has a canonical metric such that the norm of its canonical section s_{D} is given by the formula:

∥s_{D}∥ (y) = ∥σ_{Z}∥ (π(y)). (1.9)

In particular, if ℒ_{1}, …, ℒ_{r} are line bundles on X with sections s_{i} such that, as a scheme, Z = ⋂ div(s_{i}), we may take ℰ = ⨁ ℒ_{i}, σ_{Z} = (s_{i}). Assume the ℒ_{i} to be metrized and endow ℰ with the associated hermitian metric (resp. with the ℓ^{∞}-metric at non-archimedian places). The preceeding theorem implies that 𝒪_{Y} (D) may be metrized in such a way that

∥s_{D}∥^{2} (y) = ∑_{i=1}^{r} ∥s_{i}∥ (π(y)). (1.10)

In particular, if X = P^{n}, ℒ_{i} = 𝒪_{P^{n}}(n_{i}), s_{i} corresponds to a homogeneous polynomial g_{i} of degree n_{i} and, if π(y) = (x_{0} : ⋯: x_{n}),

∥s_{D}∥^{2} (y) = ∑_{i=1}^{r} \begin{matrix}|g_{i}(x_{0}, …, x_{n})| & 2 \\ ( & ∑ & {n \atop j=0} & |x_{i}| & 2 & ) & n_{i} \end{matrix} . (1.11)

As a last example, assume that X = P^{n} and Z is an integral divisor in Z_{0}. Then, the homogeneous ideal of Z is of the form (x_{0}, f(x_{1}, …, x_{n})) for some homogeneous polynomial f of degree d ⩾ 1. If π(y) = (1 : x_{1} : ⋯: x_{n}), then

∥s_{D}∥^{2} (y) = \frac{1}{1 + ∑_{j=1}^{n} |x_{j}|^{2}} + \frac{|f(x_{1}, …, x_{n})|^{2}}{(1 + ∑_{j=1}^{n} |x_{j}|^{2} )^{d}} . (1.12)

All these formulas have analogues at non-archimedian places with the sum of the squares being replaced by their maximum.

1.13. Résumé. — Let F be a number field. For 1 ⩽ k ⩽ r, choose a finite family of homogeneous polynomials g_{k,j} ∈ F [x_{0}, …, x_{n}] of degree d_{k,j} generating a prime ideal ℐ_{Z_{k}} corresponding to an integral subscheme Z_{k} ⊂ P^{n}. Let π : X → P^{n} be the blow-up of the ideal ℐ = ℐ_{Z_{1}} ⋯ ℐ_{Z_{k}}. On X, the inverse image of Z_{k} is a Cartier divisor D_{k} whose associated line bundle 𝒪_{X}(D_{k}) can be adelically metrized so that the norm of its canonical section s_{D_{k}} at a point x ∈ X mapping to (x_{0} : ⋯: x_{n}) ∈ P^{n} is given by

∥s_{D_{k}}∥_{v} (x) = max_{j} \frac{|g_{k,j}(x_{0}, …, x_{n})|_{v}}{max(|x_{0}|_{v} , …, |x_{n}|_{v})^{d_{k,j}}}

at finite places v, and by

∥s_{D_{k}}∥_{v}^{2} (x) = ∑_{j} \frac{|g_{k,j}(x_{0}, …, x_{n})|_{v}^{2}}{(|x_{0}|_{v}^{2} + ⋯ + |x_{n}|_{v}^{2})^{d_{k,j}}}

if v is an archimedian place.

We shall henceforth assume that Z_{k} is contained in the hyperplane at infinity x_{0} = 0. Then one may assume that one of the g_{k,j} = x_{0} and that the others do not depend on x_{0}. The universal property of the blow-up implies that π : X → P^{n} is an isomorphism over G_{a}^{n} ≃ {x_{0} ≠ 0} and that the action of G_{a}^{n} on P^{n} lifts to an action on X and to an action on the line bundles 𝒪_{X}(D_{k}).

The following proposition can be deduced, either through explicit computations with the formulas defining ∥s_{D_{k}}∥, or by an abstract argument involving schemes over Spec 𝔬_{F}.

Proposition 1.14. — Assume that the polynomials g_{k,j} have coefficients in 𝔬_{F} and that they generate the homogeneous ideal ℐ_{Z_{k}} ∩𝔬_{F} [x_{0}, …, x_{n}].^{(†)} Then, for each place v of F, the standard maximal compact subgroup of G_{a}^{n}(F_{v}) acts isometrically on 𝒪(D_{k}).

Let D_{0} be the strict transform of the hyperplane at infinity under π. The line bundle 𝒪_{X}(D_{0}) is the pull-back on X of the 𝒪_{P^{n}}(1) and we shall equip it with its standard metric (given by the formulas above, the family of g_{0,j} being reduced to x_{0}). By means of equations (1.5, 1.6), we then can metrize the line bundles 𝒪_{X}(D_{0}) and K_{X}^{−1}.

2. Heights, Poisson formula

2.1. Product formula and heights. — We recall some conventions concerning absolute values in number fields.

(†)This means that the subscheme V ((g_{k,j})_{j}) of P_{𝔬_{F}}^{n} is projective, surjective and flat over 𝔬_{F}.

Over R, we set |·|_{∞} to be the usual absolute value (such that |2|_{∞} = 2!). If p is a prime number, the absolute value over Q_{p} is normalized by |p|_{p} = 1/p. These absolute values extend uniquely to any algebraic extension of R or Q_{p}.

If F is a number field, we denote by Val(F) the set of places (equivalence classes of valuations) of F. If v is a place of F, we will denote by |·|_{v} be the unique extension of |·|_{∞} or |·|_{p} to F_{v} (according to v being archimedian or not). We also set m_{v} = e_{v}f_{v}, the product of the ramification index by the local degree at v. Now, for any x ∈ F and any valuation v of Q,

∏_{w|v} |x|_{w}^{m_{w}} = |N_{F/Q}(x)|_{v} .

With these normalizations, we have the product formula: for any x ∈ F^{∗},

∏_{v∈Val(F)} |x|_{v} = ∏_{v∈Val(Q)} |N_{F/Q}(x)| = 1.

Let X be a projective variety over F and ℒ a metrized line bundle on X. For any x ∈ X(F), the (exponential, absolute) height of x with respect to the metrized line bundle ℒ is defined by

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

where s is any F -rational local section of ℒ, defined and non-zero at x. The product formula implies that the height doesn’t depend on the choice of s.

2.2. Heights on blow-ups. — We keep the notations of the preceeding section. Moreover we identify G_{a}^{n} with its isomorphic inverse image in X under the blow-up π : X → P^{n}.

The metrizations above allow us to define height functions corresponding to complexified divisors D(s) = s_{0}D_{0} + ⋯ + s_{r}D_{r}. Namely, if x = (x_{1}, …, x_{n}) ∈ G_{a}^{n}(F), its exponential height is defined by

H_{D(s)}(x) = ∏_{k=0}^{r} ( ∏_{v∈Val(F)} ∥s_{D_{k}}∥_{v}^{−m_{v}} (1 : x_{1} : ⋯: x_{n}))^{s_{k}}.

This definition extends to all x ∈ G_{a}^{n}(A_{F}) and gives a pairing

H : Pic^{G}(X)_{C} × G_{a}^{n}(A_{F}) ⟶ C^{∗}

which is multiplicative as a function on Pic^{G}(X) and which is invariant under the action of the maximal compact subgroup of G_{a}^{n}(A_{F}). Such a pairing had already appeared in the context of toric varieties.

The invariance of the heights is a crucial technical ingredient for the proofs of analytic properties of the height zeta functions for toric varieties and for equivariant compactifications of G_{a}^{n} considered in the present paper.

The “height zeta function” is the series

Z(s_{0}, …, s_{r}) = ∑_{x∈G_{a}^{n}(F)} H_{D(s)}(x)^{−1}.

Its convergence in some non empty open subset of C^{r+1} is a consequence of the following (well known) lemma.

Lemma 2.3. — Let V be a projective variety over a number field F and (ℒ_{i} )_{1⩽i⩽d} a finite number of ample metrized line bundles on V. For x ∈ V (F), define H(s; x) = ∏_{i=1}^{d} H_{ℒ_{i}} (x)^{s_{i}}. Then there exists an open non-empty subset Ω of R^{d} such that the series

Z(s) = ∑_{x∈X(F)} H(s; x)^{−1}

converges absolutely and uniformly for all s ∈ C^{d} with Re(s) contained in Ω.

Moreover, any other metrization on the ℒ_{i} gives the same domain of convergence.

Proof. — The usual proof of Northcott’s theorem establishes a polynomial bound for the number of rational points of bounded exponential height. Hence, the height zeta function of (P^{n}, 𝒪(1)) converges for s ≫ 0. (There is no need to invoke Schanuel’s theorem [19] which gives the precise asymptotics. Below, we give a new proof of Schanuel’s theorem.)

Therefore, there are real numbers α_{i} such that Z(0, …, s_{i}, …, 0) converges for Re(s_{i}) ⩾ α_{i}. Now, Z(s) converges for any s = (s_{1}, …, s_{d}) ∈ C^{d} such that for each i, Re(s_{i}) ⩾ α_{i}.

2.4. Harmonic analysis on the additive group. — We recal basic facts concerning harmonic analysis on the group of adelic points G_{a}^{n}(A_{F}) (cf., for example, [22]). For any prime number p, we can view Q_{p}/Z_{p} as the p-Sylow subgroup of Q/Z. This allows us to define a local character ψ_{p} of G_{a}(Q_{p}) by setting

ψ_{p} : x_{p} ↦ exp(2πix_{p}).

At the infinite place of Q we put

ψ_{∞}: x_{∞} ↦ exp(−2πix_{∞}),

(here x_{∞} is viewed as an element in R/Z). The product of local characters gives a character ψ of G_{a}(A_{Q}) and, by composition with the trace, a character of G_{a}(A_{F}). For any a ∈ G_{a}^{n}(A_{F}) we obtain a character ψ_{a} of G_{a}^{n}(A_{F}) by

x ↦ ψ ∘ tr_{F/Q}(⟨a, x⟩).

The choice of ψ defines a self-duality of G_{a}^{n}(A_{F}) (Pontryagin duality). For v ∈ Val(F), we denote by μ_{v} the standard normalized local Haar measures on G_{a}^{n}(F_{v}) and by μ = ∏_{v} μ_{v} the self-dual measure on G_{a}^{n}(A_{F}). The precise normalization can be found in (cf. [22] or [12], p. 280); for F = Q, we have μ_{p}(Z_{p}) = 1 and μ_{∞}([0; 1]) = 1.

For a function H on G_{a}^{n}(A_{F}) we denote by Ĥ its Fourier-transform (with respect to the Haar measure μ)

Ĥ : G_{a}^{n}(A_{F}) → C, ψ ↦ ∫_{G_{a}^{n}(A_{F})} H(x)ψ(x) dμ(x),

whenever the integral converges. We shall also use the notation dx for dμ(x). We will use the Poisson formula in following form (cf. [12], p. 280).

Theorem 2.5. — Let H be a continuous function on G_{a}^{n}(A_{F}) such that both H and Ĥ are integrable and such that

∑_{a∈G_{a}^{n}(F)} H(x + a)

is absolutely and uniformly convergent on compact subsets in G_{a}^{n}(A_{F})/G_{a}^{n}(F). Then

∑_{x∈G_{a}^{n}(F)} H(x) = ∑_{a∈G_{a}^{n}(F)} Ĥ(ψ_{a}).

For s ∈ C^{r+1} and ψ ∈ G_{a}^{n}(A_{F}), we shall denote by Ĥ(s; ψ) the Fourier transform of the height function H(s; ·)^{−1} on G_{a}^{n}(A_{F}) at the character ψ. It is the product of the local Fourier transforms of the functions H(s; ·)^{−m_{v}} for all v ∈ Val(F).

Proposition 2.6. — With the above notations, for all characters ψ which are non-trivial on the maximal compact subgroup of G_{a}^{n}(A_{F}), we have that Ĥ(s, ψ) = 0.

Proof. — This follows from the invariance of the height under the maximal compact subgroups, see Prop. 1.14.

Consequently, we have the formal identity for the height zeta function:

Z(s) = ∑_{a∈G_{a}^{n}(𝔬_{F})} Ĥ(s; ψ_{a}). (2.7)

The following lemma verifies the two hypotheses of the Poisson formula 2.5 concerning H.

Lemma 2.8. — There exists a real α such that for any s ∈ C^{r+1} satisfying Re(s_{0} − s_{k}) ⩾ α and Re(s_{k}) ⩾ α, and for any compact subset K of G_{a}^{n}(A_{F})/G_{a}^{n}(F), the series

∑_{a∈G_{a}^{n}(F)} H(s; x + a)

converges absolutely and uniformly for x ∈ K.

Proof. — If s ∈ Z^{r+1}, the line bundle D(s) is ample iff all s_{k} > 0 and s_{0} > s_{1} + ⋯ + s_{r}. Moreover, the ample line bundles D = (r + 1)D_{0} + D_{1} + ⋯ + D_{r}, D +D_{1}, …, D +D_{r} provide a basis of Pic(X)_{R}. Hence, Lemma 2.3 implies the existence of a real α > 0 such that the series converges absolutely when x = 0, uniformly for all s ∈ C^{r+1} such that Re(s_{0}) > α, Re(s_{0} − s_{k}) > α.

For any x, the function H(s; x+·) is another height function for D(s), called “twisted height” in our paper [5], § 2.4, esp. proposition 2.4.3. This implies the convergence for any x. The uniformity for x ∈ K follows from the fact that the height functions can be mutually uniformly bounded.

Now, for the proof of the meromorphic continuation of the height zeta function it will be sufficient to prove that the Ĥ-series on the right-hand side of Eq. 2.7 1^{o}) converges for some Re(s_{0}) > α, Re(s_{0} − s_{k}) > α big enough, and 2^{o}) continues meromorphically.

2.9. Integrability of local height functions. — The aim of this section is to prove a general result concerning the integrability of local height functions against a measure with singularities.

Proposition 2.10. — Let X be a proper smooth variety of dimension d over a field F_{v} which is a finite extension of R or Q_{p}. Fix a finite number of metrized line bundles ℒ_{α} on X together with sections s_{α}. Assume that their divisors div(s_{α}) are smooth and that their sum is a divisor D with normal crossings and let U = X \ D. Finally, let ω ∈ Γ(U, Ω_{X/F_{v}}^{d}) be a meromorphic differential form of top degree. We assume that there are integers λ_{α} such that the divisor of ω equals ∑_{α} λ_{α} div(s_{α}). Denote by dω the associated measure on U(F_{v}).

Then, the integral

∫_{U(F_{v})} ∏_{α} ∥s_{α}∥^{m_{v}r_{α}} (x) dω

converges if and only if for all α, r_{α} > λ_{α} − 1.

Proof. — Using a partition of unity on X for the F_{v}-topology, we may assume that X is a relatively compact open subset Ω ⊂ F_{v}^{d}, with local coordinates x_{1}, …, x_{d} and that the divisor ∑_{α} div(s_{α}) is given by the equation x_{1} … x_{a} = 0 for some integer 0 ⩽ a ⩽ d. The integral is then

I_{Ω} = ∫_{Ω} ∏_{i=1}^{a} |x_{i}|_{v}^{m_{v}(r_{α(i)}−λ_{α(i)})} exp ( ∑_{α} h_{α}(x)) dx_{1} … dx_{d}

for some functions h_{α} giving the metrics in our local trivialization and which are therefore continuous and bounded.

Remark that the integral of |x|_{v}^{m_{v}s} over the unit ball of F_{v} converges if and only if s > −1. The Fubini theorem shows that the integral I_{Ω} converges if and only if for each i ∈ {1, …, a}, r_{α(i)} − λ_{α(i)} > −1. As any α appears in some chart, the proposition is proved.

2.11. The local Fourier transform in the archimedian case

When F_{v} = R or C, we want to show that the local Fourier transform of the height function as a function of ψ_{a} decreases rapidly when the norm of a ∈ F_{v}^{n} grows to infinity. The proof proceeds by integration by parts, which requires some estimates.

Lemma 2.12. — Let X be a smooth projective variety over F_{v} and Z be a smooth closed subscheme of X. Let ∂ be a global section of (Ω_{X}^{1})^{∨} ⊗ ℐ_{Z}, i.e. a derivation on X vanishing on Z. Denote by π : Y → X the blow-up of ℐ_{Z}.

1) Then the derivation ∂|_{π^{−1}(X\Z)} extends uniquely to a derivation on Y.

2) Let ℰ be a vector bundle on X equipped with a smooth hermitian metric and s a global section of ℰ whose divisor is Z. Then the function ∂ log ∥s∥ extends uniquely to a smooth function on Y.

Proof. — Choose local analytic coordinates on X such that Z is defined by x_{1} = ⋯ = x_{a} = 0. Then, Y may be embedded in P^{a−1} × A^{d} with coordinates ((t_{1} : ⋯: t_{a}), (x_{1}, …, x_{d})) and is given there by the equations t_{i}x_{j} = t_{j}x_{i} for i ∈ {1, …, a}. We consider the chart t_{a} ≠ 0. Then, local coordinates on Y are t_{1}, t_{2}, …, t_{a−1}, x_{a}, x_{a+1}, …, x_{d} and π : Y → X is given by x_{i} = t_{i}x_{a} if i < a.

On X, the derivation ∂ has the form

∂ = ∑_{i=1}^{d} h_{i} \frac{∂}{∂x_{i}} ,

for some functions h_{i} ∈ (x_{1}, …, x_{a}). Now, we have to verify that if i < a, ∂t_{i} is a regular function on Y. But

∂t_{i} = ∂(x_{a}/x_{i}) = h_{a}(x)\frac{1}{x_{i}} − h_{i}(x)\frac{x_{a}}{x_{i}^{2}} = \frac{1}{x_{a}} h_{a}(x)t_{i} − \frac{1}{x_{a}} h_{i}(x)t_{i} ∈ 𝒪_{Y}

since

h_{j}(x) ∈ (t_{1}x_{a}, …, t_{a−1}x_{a}, x_{a}) = (x_{a}).

For the statement concerning norms, we may fix the coordinates so that ∥s∥^{2} (x) = ∑_{i=1}^{a} |x_{i}|^{2}. Then,

∂ log ∥s∥ = \frac{1}{|x_{1}|^{2} + ⋯ + |x_{a}|^{2}} ( ∑_{i=1}^{a} 2x_{i}h_{i}(x))

= \frac{1}{|t_{1}|^{2} + ⋯ + |t_{a−1}|^{2} + 1}( ∑_{i=1}^{a} 2t_{i} \frac{h_{i}(x)}{x_{a}})

is regular on Y.

Proposition 2.13. — For any compact subset K ⊂ R^{r+1} where H(s; ·)^{−m} is integrable, and for any integer d ⩾ 1, there exists a constant c(d, K) such that for any a ∈ C^{n} and any s ∈ C^{r+1} with Re(s) ∈ K,

|Ĥ(s; ψ_{a})| ⩽ c(d, K) (\frac{1 + ∥ℑ(s)∥}{1 + ∥a∥})^{d} .

Proof. — The 2 preceeding lemmas imply that ∂^{α}(log ∥s_{D}∥)(x) are bounded on G_{a}^{n}(F). Moreover, ∥s_{D}∥ tends to 0 at infinity. We thus may integrate by parts d-times.

3. Projective space

This section is included to illustrate our approach in the simplest example. We give yet another proof of asymptotics for the number of rational points of bounded height on the standard projective space P^{n} over a number field F with the standard metrization of the line bundle 𝒪(1) given by the model P_{𝔬_{F}}^{n} at the finite places and by the L^{2}-norms at the archimedian places.

To keep this section as self-contained as possible, we reprove the estimates needed without referring to the general estimates of the preceeding section. For simplicity, we assume moreover that F = Q.

We will denote by A the ring of adeles A_{Q}, by p a prime number. We have the normalized valuations | · |_{p} with |p|_{p} = p^{−1} and the usual absolute value | · |_{∞}. If a ∈ G_{a}^{n}(A), we denote by ψ_{a} the corresponding character by the identification of G_{a}^{n}(A) with its Pontryagin dual.

We are interested in the height zeta function

Z(s) = ∑_{x∈G_{a}^{n}(Q)} H(x)^{−s} (3.1)

where H(x) = H_{∞}(x) ∏_{p} H_{p}(x) with

H_{v}(x) := ∥x∥_{v} = \begin{cases}(1 + ∑_{j=1}^{n} |x_{j}|_{v}^{2})^{1/2} & if v|∞ \\ max(1, max_{j} |x_{j}|_{v}) & if v is finite. \end{cases}

The series (3.1) converges absolutely and uniformly to a holomorphic function for Re(s) ≫ 0. For all s such that the both sides converge, we have the Poisson-formula identity (cf. 2.5)

Z(s) = ∑_{ψ_{a}} Ĥ(s; ψ_{a}), (3.2)

absolutely. This identity is the starting point for a meromorphic continuation of Z(s). We now compute (resp. estimate) the local Fourier transforms.

Lemma 3.3. — Let p be a prime number. For all s with Re(s) > n, H_{p}(s; ·) is integrable on Q_{p}^{n} and its Fourier transform at the trivial character ψ_{0} is given by

Ĥ_{p}(s; ψ_{0}) = \frac{1 − p^{−s}}{1 − p^{−(s−n)}} . (3.4)

Proof. — We decompose the domain of integration Q_{p}^{n} into subdomains

U(α) = {x = (x_{1}, ..., x_{n}) ; ∥x∥_{p} = p^{α}},

for α ⩾ 1 and

U(0) = {x = (x_{1}, ..., x_{n}) ; ∥x∥_{p} ⩽ 1}

Then

Ĥ_{p}(s; ψ_{0}) = ∫_{U(0)} H(x)^{−s} dx + ∑_{α⩾1}∫_{U(α)} H(x)^{−s} dx,

= 1 + ∑_{α⩾1}p^{−αs} · vol(U(α)).

One has vol U(0) = 1 and for α ⩾ 1,

vol(U(α)) = p^{αn} vol(Z_{p}^{n} \ (pZ_{p})^{n}) = p^{αn}(1 − p^{−n}).

For all s with Re(s) > n, the geometric series converges absolutely and we obtain

Ĥ_{p}(s; ψ_{0}) = 1 + (1 − \frac{1}{p^{n}}) ∑_{α⩾1}p^{−α(s−n)},

= 1 + (1 − \frac{1}{p^{n}}) · \frac{1}{p^{s−n}} · \frac{1}{1 − p^{−(s−n)}} .

Simplifying, we obtain (3.4).

For all a = (a_{1}, ..., a_{n}) ∈ Z^{n} let S(a) be the set of all primes which divide all a_{j}.

Lemma 3.5. — For all a ∈ Z^{n} \ {0}, all s with Re(s) > n and all p ∉ S(a) we have

Ĥ_{p}(s; ψ_{a}) = 1 − p^{−s}. (3.6)

Proof. — As above, we have

Ĥ_{p}(s, ψ_{a}) = 1 + ∑_{α⩾1}p^{−αs} ∫_{U(α)} ψ_{a}(x) dx.

We first integrate over the set V (α) of x ∈ Q_{p}^{n} such that ∥x∥ ⩽ p^{α}.

∫_{V (α)} ψ_{a}(x) dx = p^{αn} ∫_{Z_{p}^{n}} ψ_{a/p^{α}}(x) dx.

If α ⩾ 1, as p doesn’t divide all the a_{j}, this is the integral of a non trivial character on a compact group, hence 0. For α = 0, we get 1. Therefore, as V (0) = U(0) and U(α) = V (α) \ V (α − 1) for α ⩾ 1,

∫_{U(α)} ψ_{a}(x) dx = \begin{cases}0 & for α ⩾ 2 \\ −1 & for α = 1. \end{cases}

This implies the lemma.

Lemma 3.7. — For all ε > 0 there exist constants c and δ > 0 such that for all s with Re(s) > n + ε and all a ∈ Z^{n} \ {0} we have the estimate

| ∏_{p∈S(a)} Ĥ_{p}(s; ψ_{a})| ⩽ c · (1 + ∥a∥)^{δ}. (3.8)

Proof. — In the integral, we replace ψ_{a} by 1, s by Re(s) and use the computation in (3.3). For Re(s) ⩾ n + ε, we obtain

|Ĥ_{p}(s; ψ_{a})| ⩽ \frac{1}{1 − p^{−ε}} .

If a is a positive integer, we have an inequality

∏_{p|a} \frac{1}{p^{ε}} ≪ ln(1 + a)

which can be deduced e.g. from the Prime Number Theorem. This gives us equation (3.8).

We now turn to the estimations of the local Fourier transform for the place at infinity. For the trivial character, we can—as we could in the non-archimedian case—explicitely compute the relevant integral:

Lemma 3.9. — For all s with Re(s) > n, H_{∞}(s; ·) is integrable on R^{n} and its Fourier transform at the trivial character ψ_{0} is given by

Ĥ_{∞}(s; ψ_{0}) = π^{n/2} \frac{Γ((s − n)/2)}{Γ(s/2)}.

Lemma 3.10. — For all δ > 0 and all compacts K in the domain Re(s) > n there exists a constant c(δ, K) such that for all a ∈ Z^{n} and all s ∈ K we have

|Ĥ_{∞}(s; ψ_{a})| ⩽ c(δ, K)(1 + ∥a|)^{−δ}

Proof. — By a unitary change of variables, we may assume a = (∥a∥ , 0, …, 0). Thus,

Ĥ_{∞}(s; ψ_{a}) = ∫_{R^{n}}(1 + ∥x∥^{2})^{−s/2} exp(−2πi ∥a∥ x_{1}) dx

= ∫_{R} ∫_{R^{n−1}}(1 + |x_{1}|^{2} + ∥x′∥)^{−s/2} exp(−2πi ∥a∥ x_{1}) dx_{1} dx′

= ∫_{R}(1 + |x_{1}|^{2})^{−(s−n+1)/2} exp(−2πi ∥a∥ x_{1}) dx_{1} ∫_{R^{n−1}} \frac{dy}{(1 + ∥y∥^{2})^{s/2}}

For any k > 0, the kth derivative of t ↦ (1+t^{2})^{−s} is of the form P_{k}(t)(1+t^{2})^{−s−k} with P_{k} a polynomial of degree k whose coefficients are polynomials in s. Thus we can integrate by parts and get for any k an expression

∫_{R}(1 + t^{2})^{−(s−n+1)/2} exp(−2πi ∥a∥ t) dt = \frac{1}{(πi ∥a∥)^{k}} ∫_{R} \frac{P_{k}(t)}{(1 + t^{2})^{k− \frac{s−n+1}{2}}} dt

which imply the lemma.

Remark 3.11. — It follows from the arguments above that the Fourier transform has polynomial growth in vertical strips.

Theorem 3.12. — The series

Z(s) = ∑_{ψ_{a}} Ĥ(s; ψ_{a})

converges absolutely and uniformly to a holomorphic function for s with Re(s) > n + 1. The function Z(s) admits a meromorphic continuation to the domain Re(s) > n with exactly one simple pole at s = n + 1. The residue at this pole equals

lim_{s→n+1}(s − (n + 1))Ĥ(s; ψ_{0}) = lim_{s→n+1}(s − (n + 1)) ∫_{G_{a}^{n}(A_{Q})} H(s; x) dx.

Proof. — Choose a real number δ > n. From lemmas 4.5, 4.7 and 4.10, it follows that for any compact K ∈ ]n; +∞[, any a ∈ Z^{n} \ {0}, the product of the local Fourier transforms at the character ψ_{a} converges to a holomorphic function of s which satisfies the inequality

|Ĥ(s; ψ_{a})| ⩽ c(K)(1 + |ℑ(s)|)^{δ}(1 + ∥a∥)^{−δ}, Re(s) ∈ K.

Hence, the sum over all non-trivial ψ converges absolutely and locally uniformly to a holomorphic function in the domain Re(s) > n.

At the trivial character, we have, if Re(s) > n + 1,

Ĥ(s; ψ_{0}) = \frac{ζ(s − n)Γ((s − n)/2)}{ζ(s)Γ(s/2)}.

This has a simple pole at s = n and extends meromorphically to the domain Re(s) > n, with no other pole there.

The identification of the residue and Peyre’s Tamagawa constant in [14] is straightforward, granted the meromorphic continuation of Ĥ(s; ψ_{0}).

4. Blow-ups of P^{2}

4.1. Preliminaries. — We continue to work over Q and we keep the notations of previous sections.

Let us consider the projective plane P^{2} with coordinates (x_{0}, x_{1}, x_{2}) and its Zariski open subset U ⊂ P^{2} given by x_{0} ≠ 0. Denote by X the blow-up of P^{2} in r distinct points Z_{1}, …, Z_{r} which are contained in the line at infinity Z_{0} ⊂ P^{2} which is given by x = 0.

For all k ∈ {1, …, r}, there is a linear form ℓ_{k} ∈ Z[x_{1}, x_{2}] with primitive coefficients such that Z_{k} = 𝒱 (x_{0}, ℓ_{k}). For k = 1, …, r, we denote by D_{k} the inverse image of Z_{k} in X and by D_{0} the strict transform of the line Z_{0}. The variety X is smooth; the anticanonical class is given by

−[K_{X}] = 3[D_{0}] + 2 ∑_{k=1}^{r−1}[D_{k}].

In the sequel, we shall identify a point x ∈ G_{a}^{2} with the point with homogeneous coordinates (1 : x) in P^{2} or with its pre-image in the blow-up. It follows from the general theory of height functions on blow-ups that for all k ∈ {1, …, r}, a local height function for the divisor D_{k} at such a point x is given by

H_{k,p}(x) = \frac{max(1, ∥x∥_{p})}{max(1, |ℓ_{k}(x)|_{p})}

at a finite place p, and by an analogous formula where max(1, ·) is replaced by \sqrt{1 + ·^{2}} at the infinite place. For D_{0}, we have

H_{0,p}(x) = max(1, ∥x∥_{p}) ∏_{k=1}^{r} H_{k,p}^{−1}(x)

(with the same convention if v = ∞). The global height is given by

H_{k}(x) = H_{k,∞}(x) · ∏_{p} H_{k,p}(x)

and for s = (s_{0}, …, s_{r}) ∈ C^{r}, we define

H(s; x) := ∏_{k=0}^{r} H_{k}(x)^{s_{k}}

the global height corresponding to the complexified line bundle D(s).

From 2.5, we see that the height zeta function for X has the following formal “Fourier expansion”:

∑_{x∈Q^{2}} H(s; x)^{−1} = ∑_{a∈Z^{2}} Ĥ(s; ψ_{a})

We have the decomposition

Ĥ(s; ψ_{a}) = Ĥ_{∞}(s; ψ_{a,∞}) · ∏_{p} Ĥ_{p}(s; ψ_{a,p}).

As in the case of P^{n}, we compute the local Fourier transforms for almost all places and estimate them at the remaining bad places.

Let S be the set of primes of bad reduction of the schematic closure of ⋃_{k} Z_{k} in P_{Z}^{2}. A prime p belongs to S if there exist two linear forms ℓ_{k} and ℓ_{j} such that p divides det(ℓ_{k}, ℓ_{j}).

4.2. Decomposition of the domain. — Fix a prime p ∉ S which we may omit from the notations for norms, etc. Define subsets of Q_{p}^{2} as follows:

– U(0) = Z_{p}^{2};

– if 1 ⩽ β ⩽ α and k ∈ {1, …, r}, U_{k}(α, β) is the set of x ∈ Q_{p}^{2} such that ∥x∥ = p^{α} and |ℓ_{k}(x)| = p^{α−β};

– if α ⩾ 1 and k ∈ {1, …, r}, U_{k}(α) is the set of x ∈ Q_{p}^{2} such that ∥x∥ = p^{α} and |ℓ_{k}(x)| ⩽ 1;

– if α ⩾ 1, U(α) is the set of x ∈ Q_{p}^{2} such that ∥x∥ = p^{α} and all |ℓ_{j}(x)| = p^{α}. As p ∉ S, these sets furnish a partition of Q_{p}^{2}. This decomposition is well

adjusted to our local heights since they are constant on each subset: – on U(0), all H_{k}’s are 1;

– on U_{k}(α, β), H_{k} = p^{β}, the other H_{j} with j ⩾ 1 are 1 and H_{0} = p^{α−β};

– on U_{k}(α), H_{k} = p^{α}, the other H_{j} are 1;

– on U(α), H_{0} = p^{α} and all other are 1.

In other words,

(4.3) H(s; x) = \begin{cases}1 & if x ∈ U(0); \\ p^{αs_{0}} & if x ∈ U(α), α ⩾ 1; \\ p^{αs_{0}+β(s_{k}−s_{0})} & if x ∈ U_{k}(α, β), 1 ⩽ β < α; \\ p^{αs_{k}} & if x ∈ U_{k}(α, β), β ⩾ α. \end{cases}

The table on p. 29 summarizes this information.

4.4. Some integrals of characters. — We identify a ∈ Z^{2} with the linear form it defines on G_{a}^{2} as well as with the character ψ_{a} of G_{a}^{2}(A_{Q}) it determines.

We will say that a character is generic if a is not proportional to any ℓ_{k}. A non trivial character is special if it is proportional to some (necessarily unique) ℓ_{k}.

If a is generic, S(a) is the set of primes p such that p divides some determinant det(ℓ_{k}, a).

If a is special for ℓ_{k}, S(a) is the set of primes such that p divides some determinant det(ℓ_{j}, a) for j ≠ k.

Note that S ⊂ S(a) for any non-trivial a and that if p|a, then p ∈ S(a). We now compute the integral of ψ_{a} over the subsets defined in the previous

subsection, at least in the cases a = 0, a special and a generic.

Remark that for any a, ∫_{U(0)} ψ_{a}(x) dx = 1.

Lemma 4.5 (Trivial character). — Let p be a prime not in S. Then,

(4.5a)  vol U_{k}(α, β) = p^{2α−β} \frac{(p − 1)^{2}}{p^{2}} ;
(4.5b)  vol U_{k}(α) = p^{α} \frac{p − 1}{p};
(4.5c)  vol U(α) = p^{2α} \frac{(p − 1)(p + 1 − r)}{p^{2}}.

Lemma 4.6 (Generic character). — Let a be a generic character and p ∉ S(a). Then,

(4.6a)  ∫_{U_{k}(α,β)} ψ_{a} = 0

(4.6b)  ∫_{U_{k}(α)} ψ_{a} = \begin{cases}−1 & if α = 1 \\ 0 & else; \end{cases}

(4.6c)  ∫_{U(α)} ψ_{a} = \begin{cases}−1 + r & if α = 1 \\ 0 & else; \end{cases}

Lemma 4.7 (Special character). — Let a a character which is special for ℓ_{k}. If p ∉ S(a) and j ≠ k, one has

(4.7a)  ∫_{U_{j}(α,β)} ψ_{a} = 0

(4.7b)  ∫_{U_{j}(α)} ψ_{a} = \begin{cases}−1 & if α = 1 \\ 0 & else; \end{cases}

(4.7c)  ∫_{U_{k}(α,β)} ψ_{a} = \begin{cases}−p^{α} \frac{p−1}{p} & if β = α − 1 \\ 0 & else; \end{cases}

(4.7d)  ∫_{U_{k}(α)} ψ_{a} = p^{α} \frac{p − 1}{p}

(4.7e)  ∫_{U(α)} ψ_{a} = \begin{cases}−(p + 1 − r) & if α = 1 \\ 0 & else; \end{cases}

Proof. — We prove the three lemmas simultaneously. By a unitary change of variables, we may assume that ℓ_{k}(x) = x_{1}. Then one has

U_{k}(α, β) = p^{β−α}Z_{p}^{∗} × p^{−α}Z_{p}^{∗}

and

U_{k}(α) = Z_{p} × p^{−α}Z_{p}^{∗},

hence their volumes as in formulas (4.5a) and (4.5b).

If p doesn’t divide det(ℓ_{k}, a), we may change variables and even assume that a = (0, 1). Then,

∫_{U_{k}(α,β)} ψ_{a}(x) dx = p^{2α−β} \frac{p − 1}{p} ∫_{Z_{p}^{∗}} exp(2πiu/p^{α}) du

and the last integral has already been calculated when we studied the case of P^{n} : one finds 0 if α ⩾ 2 and −1/p if α = 1. But α > β ⩾ 1, so α ≠ 1. This proves formulas (4.6a) and (4.7a).

Similarly,

∫_{U_{k}(α)} ψ_{a}(x) dx = p^{α} ∫_{Z_{p}^{∗}} exp(2πiu/p^{α}) du

is −1 for α = 1 and 0 else. Formulas (4.6b) and (4.7b) are therefore proved.

We now treat the case of a character a which is special for ℓ_{k}. A unitary change of variables allows to assume ℓ_{k}(x) = x_{1} and a = (1, 0). Then,

∫_{U_{k}(α,β)} ψ_{a}(x) dx = p^{2α−β} \frac{p − 1}{p} ∫_{Z_{p}^{∗}} exp(2πix/p^{α−β})

is 0 if α − β ≠ 1 and is equal to

p^{2α−α+1} \frac{p − 1}{p} \frac{(−1)}{p} = −p^{α} \frac{p − 1}{p}

if α = β − 1, as stated in (4.7c). Equation (4.7d) follows from

∫_{U_{k}(α)} ψ_{a}(x) dx = ∫_{Z_{p}×p^{−α}Z_{p}^{∗}} exp(2πix_{1}) dx = p^{α} \frac{p − 1}{p}.

To compute the volume of U(α), it is useful to remark that U(α) is p^{−α} times the complementary subset in Z_{p}^{2} of 1 + (p − 1)r disjoint balls of radius 1/p. Therefore,

vol U(α) = p^{2α}(1 − \frac{1 + (p − 1)r}{p^{2}}) = p^{2α} \frac{(p − 1)(p + 1 − r)}{p^{2}},

as in formula (4.5c).

If p ∤ a, remark that the integral of ψ_{a} over p^{−α}(Z_{p}^{2} \ pZ_{p}^{2}) is −1 for α = 1 and 0 for α ⩾ 2. We now need to substract the integrals over the disjoint subsets U_{k}(α, β) and U_{k}(α).

For a generic character, one gets 0 if α ⩾ 2 and −1 + r if α = 1; this establishes formula (4.6c). Finally, if a is special for ℓ_{k}, one has

∫_{U(1)} ψ_{a}(x) dx = −1 + (r − 1) − (p − 1) = −(p + 1 − r)

and ∫_{U(α)} ψ_{a} = 0 for α ⩾ 2, as claimed in (4.7e).

4.8. The local Fourier transform at ψ_{0}. — We still assume p ∉ S and compute the local Fourier transform at the trivial character ψ_{0}. By the general result 2.10 on Fourier transforms of height functions, H(s; ·)^{−1} is integrable on G_{a}^{n}(Q_{p}) as soon as Re(s_{0}) > 2 and all Re(s_{k}) > 1. We then have:

Ĥ(s; ψ_{0}) = ∫_{U(0)} + (∑_{k=1}^{r} ∑_{1⩽β<α}∫_{U_{k}(α,β)} + ∑_{α=1}^{∞} ∫_{U_{k}(α)}) + ∑_{α=1}^{∞} ∫_{U(α)}

and we compute each sum separately. The integral over U(0) is 1. Now, for a fixed k, the integral over all U_{k}(α, β) is

∑_{1⩽β<α}∫_{U_{k}(α,β)} = \frac{(p − 1)^{2}}{p^{2}} ∑_{1⩽β<α}p^{−αs_{0}}p^{−β(s_{k}−s_{0})}p^{2α}p^{−β}

= \frac{(p − 1)^{2}}{p^{2}} ∑_{β=1}^{∞} p^{−β(s_{k}−s_{0}+1)} ∑_{α=β+1}^{∞} p^{−α(s_{0}−2)}

= \frac{(p − 1)^{2}}{p^{2}} ∑_{β=1}^{∞} p^{−β(s_{k}−s_{0}+1)}p^{−β(s_{0}−2)} \frac{1}{p^{s_{0}−2} − 1}

= \frac{(p − 1)^{2}}{p^{2}} \frac{1}{p^{s_{0}−2} − 1} ∑_{β=1}^{∞} p^{−β(s_{k}−1)}

= \frac{(p − 1)^{2}}{p^{2}} \frac{1}{p^{s_{0}−2} − 1} \frac{1}{p^{s_{k}−1} − 1}

The sum over all U_{k}(α) (k fixed) equals

∑_{α=1}^{∞} ∫_{U_{k}(α)} = \frac{p − 1}{p} ∑_{α=1}^{∞} p^{−αs_{k}}p^{α} = \frac{p − 1}{p} ∑_{α=1}^{∞} p^{−α(s_{k}−1)} = \frac{p − 1}{p} \frac{1}{p^{s_{k}−1} − 1}

Finally, the sum over all U(α) is

∑_{α=1}^{∞} ∫_{U(α)} = \frac{(p − 1)(p + 1 − r)}{p^{2}} ∑_{α=1}^{∞} p^{−αs_{0}}p^{2α} = \frac{(p − 1)(p + 1 − r)}{p^{2}} ∑_{α=1}^{∞} p^{−α(s_{0}−2)}

= \frac{(p − 1)(p + 1 − r)}{p^{2}} \frac{1}{p^{s_{0}−2} − 1} = \frac{(p − 1)(p + 1 − r)}{p^{2}} \frac{1}{p^{s_{0}−2} − 1}

Putting all this together, we have

Ĥ(s; ψ_{0}) = 1 + \frac{p − 1}{p^{2}} \frac{1}{p^{s_{0}−2} − 1} ∑_{k=1}^{r} \frac{1}{p^{s_{k}−1} − 1} ((p − 1) + p(p^{s_{0}−2} − 1))

+ \frac{(p − 1)(p + 1 − r)}{p^{2}} \frac{1}{p^{s_{0}−2} − 1}

= 1 + \frac{p − 1}{p^{2}} \frac{p^{s_{0}−1} − 1}{p^{s_{0}−2} − 1} ∑_{k=1}^{r} \frac{1}{p^{s_{k}−1} − 1} + \frac{(p − 1)(p + 1 − r)}{p^{2}} \frac{1}{p^{s_{0}−2} − 1}

= 1 + \frac{p − 1}{p^{2}} \frac{1}{p^{s_{0}−2} − 1} ((p + 1 − r) + ∑_{k=1}^{r} \frac{p^{s_{0}−1} − 1}{p^{s_{k}−1} − 1})

= 1 + \frac{p^{2} − 1}{p^{s_{0}} − p^{2}} + \frac{p − 1}{p^{s_{0}} − p^{2}} ∑_{k=1}^{r} \frac{p^{s_{k}−1} − p^{s_{0}−1}}{p^{s_{k}−1} − 1} (4.9)

We remark that if (s_{0}, s_{1}, …, s_{r}) = (3, 2, …, 2), corresponding to the anticanonical class K_{X}^{−1}, this gives

Ĥ(K_{X}^{−1}, ψ_{0}) = 1 + \frac{p^{2} − 1}{p^{3} − p^{2}} + r\frac{p − 1}{p^{3} − p^{2}} \frac{p^{2} − p}{p − 1} = 1 + \frac{r + 1}{p} + \frac{1}{p^{2}} = \frac{1}{p^{2}} #X(F_{p}),

the expected local density at p.

4.10. The local Fourier transform at a generic character. — Let a be a generic character and p ∉ S(a). In that case, the summation is easier. The integrals over U_{k}(α, β) are 0, as are the integrals over U_{k}(α) or U(α) if α ⩾ 2. Therefore

Ĥ(s; ψ_{a}) = 1 − ∑_{k=1}^{r} p^{−s_{k}} + (r − 1)p^{−s_{0}}. (4.11)

For K_{X}^{−s}, this specializes to

Ĥ(K_{X}^{−s}, ψ_{a}) = 1 − rp^{−2s} + (r − 1)p^{−3s}.

4.12. The local Fourier transform at a special character. — If a is special for ℓ_{k} and p ∉ S(a), it behaves as if it were generic for the other ℓ_{j}. Besides U(0), U(1) and U_{j}(1) for j ≠ k, remain the integrals over U_{k}(α, α − 1) for α ⩾ 2 and the one over U(α) for α ⩾ 2.

Ĥ(s; ψ_{a}) = 1 − ∑_{j≠k} p^{−s_{j}} + (r − p − 1)p^{−s_{0}}

+ ∑_{α=1}^{∞} p^{α} \frac{p − 1}{p}p^{−αs_{k}} − ∑_{α⩾2}p^{α} \frac{p − 1}{p}p^{−αs_{0}−(α−1)(s_{k}−s_{0})}

= 1 − ∑_{j≠k} p^{−s_{j}} + (r − p − 1)p^{−s_{0}}

+ \frac{p − 1}{p} \frac{1}{p^{s_{k}−1} − 1} − \frac{p − 1}{p}p^{s_{k}−s_{0}} \frac{p^{−s_{k}−1}}{p^{s_{k}−1} − 1}

= 1 − ∑_{j≠k} p^{−s_{j}} + (r − p − 1)p^{−s_{0}} + (p − 1)(1 − p^{1−s_{0}})\frac{1}{p^{s_{k}} − p} (4.13)

For K_{X}^{−s}, one has

Ĥ(K_{X}^{−s}, ψ_{a}) = 1 − (r − 1)p^{−2s} + (r − p − 1)p^{−3s} + (p − 1)\frac{1 − p^{1−3s}}{p^{2s} − 1} .

4.14. Bad reduction, general estimates. — If p ∈ S(a), then the previous analysis doesn’t say anything about the behaviour of Ĥ(s; ψ_{a}). However, for any compact contained in the domain of integrability of the height function, there is a uniform estimate

|Ĥ_{p}(s; ψ_{a})| ⩽ C

where the constant C doesn’t depend on p ∈ S. For p ∈ S(a) \ S, we replace ψ_{a} by 1 and insert the estimates for the trivial character.

It follows that

∏_{p∈S(a)} |Ĥ_{p}(s; ψ_{a})| ⩽ C′(1 + ∥a∥)^{κ}

for some constant C′ and some κ > 0.

4.15. Meromorphic continuation. — We split the sum over all characters in r + 2 parts: the trivial character is treated separately; then the generic characters; then the characters which are special for ℓ_{k}, k varying from 1 to r:

∑_{a} Ĥ(s; ψ_{a}) = Ĥ(s; ψ_{0}) + ∑_{a generic} Ĥ(s; ψ_{a}) + ∑_{k=1}^{r} ∑_{a special for ℓ_{k}} Ĥ(s; ψ_{a}).

Let Z_{0}, Z_{gen} and Z_{k} (for 1 ⩽ k ⩽ r) be the functions defined by the corresponding series.

Each global Fourier transform at a generic character defines a holomorphic function of s in the domain Re(s_{0}) > 2 and Re(s_{k}) > 1 for all 1 ⩽ k ⩽ r. Moreover, the estimate at infinity 2.13 ensures that the sum over all generic characters converges locally uniformly. Therefore, Z_{gen} extends to a holomorphic function in that domain.

For the characters which are special for ℓ_{k}, the product of the local Fourier transform defines a holomorphic function of s in the domain Re(s_{0}) > 2, Re(s_{j}) > 1 if j ≠ k and Re(s_{k}) > 2. It extends to a meromorphic function in the domain Re(s_{0}) > 2, Re(s_{j}) > 1 and Re(s_{k}) > 1 with a simple pole along the hypersurface s_{k} = 2.

And finally, for the trivial character, we have absolute convergence for Re(s_{0}) > 3 and Re(s_{k}) > 2 for 1 ⩽ k ⩽ r, and meromorphic continuation to the domain Re(s_{0}) > 2, Re(s_{k}) > 1, with principal part 1/(s_{0}−3)(s_{1}−2) … (s_{r}−2).

The estimates of Prop. 2.13 as well as standard estimates for the growth of the Riemann zeta function in vertical strips imply that (away from poles) Z(s) has polynomial growth in vertical strips.

Therefore, we have proven the following theorem:

Theorem 4.16. — The height zeta function Z(s) converges in the domain Re(s_{0}) > 3, Re(s_{k}) > 2. Moreover, there exists a holomorphic function g in the domain Re(s_{0}) > 2, Re(s_{k}) > 1 such that

Z(s) = g(s)\frac{1}{(s_{0} − 3)(s_{1} − 2) … (s_{r} − 2)}

and g(K_{X}^{−1}) ≠ 0. Moreover, g has polynomial growth in vertical strips.

Corollary 4.17. — There exist a polynomial P_{X} of degree r and an α > 0 such that the number of rational points in U of anticanonical height ⩽ B satisfies

N(U, −K_{X}, B) = BP_{X}(log B) + O(B^{1−α}).

POINTS OF BOUNDED HEIGHT 29

Moreover, if τ(K_{X}) denotes the Tamagawa number, the leading coefficient of P_{X} is equal to

\frac{1}{r!} \frac{τ(K_{X})}{3 · 2^{r}},

as predicted by Peyre’s refinement of Manin’s conjecture.

Table 1

U(0)  U_{k}(α, β)  U_{k}(α)  U(α)
volume  1  p^{2α−β} \frac{(p−1)^{2}}{p^{2}}  p^{α} \frac{p−1}{p} p^{2α} \frac{(p−1)(p+1−r)}{p^{2}}
H_{0}  1  p^{α−β}  1  p^{α}
H_{j} (j ≠ k)  1  1  1  1
H_{k}  1  p^{β}  p^{α}  1
H(s; ·)  1  p^{αs_{0}+β(s_{k}−s_{0})}  p^{αs_{k}}  p^{αs_{0}}

Integrals of a generic character ψ_{a}

α = 1  1  −1  −1 + r
α ⩾ 2  1  0  0  0

Integrals of a character ψ_{a} special for ℓ_{k}

any α 1 p^{α} \frac{p−1}{p}
α = 1  −(p + 1 − r)
α ⩾ 2  0
α = β − 1  −p^{α} \frac{p−1}{p}

References

[1] V. V. Batyrev & Yu. I. Manin – “Sur le nombre de points rationnels de hauteur bornée des variétés algébriques”, Math. Ann. 286 (1990), p. 27–43.

[2] V. V. Batyrev & Yu. Tschinkel – “Tamagawa numbers of polarized algebraic varieties”, in Nombre et répartition des points de hauteur bornée [16].

[3] B. J. Birch – “Forms in many variables”, Proc. London Math. Soc. 265A (1962), p. 245–263.

[4] A. Chambert-Loir & Yu. Tschinkel – “ Fonctions zêta des hauteurs des espaces fibrés”, En préparation, 1998.

[5] , “ Torseurs arithmétiques et espaces fibrés”, prépublication éléctronique, math.AG/9901006, 1999.

[6] H. Davenport – “Cubic forms in sixteen variables”, Proc. Roy. Soc. London Ser. A 272 (1963), p. 285–303.

[7] J. Franke, Yu. I. Manin & Yu. Tschinkel – “Rational points of bounded height on Fano varieties”, Invent. Math. 95 (1989), no. 2, p. 421–435.

[8] R. Hartshorne – Algebraic geometry, Graduate Texts in Math., no. 52, Springer Verlag, 1977.

[9] B. Hassett & Yu. Tschinkel – “Geometry of equivariant compactifications of G_{a}^{n}”, Tech. report, 1999.

[10] R. Heath-Brown – “Cubic forms in 10 variables”, Proc. London Math. Soc. 47 (1983), no. 2, p. 225–257.

[11] C. Hooley – “On nonary cubic forms”, J. Reine Angew. Math. 386 (1988), p. 32–98.

[12] Yu. I. Manin & A. Panchishkin – Number theory I. Introduction to number theory, Springer, Berlin, 1995.

[13] D. Mumford, J. Fogarty & F. Kirwan – Geometric invariant theory, Ergeb., no. 34, Springer Verlag, 1994.

[14] E. Peyre – “Hauteurs et nombres de Tamagawa sur les variétés de Fano”, Duke Math. J. 79 (1995), p. 101–218.

[15] , “Torseurs et méthode du cercle”, Tech. report, Isaac Newton Institute for Mathematical Sciences, 1998.

[16] (ed.) – Nombre et répartition des points de hauteur bornée, Astérisque, 1999.

[17] , “Terme principal de la fonction zêta des hauteurs et torseurs universels”, in Nombre et répartition des points de hauteur bornée [16].

[18] P. Salberger – “Tamagawa measures on universal torsors and points of bounded height on Fano varieties”, in Nombre et répartition des points de hauteur bornée [16].

[19] S. Schanuel – “Heights in number fields”, Bull. Soc. Math. France 107 (1979), p. 433–449.

[20] W. Schmidt – “The density of integer points on homogeneous varieties”, Acta Math. 154 (1985), no. 3–4, p. 243–296.

[21] M. Strauch & Yu. Tschinkel – “Height zeta functions of toric bundles over flag varieties”, Selecta Math. (1999), to appear.

[22] J. T. Tate – “Fourier analysis in number fields, and Hecke’s zeta-functions”, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, p. 305–347.

April 9, 1999 Antoine Chambert-Loir, Institut de mathématiques de Jussieu, Boite 247, 4, place

Jussieu, F-75252 Paris Cedex 05 • E-mail : chambert@math.jussieu.fr Yuri Tschinkel, Dept. of Mathematics, U.I.C., Chicago, (IL) 60607-7045, U.S.A.

E-mail : yuri@math.uic.edu