On the distribution of points of bounded height on equivariant compactifications of vector groups

We prove asymptotic formulas for the number of rational points of bounded height on smooth equivariant compactifications of the affine space. (Nous \'etablissons un d\'eveloppement asymptotique du nombre de points rationnels de hauteur born\'ee sur les compactifications \'equivariantes lisses de l'espace affine.)


Introduction
A theorem of Northcott asserts that for any real number B there are only finitely many rational points in the projective space P n with height smaller than B. An asymptotic formula for this number (as B tends to infinity) has been proved by Schanuel [19]. Naturally, it is interesting to consider more general projective varieties and there are indeed a number of results in this direction. The techniques employed can be grouped in three main classes: -the classical circle method in analytic number theory permits to treat complete intersections of small degree in projective spaces of large dimension (cf. for example [5]); -harmonic analysis on adelic points of reductive groups leads to results for toric varieties [2], flag varieties [12] and horospherical varieties [20]; -elementary (but nontrivial) methods for del Pezzo surfaces of degree 4 or 5 cf. de la Bretèche [7], and for cubic surfaces (Salberger, Swinnerton-Dyer, Heath-Brown, cf. [18] and the references therein; This research has been stimulated by a conjecture put forward by Batyrev and Manin. They proposed in [1] an interpretation of the growth rate in terms of the mutual positions of the class of the line bundle giving the projective embedding, the anticanonical class and the cone of effective divisors in the Picard group of the variety. Peyre refined this conjecture in [17] by introducing an adelic Tamagawa-type number which appears as the leading constant in the expected asymptotic formula for the anticanonical embedding. Batyrev and the second author proposed an interpretation of the leading constant for arbitrary ample line bundles, see [3]. In this paper we consider a new class of varieties, namely equivariant compactifications of vector groups. On the one hand, we can make use of harmonic analysis on the adelic points of the group. On the other hand, such varieties have a rich geometry. In particular, in contrast to flag varieties and toric varieties, they admit geometric deformations, and in contrast to the complete intersections treated by the circle method their Picard group can have arbitrarily high rank. Their geometric classification is a difficult open problem already in dimension 3 (see [13]).
A basic example of such algebraic varieties is of course P n endowed with the action of G n a ⊂ P n by translations. A class of examples is provided by the following geometric construction: Take X 0 = P n endowed with the translation action of G n a . Let Y be a smooth subscheme of X 0 which is contained in the hyperplane at infinity. Then, the blow-up X = Bl Y (X 0 ) contains the isomorphic preimage of G n a and the action of G n a lifts to X. The rank of Pic(X) is equal to the number of irreducible components of Y . Using equivariant resolutions of singularities, one can produce even more complicated examples, although less explicitly.
Our first steps towards this paper are detailed in [8] and [9]. There we studied the cases n = 2 with Y a finite union of Q-rational points and n > 2 with Y a smooth hypersurface contained in the hyperplane at infinity.
We now describe the main theorem of this article. Let X be an equivariant compactification of the additive group G n a over a number field F . Unless explicitly stated, we shall always assume that X is smooth and projective. The boundary divisor D = X \ G n a is a sum of irreducible components D α (α ∈ A ). We do not assume that they are geometrically irreducible.
The Picard group of X is free and has a canonical basis given by the classes of D α . The cone of effective divisors consists of the divisors α∈A d α D α with d α 0 for all α. Denote by K −1 X the anticanonical line bundle on X and by ρ = (ρ α ) its class in Pic(X). Let λ = (λ α ) be a class contained in the interior of the cone of effective divisors Λ eff (X) ⊂ Pic(X) and L λ the corresponding line bundle, equipped with a smooth adelic metric (see 3.1 for the definition). With the above notations, we have λ α > 0 for all α. Denote by H L λ the associated exponential height on X(F ). Let a λ = max(ρ α /λ α ) and let b λ be the cardinality of For example, one has a ρ = 1 and b ρ = rk Pic(X). We denote by τ (K X ) the Tamagawa number as defined by Peyre in [17]. c) For λ = ρ we have τ ρ = τ (K X ).  The proof proceeds as follows. First, we extend the height function to the adelic space G n a (A F ). Next we apply the additive Poisson formula and find a representation of Z λ (s) as a sum over the characters of G n a (A F )/G n a (F ) of the Fourier transforms of H. The Fourier transforms of H decompose as products over all places v of "global" integrals on X(F v ) which are reminiscent of Igusa zeta functions.
The most technical part of the paper is devoted to the evaluation of these products: meromorphic continuation and control of their growth in vertical strips. For this, we need to consider the special case where D has strict normal crossings: this means that over an algebraic closure of F , D is a sum of smooth irreducible divisors meetings transversally. Then, at almost all nonarchimedean places, we compute explicitly the local Fourier transforms in terms of the reduction of X modulo the corresponding prime. The obtained formulas resemble Denef's formula in [10] for Igusa's local zeta function. For the remaining nonarchimedean places we find estimates. This leads to a proof of the meromorphic continuation of the Fourier transforms at each character.
To consider the general cas, where D is not assumed to have strict normal crossings, we then introduce a proper modification π :X → X which is a composition of equivariant blow-ups with smooth centers lying on D such thatX is a smooth projective equivariant compactification of G n a whose boundary divisorD has strict normal crossings: over an algebraic closure of F ,D is a sum of smooth irreducible divisors meeting transversally. Considering π * λ, all the computations shall be made onX. We also prove in Lemma 6.1 that a π * λ = a λ and b π * λ = b λ .
In the Poisson formula, invariance properties of the height reduce the summation over G n a (F ) to one over a lattice. The meromorphic continuation in part a) of the theorem follows then from additional estimates for the Fourier transforms at the infinite places.
At this stage one has a meromorphic continuation of Z λ (s) to the domain Re(s) > a λ −δ for some δ > 0, with a single pole at s = a λ whose order is less or equal than b λ . It remains to check that the order of the pole is exactly b λ ; we need to prove that the limit For λ = ρ this is more or less straightforward: the main term is given by the summand corresponding to the trivial character and the Tamagawa number defined by Peyre appears naturally in the limit. For other λ we use the Poisson formula again and relate the limit to an integral of the height over some subspace which is shown to be strictly positive.
Part b) follows by a Tauberian theorem.

Geometry
Let X be a smooth projective equivariant compactification of the additive group G n a of dimension n over a field F : X is a (smooth, projective) algebraic variety endowed with an action of G n a with a dense orbit isomorphic to G n a . We will always assume that X is smooth and projective. Let D = X \ G n a denote the boundary divisor. We have a decomposition in irreducible components where A is a finite set and for each α ∈ A , D α is an integral divisor in X. We do not assume that the D α are geometrically irreducible, nor that they are smooth.
There is however a similar description over any extension of F . In particular, letF be a separable closure of F and let DF = α∈AF DF ,α be the decomposition of DF in irreducible components. The natural action of Γ F = Gal(F /F ) onX induces an action of Γ F on AF such that for any α ⊂ AF , g(D α ) = D g(α) .
where Λ eff (X) is the monoid of classes of effective divisors of X.
Subsequently, we identify the divisors D α (and the corresponding line bundles) with their classes in the Picard group.
Proof. -These maps are equivariant under the action of Γ F . Hence, it remains to show injectivity and surjectivity.
For this we may assumeF = F . Let L be a line bundle on X. As X is smooth and Pic(G n a ) = 0, there is a divisor D in X not meeting G n a such that L ≃ O X (D). Such a divisor D is a sum α∈A n α D α . Moreover, such a D is necessarily unique. If is a rational function on X without zeroes nor poles on G n a . Rosenlicht's lemma (see lemma 1.2 below) implies that s D /s D ′ is constant, so that D = D ′ .
We remark that any effective cycle Z on X is rationally equivalent to a cycle that does not meet G n a . Indeed, if t is the parameter of a subgroup of G n a isomorphic to G a , we can consider the specialization of the cycles t + Z when t → ∞.  The Γ F -module Pic(X) is a permutation module. In particular, H 1 (Γ F , Pic(X)) = 0.
2) Pic(X) = Pic(X) Γ F is a free Z-module of finite rank equal to the number of Γ F -orbits in AF .
For any α ∈ A , we shall denote by by F α the algebraic closure of F in the function field of D α . Over F α , the irreducible components of D α are geometrically irreducible. We also denote by ζ Fα the Dedekind zeta function of the number field F α .
Let f be a nonzero linear form on G n a viewed as an element of F (X). Its divisor can be written as where E(f ) is the unique irreducible component of {f = 0} that meets G n a and d α (f ) are integers. (The divisor E(f ) can also be seen as the closure in X of the hypersurface of G n a defined by f .) Since E(f ) is rationally equivalent to α∈A d α (f )D α , the preceding proposition implies the following lemma: -Let X be a normal equivariant compactification of G n a over F . Every line bundle L on X admits a unique G n a -linearization. If L is effective then H 0 (X, L ) has a unique line of G n a -invariant sections.
Proof. -Since G n a has no nontrivial characters, it follows from Proposition 1.4 and from the proof of Proposition 1.5 in [16], Chapter 1, that any line bundle on X admits a unique G n a -linearization. Assume that H 0 (X, L ) = 0 and consider the induced action of G n a on the projectivization P(H 0 (X, L )). Borel's fixed point theorem ( [6], Theorem 10.4) implies that there exists a nonzero section s ∈ H 0 (X, L ) such that the line F s is fixed under this action. As G n a has no nontrivial characters, s itself is fixed. The divisor div(s) is G n a -invariant; therefore, div(s) does not meet G n a and is necessarily a sum d α D α such that L ≃ O X ( d α D α ). Because of Proposition 1.1, every other such section will be proportional to s.

Vector fields
We now recall some facts concerning vector fields on equivariant compactifications of algebraic groups. Let G be a connected algebraic group over F and g its Lie algebra of invariant vector fields. Let X be a smooth equivariant compactification of G. Denote by D = X \ G the boundary. We assume that D is a divisor with strict normal crossings. Let T X be the tangent bundle of X. Evaluating a vector field at the neutral element 1 of G induces a "restriction map" Conversely, given ∂ ∈ g, there is a unique vector field ∂ X such that for any open subset U of X and any f ∈ O X (U), ∂ X (f )(x) = ∂ g f (g · x)| g=1 . The map ∂ → ∂ X is a section of the restriction map. (This is of course peculiar to the additive group. For a general algebraic group G, vector fields on G of the form ∂ X | G need not be invariant.) Proof. -By purity, it is sufficient to prove this when D is smooth in a neighborhood of x (since X is smooth, a function which is regular in the complement to a codimension 2 subscheme is regular everywhere). Then, we can choose (étale) local coordinates x 1 , . . . , x n in U (i.e., elements of O X (U), Ω 1 X | U is free over O U with basis dx 1 , . . . , dx n ) such that s D = x 1 . Moreover, we can write uniquely Necessarily, G stabilizes D x so that the function g → x 1 (g · x) is identically 0 in a Zariski neighborhood of 1 in G and a fortiori, The lemma is proved.
, the Lie algebra g has a canonical basis ∂, given by the local parameter x at the neutral element (respectively 0 and 1) of G. If we embed G in P 1 equivariantly, we get ∂ X = ∂/∂x for G = G a and x∂/∂x for G = G m . We see that it vanishes at infinity (being {∞} or {0; ∞}, accordingly). This is a general fact, as the following lemma shows.

Lemma 2.4 (Hassett/Tschinkel
). -There exist integers ρ α 1 such that If G = G n a , then for each α, ρ α 2. Proof. -We only prove that ρ α 1, because the sharper bound which is valid in the case of G = G n a won't be used below. We refer to [13], Theorem 2.7 for its proof. Let ∂ 1 , . . . , ∂ n be a basis of g. Then, δ : , hence we have to prove that these integers are positive.
Fix any x in the smooth part of X \G, so that there exists a unique α ∈ A such that x ∈ D α . Pick local coordinates x 1 , . . . , x n in a neighborhood U of x in such a way that in U, D is defined by the equation

Metrizations
Let F be a number field and o F its ring of integers. Denote by F v the completion of F at a place v, by o v the ring of integers in F v if v is nonarchimedean, by A F the ring of adeles of F and by A fin the (restricted) product of F v over the nonarchimedean places v. The valuation in F v is normalized in such a way that for any Haar measure µ v on F v and any measurable subset Let X be an equivariant compactification of G n a as above and L a line bundle on X, endowed with its canonical linearization.
Definition 3.1. -A smooth adelic metric on L is a family of v-adic norms · v on L for all places v of F satisfying the following properties: , the norm of any local nonvanishing section is C ∞ , resp. locally constant) (c) there exists an open dense subset U ⊂ Spec(o F ), a flat projective U-scheme X /U extending X together with an action of G n a /U extending the action of G n a on X and a linearized line bundle L on X /U extending the linearized line bundle on X, such that for any place v lying over U, the v-adic metric on L is given by the integral model. If m and p 2 denote the action and the second projection G×X → X, respectively, endow the trivial line bundle m * L ⊗ p * 2 L −1 on G n a × X with the tensor-product metric. This is a locally constant metric on the trivial line bundle. The function on G n is the norm of the canonical basis 1 and therefore extends to a locally constant function . This proves the lemma in the effective case.
In the general case, we write L = L 1 ⊗ L −1 2 for two effective line bundles (each having a nonzero global section). We can endow L 2 with any locally constant v-adic metric and L 1 with the unique (necessarily locally constant) v-adic metric such that the isomorphism L 1 ≃ L ⊗ L 2 is an isometry. By the previous case, there exist two compact open subgroups K 1,v and K 2,v contained in G n a (o v ) which act isometrically on L 1 and L 2 , respectively. Their intersection acts isometrically on L .
. From now on we choose adelic metrics on the line bundles L in such a way that the tensor product of two line bundles is endowed with the product of the metrics (this can be done by fixing smooth adelic metrics on a Z-basis of Pic(X) and extending by linearity). We shall denote by K the compact open subgroup of G n a (A fin ) stabilizing all these metrized line bundles on X . We also fix an open dense U ⊂ Spec(o F ), a flat and projective model X /U over U and models of the line bundles L such that the chosen v-adic metrics for all line bundles on X are given by these integral models.

Heights
Let X be an algebraic variety over F and (L , · v ) an adelically metrized line bundle on X. The associated height function is defined as where s is any F -rational section of L not vanishing at x. The product formula ensures that H L does not depend on the choice of the F -rational section s (though the local heights H L ,v (x) do).
In Section 3 we have defined simultaneous metrizations of line bundles on equivariant compactifications of G n a . This allows to define compatible systems of heights Fix for any α ∈ A some non zero G n a -invariant section s α of O X (D α ). We can then extend the height pairing H to a pairing Recall that K denotes the compact open subgroup of G n a (A F ) stabilizing all line bundles on X together with their chosen metrization. By construction, we have the following proposition.  Proof. -Let M be an ample line bundle on X and let ν be a sufficiently large integer such that L ν ⊗ M −1 is effective. It follows from the preceding section that L ν ⊗ M −1 has a section s which does not vanish on G n a . This implies that the function x → − log H(x; L ν ⊗ M −1 ) is bounded from above on G n a (F ). Therefore, there exists a constant C > 0 such that for any x ∈ G n a (F ), We may now apply Northcott's theorem and obtain the desired finiteness.  Proof. -Fix a basis (L j ) j of Pic(X) consisting of (classes of) line bundles lying in the interior of the effective cone of Pic(X). Let Ω t denote the open set of all linear combinations t j L j ∈ Pic(X) R such that for some j, t j > t. Fix some ample line bundle M . It is well known that the height zeta function of X relative to M converges for Re(s) big enough, say Re(s) > σ 0 . In the proof of Proposition 4.3 we may choose some ν which works for any L j , so that for any j Therefore, the height zeta function converges absolutely and uniformly on the tube domain The following sections are devoted to the study of analytic properties of Z(s).

The Poisson formula
We recall basic facts concerning harmonic analysis on the group G n a over the adeles A = A F (cf., for example, [21]). For any prime number p, we can view Q p /Z p as the p-Sylow subgroup of Q/Z and we can define a local character ψ p of G a (Q p ) by setting At the infinite place of Q we put (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 ).
If a ∈ G n a (A F ), recall that f a = ·, a is the corresponding linear form on G n a (A F ) and let ψ a = ψ • f a . This defines a map G n a (A F ) → G n a (A F ) * . It is well known that this map is a Pontryagin duality. The subgroup G n a (F ) ⊂ G n a (A F ) is discrete, cocompact and we have an induced Pontryagin duality G n a (A F ) → (G n a (A F )/G n a (F )) * . We fix selfdual Haar measures dx v on G a (F v ) for all v. We refer to [21] for an explicit normalization of these measures. We will use the fact that for all but finitely many v the volume of G a (o v ) with respect to dx v is equal to 1. Thus we have an induced selfdual Haar measure dx on G a (A F ) and the product measure dx on G n a (A F ). The Fourier transform (in the adelic component) of the height pairing on G n a (A F ) × Pic(X) C is defined bŷ We will use the Poisson formula in the following form (cf. [15], p. 280).
converges absolutely and uniformly when b belongs to G n a (A F )/G n a (F ). Then, The following lemma (a slight strengthening of Proposition 4.5) verifies the two hypotheses of the Poisson formula 5.1 concerning H. Indeed, the quotient H(x + b; L )/H(x; L ) defines a bounded continuous function of (x, b) ∈ G n a (A F ) × K. (Only at a finite number of places do they differ, and at these places v, a comparison is provided by the compactness of X(F v ).) As G n a (A F )/G n a (F ) is compact, one may take for K any compact set containing a fundamental domain. This implies the uniform convergence of the series once s belongs to the tube domain of Proposition 4.5.
The integrability follows at once : for such s, The following proposition follows from the invariance of the height under the action of the compact K. Consequently, the set d X of all a ∈ G n a (F ) such that ψ a is trivial on K is is a sub-o Fmodule of G n a (F ), commensurable with G n a (o F ). Consequently, provided the Poisson formula applies and we have a formal identity for the height zeta function: The last four Sections of the paper are concerned with analytic arguments leading to the evaluation ofĤ and to the fact that Equation (5.4) holds for all s in some tube domain. It will be necessary to assume that the boundary divisor D has strict normal crossings, which means that over the separable closureF of F , D is a union of smooth irreducible components meeting transversally.
We sum up these Sections in the following Proposition. Recall that for a ∈ G n a (F ), A 0 (a) is the set of α ∈ A such that d α (f a ) = 0. In particular, A 0 (0) = A . Also, for any t ∈ R, we denote by Ω t the open tube domain in Pic(X) C defined by the inequalities s α > ρ α + t. Finally, let · ∞ denote any norm on the real vector space G n a (F ⊗ Q R) ≃ v|∞ F n ∞ . Proposition 5.5. -Assume that D has strict normal crossings. a) For any s ∈ Ω 0 , H(·; s) is integrable on G n a (A F ). b) For any ε > 0 and a ∈ d X there exists a holomorphic bounded function ϕ(a; ·) on Ω −1/2+ε such that for any s ∈ Ω 0 H(ψ a ; s) = ϕ(a; s) c) For any N > 0 there exist constants N ′ > 0 and C(ε, N) such that for any s ∈ Ω −1/2+ε and any a ∈ d X one has the estimate |ϕ(a; s)| C(ε, N)(1 + Im(s) ) N ′ (1 + a ∞ ) −N .

Meromorphic continuation of the height zeta function
For the proof of our main theorem, we need to extend the estimates of Proposition 5.5 when D is not assumed to have strict normal crossings. Proof. -Equivariant resolution of singularities (in char. 0, see [11] or [4]) shows that there exists a composition of equivariant blow-up with smooth centers π :X → X such that -X is a smooth projective equivariant compactification of G n a ; -π is equivariant (hence induces an isomorphism on the G n a ); -the boundary divisorD =X \ G n a has strict normal crossings. The map π induces a map from the setÃ of irreducible components ofD to A . Moreover, π has a canonical section obtained by associating to a component of D its strict transform inX. This allows to view A as a subset ofÃ , the complementary subset E consisting of exceptional divisors. Via the identifications of Pic(X) with Z A and PicX with ZÃ , the map π * on line bundles induces a linear map i : Z A → ZÃ which is a section of the first projection ZÃ = Z A ⊕ Z E → Z A . (We also extend i by linearity to the vector spaces of complexified line bundles.) Concerning metrizations, it is possible to endow line bundles onX with adelic metrics which extend the metrics coming from X via π * . Remark that with such a choice, the height function on G n a (A F ) and its Fourier transform onX extend those on X. For instance, for any s ∈ Pic(X) C and any x ∈ G n a (A F ), one has H(x; s) =H(x; i(s)).
As i maps N * A into N * Ã , it follows in particular that H(·; s) is integrable for s ∈ Ω 0 and that for any a ∈ G n a (A F ) and any s ∈ Ω 0 , one haŝ H(ψ a ; s) =Ĥ(ψ a ; i(s)).
Unless π is an isomorphism, π is not smooth and ωX ⊗ π * ω −1 X has a nonzero coefficient at each exceptional divisor. For any effective λ, this implies that i(λ) − a λρ lies on the boundary of Λ eff (X). (Recall that the classes of ω −1 X and ω −1 X in Z A and ZÃ are denoted ρ andρ.) It follows that a i(λ) = a λ and that B(i(λ)) = {α ∈Ã ; i(λ) = a i(λ)ρ } can be identified with B(λ).
Consequently, any "exceptional" factor (s α −ρ α ) −1 for α ∈ E extends to a holomorphic bounded function on i(Ω −1/2 ) This shows that Proposition 5.5 still stands true without the hypothesis that D has strict normal crossings.
One has also the following result which shall allow for the determination of the constant in the asymptotics. Proposition 6.2. -Denote by τ (K X ) the Tamagawa number of X defined by the adelic metrization on K X . One then has the equality ϕ(0; ρ) = τ (K X ).
In particular, it is a strictly positive real number.
Proof. -Recall first Peyre's definition [17] of the Tamagawa measure in this context, which extends Weil's definition in the context of linear algebraic groups [22]. For any place v of F and any local non-vanishing differential form ω on X(F v ), the self-dual measure on G n a (F v ) induces a local measure |ω| / ω on X(F v ). These local measures glue to define a global measure dµ v on X(F v ).
Recall that Artin's L-function of Pic(XF ) is equal to α ζ Fα . Using Deligne's theorem on Weil's conjectures (but it will also follow from our calculations), Peyre has shown that the Euler product converges absolutely and he defined the Tamagawa number of X as (Note however that Peyre apparently doesn't use the selfdual measure for his definition in loc. cit., but inserts the appropriate correcting factor ∆ − dim X/2 F

.)
On the open subset G n a (F v ), we can define the local measure with the differential form ω = dx 1 ∧ · · · ∧ dx n induced by the canonical coordinates of G n a . Let c ∈ F × be such that ω = c α s α −ρα . On G n a (F v ), we thus have so that, the boundary subset being of measure 0, Moreover, it follows from these results and the absolute convergence of the Euler products of Dedekind zeta functions for Re(s) > 1 that for any s > 1, H(·; sρ) is integrable on X(A F ) and that one has The estimates of Proposition 9.5 below allow us to interchange the limit lim s→1 and the infinite product v , so that because of the product formula.
It follows from these results that the Poisson formula (5.4) applies for any s ∈ Ω 0 and that Hence, This last series is a sum of holomorphic functions on Ω −1/2 and taking N > n[F : Q], the estimate in Proposition 5.5 implies that the series converges locally uniformly in that domain. Therefore, we have shown that the function has a holomorphic continuation to Ω −1/2 with polynomial growth in vertical strips. The restriction of Z(s) to a line C(λ α ) (with λ α > 0 for all α) gives where h λ is a holomorphic function for s ∈ C such that providing a meromorphic continuation of Z(sλ) to this domain. The rightmost pole is at and its order is less or equal than b λ . Theorem 6.3. -Let X be a smooth projective equivariant compactification of G n a over a number field F . For any strictly effective λ ∈ Pic(X), the height zeta function Z(sλ) converges absolutely for Re(s) > a λ , has a meromorphic extension to the left of a λ , with a single pole at a λ of order b λ .
When λ = ρ, a ρ = 1, b ρ = rk Pic(X) = r and Proposition 6.2 implies the equality hence the order of the pole is exactly r. Using a standard Tauberian theorem, we deduce an asymptotic expansion for the number of rational points in G n a of bounded anticanonical height.
Theorem 6.4. -Let X be a smooth projective equivariant compactification of G n a over a number field F .
There exists a real number δ > 0 and a polynomial P ∈ R[X] of degree r − 1 such that the number of F -rational points on G n a ⊂ X of anticanonical height B satisfies

Asymptotics
Let λ ∈ Λ eff (X) be an effective class and L λ the corresponding line bundle on X equipped with a smooth adelic metric as in Section 3.
The function s → Z(sλ) is holomorphic for Re(s) > a λ = max(ρ α /λ α ) and admits a meromorphic continuation to the left of a λ . Recall from the previous section that it has a pole of order b λ at a λ , where b λ is the cardinality of (Geometrically, the integer b λ is the codimension of the face of the effective cone Λ eff (X) containing the class a λ λ − ρ.) In this section we prove that the order of the pole of the height zeta function Z(sλ) at s = a λ is exactly b λ and derive our main theorem.
Denote by Z λ (s) the sub-series It follows from the calculations above that since all other terms converge to zero when s → a λ , with uniform convergence of the series.
The set V λ of a ∈ G n a (F ) such that A 0 (a) contains B λ is a sub-vector space. Let G λ ⊂ G n a be the F -sub-vector space defined by the corresponding linear forms a, · .
Then, the autoduality on G n a (A F ) identifies V λ with the dual of G λ (A F )G n a (F ). We apply the Poisson summation formula and obtain H(x + y; sλ) −1 dy.
(The justification of Poisson summation formula is as in Section 5.) Proof. -It is sufficient to prove the proposition when x = 0 as the integrals for different values of x are comparable. The Zariski closure Y of G λ in X need not be smooth. Nevertheless, we may introduce an equivariant proper modification π :X → X such that the Zariski closureỸ ⊂X of G λ is a smooth equivariant compactification whose boundary is a divisor with strict normal crossings obtained by intersecting the components of the boundary ofX withỸ . The arguments of Lemma 6.1 show that we do not lose any generality by assuming this on X itself.
Proof. -We prove this by induction on the codimension of Y in X. If G λ = div(f a ), the adjunction formula shows that For a ∈ V λ the nonnegative integer d α (a) is equal to 0 if and only if α ∈ B λ . The lemma is proved.
End of the proof of Proposition 7.2. -Let dτ ′ be the (nonrenormalized) Tamagawa measure on Y (A F ): by definition, on G λ (A F ), dτ = H(y; ρ ′ ) −1 dy. We have to estimate The integral in the limit is of the type studied in Prop 5.5, but on the subgroup Y . Denote therefore by primes ′ restrictions of objects from X to Y . Thanks to the previous lemma, one has a λ ′ = a λ , B(λ ′ ) = B(λ) and and b λ ′ = b λ . Therefore, with notations from Proposition 5.5, The proposition is proved.
We now can conclude as in the case of the anticanonical line bundle: Theorem 7.4. -Let λ ∈ Pic(X) be the class of a line bundle contained in the interior of the effective cone Λ eff (X), and equip the corresponding line bundle L λ with a smooth adelic metric as in Section 3. There exist a polynomial P λ ∈ R[X] of degree b λ − 1 and a real number ε > 0 such that the number N(L λ , B) of F -rational points on G n a of L λ -height B satisfies The leading term of P λ is equal to It is compatible with the description by Batyrev and Tschinkel in [3].

General estimates
In the remaining sections we have to prove Prop. 5.5. First we define a set of bad places S: as in Section 3, fix a good model X /U (flat projective U-scheme) over a dense open subset U ⊂ Spec(o F ). In particular, we assume that X /U extends X as an equivariant compactificatation of G n a whose boundary consists of the schematic closures of the D α . Moreover, we can assume that for any α, the chosen section s α of O(D α ) extends to X /U and that its norm coincides with the one that can be defined with this model at all finite places dominating U. In particular, at such places v, the local height functions restricted to G n a (F v ) are invariant under G n a (o v ). We can also assume that for any finite place v dominating U, the (nonrenormalized) Tamagawa measure dµ v is given on G n a (F v ) by H v (x; ρ) dx.
Let S be the set of places v of F such that either -v is archimedean, or -v doesn't dominate U, or -the residual characteristic of v is 2 or 3, or -the volume of G n a (o v ) with respect to dx v is not equal to 1 (equivalently, F is ramified at v), or -over the local ring o v , the union α D α is not a transverse union of smooth relative divisors over U. For a ∈ d X , let f a be the corresponding linear form on G n a , considered as an element of F (X) or of F (X /U ) and div(f a ) its divisor. We denote by S(a) the set of all valuations v such that either Finally, recall that for any t ∈ R, Ω t is the tube domain in Pic(X) C consisting of classes (s α ) such that for any α, Re(s α ) > ρ α .
In Section 8 we prove estimates for finite products v∈S(a)Ĥ v (ψ a ; s). In Sections 9 and 10 we compute explicitly the local Fourier transformsĤ v (ψ a ; s) for all v / ∈ S(a). In the following four sections we temporarily assume that the irreducible components of the boundary of X are geometrically irreducible. In Section 11, we shall explain how our results extend to the general case, (still assuming that D has strict normal crossings). where s = Re(s) + Im(s) .
We subdivide the proof of this proposition into a sequence of lemmas.
if and only if s ∈ Ω −1 ( i.e., Re(s α ) > ρ α − 1 for all α ∈ A ). Moreover, for all ε > 0 and all nonarchimedean places v, there exists a constant C v (ε) such that for all s ∈ Ω −1+ε and all a ∈ d X one has the estimate Proof. -Without loss of generality, we can assume that ψ a (x) ≡ 1. Now, using an analytic partition of unity on X(F v ) and the assumption that the boundary D = α∈A D α is a divisor with strict normal crossings, we see that it suffices to compute the integral over a relatively compact neighborhood of the origin in F n v (denoted by B) on which we have a set coordinates x 1 , . . . , x n so that in B the divisor D is defined by the equations x 1 · · · x a = 0 for some a ∈ {1, . . . , n}. In B there exist continuous bounded functions h α The integral over B is comparable to an integral of the same type with functions h α replaced by 0. The lemma is now a consequence of the following well-known lemma. Proof. -We may assume s ∈ R. Choose c ∈ ]0; 1[ such that the annulus c < |x| 1 has positive measure in K and let I 0 = c<|x| 1 |x| s dx. Then, we have It follows that the integral over K converges if and only if the geometric series c n(s+1) converges, that is if s + 1 > 0.
Proof. -Let us assume for the moment that F v = R. We shall write a v = (a 1 , ..., a n ), x v = (x 1 , ..., x n ) and dx v = dx. Then, in the domain Ω −1 one haŝ Now integrate by parts: for any j ∈ {1, . . . , n} we have For any α, let us define h α = log s Dα ; it is a C ∞ function on G n a (R). Let x be a point of X(R) and let A be the set of α ∈ A such that x ∈ D α . If t α is a local equation of D α in a neighborhood U of some R-point of D α , then there is a C ∞ function ϕ α on U such that for x ∈ U ∩ G n a (R) we have h α (x) = log |t α (x)| + ϕ α (x). It then follows from Propositions 2.1 and 2.2 that for each α, extends to a C ∞ function on X(R).
From the equality we deduce by induction the existence of an isobaric polynomial P N ∈ R[X (1) α , . . . , X (N ) α ] of degree N (with the convention that each X (p) α has weight p) and such that Choosing j such that |a j | is maximal gives |a j | a / √ n, hence an upper bound where ψ 0 is the trivial character and C ′ v (ε, N) some positive constant. To conclude the proof it suffices to remark that for any ε > 0,Ĥ v is bounded on the set Ω −1+ε (but the bound depends on ε).
The case F v = C is treated using a similar integration by parts. (The exponent 2 on the numerator comes from the fact that for a complex place v, · v is the square of a norm.)

Nonarchimedean computation at the trivial character
In this section we consider only v / ∈ S. Let m v ⊂ o v be the maximal ideal, k v = o v /m v and q = #k v . Recall that we have fixed a good model X /U over some U ⊂ Spec o F . To simplify notations we will write x = x v , dx = dx v etc.
The following formula is an analogue of Denef's formula in [10, Thm 3.1] for Igusa's local zeta function.
Remark 9.2. -For s = ρ we get #X /U (k v )/q dim X , the expected local density at v.
Proof. -We split the integral along residue classes mod m v . Letx ∈ X (k v ) and A = {α ∈ A ;x ∈ D α } so thatx ∈ D • A . We can introduce local (étale) coordinates x α (α ∈ A) and y β (β ∈ B, #A + #B = dim X) aroundx such that locally, the divisor D α is defined by the vanishing of x α . Then, the local Tamagawa measure identifies with the measure dx α × dy β on m A v × m B v . If dx denotes the fixed measure on G n a (F v ), one has the equality of measures on G n a (F v ) ∩ red −1 (x): Consequently, Summing these equalities forx ∈ X /U (k v ) gives the desired formula.
To estimate furtherĤ v (ψ 0 ; s) and the product over all places, we need to give an estimate for the number of k v -points in D α which is uniform in v.
Lemma 9.4. -There exists a constant C(X) such that for all v / ∈ S and all A ⊂ A we have the estimates: Proof. -We use the fact that for any projective variety Y of dimension dim Y and degree deg Y the number of k v -points is estimated as (see, e.g., Lemma 3.9 in [9]). Since X is projective, all D A can be realized as subvarieties in some projective space. This proves the second part. To prove the first part, we apply Lang-Weil's estimate [14] to the geometrically irreducible smooth U-scheme D α . Proposition 9.5. -For all ε > 0 there exists a constant C(ε) such that for any s ∈ Ω − 1 2 +ε and any finite place v ∈ S, Proof. -Using the uniform estimates from Lemma 9.4 we see that in the formula for H v , each term with #A 2 is O(q −( 1 2 +ε)#A ) = O(q −1−2ε ), with uniform constants in O. Turning to the remaining terms, we get Finally, we have the desired estimate.
For X as above and s = (s α ) ∈ Pic(X) C , the (multi-variable) Artin L-function is given by From standard properties of Dedekind zeta functions, we conclude that L(Pic(X); s) admits a meromorphic continuation and that it has polynomial growth in vertical strips. For s = (s, . . . , s) we get the usual Artin L-function of Pic(X)-here a power of the Dedekind zeta function-which has been used in the regularization of the Tamagawa measure in Prop. 6.2.

It suffices to define
The polynomial growth of ϕ(0; ·) in vertical strips follows from the boundedness of f in Ω −1/2+ε and from the fact that Dedekind zeta functions have a polynomial growth in such vertical strips.

Other characters
Our aim here is to compute as explicitely as possible the Fourier transforms of local heights with character ψ a . In general, this will be only possible up to some error term.
The calculations in the preceding Section allow us to strengthen Proposition 8.1. For a ∈ G n a (F ), we denote by a ∞ a norm of a in the real vector space defined by extension of scalars via the diagonal embedding F ֒→ F ⊗ Q R = v|∞ F v .
Proposition 10.1. -For any ε > 0, there exist an integer κ 0 and for any N 0, a constant C(ε, N) such that for any s ∈ Ω −1/2+ε and all a ∈ d X we have v∈S(a) Proof. -For v ∈ S the local integrals converge absolutely in the domain under consideration and are bounded as in Proposition 8.1. For v ∈ S, we have shown that there exists a constant c such that for all a ∈ d X and all s ∈ Ω −1/2+ε one has the estimate This implies that Ĥ v (ψ a ; s) is bounded independently of a, v ∈ S(a)\S and s ∈ Ω −1/2+ε . For any nonzero a ∈ o F , there is a trivial estimate which implies that for some constant κ, For the explicit calculation at good places, let us recall some notations: m v ⊂ o v is the maximal ideal, π = π v a uniformizing element, k v the residue field, As in Section 1, let f = f a be a linear form on G n a and write div(f ) = E − α d α D α . We have also defined A 0 (a) = {α ; d α (f a ) = 0} and A 1 (a) = {α ; d α (f a ) = 1}.
Proposition 10.2. -There exists a constant C(ε) independent of a ∈ d X such that for any v ∈ S(a), Similarly to the proof of Proposition 9.5 in the preceding Section, this proposition is proved by computing the integral on residue classes.
Letx ∈ X(k v ) and A = {α ;x ∈ D α }. We now consider three cases: -The sum of all these contribution is equal to the integral over G n a (o v ): -We can introduce analytic coordinates x α and y β aroundx such that locally f (x) = ux −dα α with u ∈ o * v . Then, we compute the integral of H v (x; s) −1 ψ a (x) as Proof.
We can find such an e if and only if 2(nd − r − 1) nd, i.e. nd 2r + 2. If r = 0, this is true since d 2. If r 1, one has nd p r 2r + 2 since we assumed p 5.
This lemma implies the following trichotomy: Case 3. #A 2 or #A = 1 andx ∈ E. -Under some transversality assumption, we could compute explicitely the integral as before. We shall however content ourselves with the estimate obtained by replacing ψ by 1 in the integral.
The total contribution of these points will therefore be smaller than Finally, the Fourier transform is estimated as follows: H v (ψ a ; s) = 1 + q −n α∈A 0 (a) with an "error term" ET smaller than the expression in (10.4). It is now a simple matter to rewrite this estimate as in the statement of 10.2.

The nonsplit case
In this section we extend the previous calculations of the Fourier transform to the nonsplit case, i.e. when the geometric irreducible components of X \ G n a are no longer assumed to be defined over F . Lemma 11.1. -Let x ∈ X(F ) and A = {α ∈ A ; x ∈ D α }. Fix for any orbitᾱ ∈ A/Γ F some element α and let F α be the field of definition of D α . Then there exist an open neighborhood U of x,étale coordinates around x overF , x α (α ∈ A) and y β such that x α is a local equation of D α and such that the induced morphism UF → A n /F descends to ań etale morphism An analogous result holds over the local fields F v and also on o v , v being any finite place of F such that v ∈ S. Proof. -Chose some element α in each orbitᾱ and fix a local equation x α for the corresponding D α which is defined over the number field F α . Then if α ′ = gα (for some g ∈ Γ F ) is another element in the orbit of α, set x α ′ = g · x α . This is well defined since we assumed that the equation x α is invariant under Γ Fα .
Finally, add F -rational localétale coordinates corresponding to a basis of the subspace in Ω 1 X,x which is complementary to the span of dx α for α ∈ A. Let v be a place of F . The above lemma allows us to identify a neighborhood of x in X(F v ) (for the analytic topology) with a neighborhood of 0 in the product α∈A/Γv F v,α × F n−a v , the local heights α∈α H α,v (ξ) being replaced by where h α,v is a smooth function.
Similarly, for good places v we identify red −1 (x) with m v,α × m n−a and the functions h α,v are equal to 1.
The assertions of Section 8 still hold in this more general case and the proofs require only minor modifications. However the calculations of Sections 9 and 10 have to be redone.