Height zeta functions of toric bundles over flag varieties

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

1 Introduction 1.1 Let X be a nonsingular projective algebraic variety over a number field F . Let L = (L, ( · v ) v ) be a metrized line bundle on X, i.e., a line bundle L together with a family of v-adic metrics, where v runs over the set Val(F ) of places of F . Associated to L there is a height function on the set X(F ) of F -rational points of X (cf. [24,18] for the definitions of v-adic metric, metrized line bundle and height function). For appropriate subvarieties U ⊂ X and line bundles L we have for all H (e.g., this holds for any U if L is ample). We are interested in the asymptotic behavior of this counting function as H → ∞. It is expected that the behavior of such asymptotics can be described in geometric terms ( [1,9]). More precisely, suppose that Z U (L, s) converges for Re(s) ≫ 0, that it has an abscissa of convergence a > 0 and that it can be continued meromorphically to a half-space beyond the abscissa of convergence. Suppose further that there is a pole of order b at s = a and that there are no other poles in this half-space. Then It is conjectured that for appropriate U and L one has a = a(L) and b = b(L) (cf. [1,9]). Moreover, there is a conjectural framework how to interpret the constant c (cf. [18,6]).
There are examples which demonstrate that this geometric "prediction" of the asymptotic cannot hold in complete generality, even for smooth Fano varieties (cf. [5]). Our goal is to show that the conjectures do hold for a class of varieties closely related to linear algebraic groups. Our results are a natural extension of corresponding results for flag varieties (cf. [9]) and toric varieties (cf. [3,4]). We proceed to describe the class of varieties under consideration.
1.2 Let G be a semi-simple simply connected split algebraic group over F and P ⊂ G an F -rational parabolic subgroup of G. Let T be a split algebraic torus over F and X a projective nonsingular equivariant compactification of T . A homomorphism η : P → T gives rise to an action of P on X × G and the quotient Y := (X × G)/P is again a nonsingular projective variety over F . There is a canonical morphism π : Y → W := P \G such that Y becomes a locally trivial fiber bundle over W with fiber X. Corresponding to a character λ ∈ X * (P ) there is a line bundle L λ on W and the assignment λ → L λ gives an isomorphism X * (P ) → Pic(W ). The toric variety X can be described combinatorially by a fan Σ in the dual space of the space of characters X * (T ) R . Let P L(Σ) be the group of Σ-piecewise linear integral functions on the dual space of X * (T ) R . Any ϕ ∈ P L(Σ) defines a line bundle L ϕ on X which is equipped with a canonical T -linearization and we get an isomorphism P L(Σ) ≃ Pic T (X). There is a canonical exact sequence 0 → X * (T ) → P L(Σ) → Pic(X) → 0.
The T -linearization of L ϕ allows us to define a line bundle L Y ϕ on Y and this gives a homomorphism P L(Σ) → Pic(Y ). One can show that there is an exact sequence 0 → X * (T ) → P L(Σ) ⊕ X * (P ) → Pic(Y ) → 0.

1.3
By means of a maximal compact subgroup in the adelic group G(A) we can introduce metrics on the line bundles L λ . The corresponding height zeta functions are Eisenstein series: On the other hand, for any ϕ ∈ P L(Σ) there is a function such that H Σ (x, ϕ) −1 is the height of x ∈ T (F ) with respect to a metrization L ϕ of L ϕ . This metrization induces a metrization L Y ϕ of the line bundle L Y ϕ on Y .
Let (x, γ) ∈ T (F ) × G(F ) and let y be the image of (x, γ) in Y (F ). Then there is a p γ ∈ P (A) such that H L Y ϕ (y) = H Σ (xη(p γ ), ϕ) −1 . Hence we may write formally Now we apply Poisson's summation formula for the the torus and get whereĤ Σ ( · , sϕ) denotes the Fourier transform of H Σ ( · , sϕ) and (T (A)/T (F )) * is the group of unitary characters of T (A) which are trivial on T (F ) equipped with the orthogonal measure dχ. Actually, it is sufficient to consider only those characters which are trivial on the maximal compact subgroup K T of T (A), because the function H Σ ( · , ϕ) is invariant under K T . The expression (3) can now be put into (2), and after interchanging summation and integration the result is (4) where E G P (sλ − ρ P , ξ) = E G P (sλ − ρ P , ξ, 1 G ) is the Eisenstein series twisted by a character ξ of P (A). This is the starting point for the investigation of the height zeta function. To get an expression which is more suited for our study we decompose the group of characters (T (A)/T (F )K T ) * into a continuous and a discrete part, i.e., where X * (T ) R is the continuous part and U T is the discrete part. The right-hand side of (4) is accordingly beyond the abscissa of convergence. To achieve this we need more information on the function under the integral sign in (5). First we have to determine the singularities of (ϕ, λ) →Ĥ Σ (χ, ϕ)E G P (λ − ρ P , (χ • η) −1 ) near the cone of absolute convergence. This is possible becauseĤ Σ can be calculated rather explicitly and it is not so difficult to determine the singular hyperplanes of the Eisenstein series with characters. The next step consists in an iterated application of Cauchy's residue formula to the integral over the real vector space X * (T ) R . This can be done only if one knows that satisfies some growth conditions when x ∈ X * (T ) R tends to infinity. This is true for the function x →Ĥ Σ (χ, sϕ + ix) thanks to the explicit expression mentioned above. The absolute value of the Eisenstein series E G P (sλ−ρ P −iη(x), (χ•η) −1 ) will in general increase for x → ∞ if Re(s)λ − ρ P is not contained in the cone of absolute convergence. However, if Re(s)λ − ρ P is sufficiently close to the boundary of that cone, this increasing behavior is absorbed by the decreasing behavior ofĤ Σ (χ, sϕ + ix).
Therefore, we may apply Cauchy's residue theorem and show that (4) can be continued meromorphically to a larger half-space and that there are no poles (in s) with non-zero imaginary part.
The Tauberian Theorem can now be used to prove asymptotic formulas for the counting function N Y o (L Y ϕ ⊗ π * L λ , H) provided that one knows the order of the pole of the height zeta function. This problem can be reduced to the question whether the "leading term" of the Laurent series of (6) does not vanish. That this is indeed so will be shown in section 6.

1.4
We have restricted ourself to the case of split tori and split groups because this simplifies some technical details. The general case can be treated similarly.
We consider these results as an important step towards an understanding of the arithmetic of spherical varieties. For example, choosing P = B a Borel subgroup, T = B/U where U is the unipotent radical of B and η : B → T the natural projection, we obtain an equivariant compactification of U\G, a horospherical variety.
We close this introduction with a brief description of the remaining sections. Section 2 recalls the relevant facts we need concerning generalized flag varieties, i.e., description of line bundles on W = P \G, the cone of effective divisors in Pic(W ) R , metrization of line bundles, height zeta functions. The exposition is based entirely on the paper [9].
The next section contains the corresponding facts for toric varieties. It is a summary of a part of [2]. We give the explicit calculation of the Fourier transformĤ Σ ( · , ϕ) and show that Poisson's summation formula can be used to give an expression of the height zeta function Z T (L ϕ , s) .
In section 4 we introduce twisted products, discuss line bundles on these, the Picard group (cf. (1)), metrizations of line bundles etc. It ends with the formula (4) for the height zeta function Z Y o (L Y ϕ ⊗ π * L λ , s) in the domain of absolute convergence. The first part of section 5 explains the method for the proof that the height zeta function can be continued meromorphically to a half-space beyond the abscissa of absolute convergence. Moreover, we state a theorem which gives a description of the coefficient of the Laurent series at the pole in question. This coefficient will be the leading one, provided that it does not vanish. One can relate the coefficient to arithmetic and geometric invariants of the pair (U, L) but we decided not to pursue this, since there are detailed expositions of all the necessary arguments in [18,2,6].
These two theorems (meromorphic continuation of certain integrals and the description of the coefficient) will be proved in a more general context in section 7. The second part of section 5 contains the proof that the hypothesis of these theorems are fulfilled in our case. It ends with the main theorem on the asymptotic behavior of the counting function N Y o (L, H), assuming that the coefficient of the Laurent series mentioned above does not vanish. Section 6 is devoted to the proof of this fact. In the last section we prove some statements on Eisenstein series (well-known to the experts) which are used in section 5.

Some notations.
In this paper F always denotes a fixed algebraic number field. The set of places of F will be denoted by Val(F ) and the subset of archimedean places by For any place v of F we denote by F v the completion of F at v and by O v the ring of v-adic integers (for v ∤ ∞). The local absolute value | · | v on F v is the multiplier of the Haar measure, i.e., d(ax v ) = |a| v dx v for some Haar measure dx v on F v . Let q v be the cardinality of the residue field of F v for non-archimedean valuations and put q v = e for archimedean valuations. We denote by A the adele ring of F . For any algebraic group G over F we denote by X * (G) the group of (algebraic) characters which are defined over F .
2 Generalized flag varieties 2.1 Let G be a semi-simple simply connected linear algebraic group which is defined and split over F . We fix a Borel subgroup P 0 over F and a Levi decomposition P 0 = S 0 U 0 with a maximal F -rational torus S 0 of G. Denote by P a standard (i.e., containing P 0 ) parabolic subgroup and by W = P \G the corresponding flag variety. The quotient morphism G → W will be denoted by π W . Any character λ ∈ X * (P ) defines a line bundle L λ on W by The assignment λ → L λ gives an isomorphism (because G is assumed to be simply connected) X * (P ) → Pic(W ) (cf. [21], Prop. 6.10). The anti-canonical line bundle ω ∨ W corresponds to 2ρ P (the sum of roots of S 0 occurring in the unipotent radical of P .) 2.2 These line bundles will be metrized as follows. Choose a maximal compact subgroup For any local section s of L λ at w we define This gives a v-adic norm · w : w * L λ → R and we see that the family · v := ( · w ) w∈W (Fv) is a v-adic metric on L λ . The family ( · v ) v∈Val(F ) will then be an adelic metric on L λ (cf. [18] for λ = 2ρ P and [24] for the definitions of "v-adic metric" and "adelic metric"). The metrized line bundle (L λ , ( · v ) v ) will be denoted by L λ .

Define a map
The height zeta function is therefore an Eisenstein series To describe the domain of absolute convergence of this series we let ∆ 0 be the basis of positive roots of the root system Φ(S 0 , G) which is determined by P 0 . For any α ∈ ∆ 0 denote byα the corresponding coroot. For λ ∈ X * (P ) = X * (S 0 ) we define λ, α by (λ •α)(t) = t λ,α and extend this linearly in λ to X * (P 0 ) C . Restriction of characters defines an inclusion X * (P ) → X * (P 0 ). Let By [10], Théorème 3, the Eisenstein series converges absolutely for Re(λ) − ρ P in X * (P ) + and it can be meromorphically continued to X * (P ) C (cf. [16], IV, 1.8). The closure of the image of X * (P ) + in Pic(W ) R is the cone Λ eff (W ) generated by the effective divisors on W ( [13], II, 2.6).
3 Toric varieties 3.1 Let T be a split algebraic torus of dimension d over F . We put M = X * (T ) and N = Hom(M, Z). Let Σ be a complete regular fan in N R such that the corresponding smooth toric variety X = X Σ is projective (cf. [2,17]). The variety X is covered by affine open sets where σ runs through Σ andσ is the dual coně Denote by P L(Σ) the group of Σ-piecewise linear integral functions on N R . By definition, a function ϕ : N R → R belongs to P L(Σ) if and only if ϕ(N) ∈ Z and the restriction of ϕ to every σ ∈ Σ is the restriction to σ of a linear function on N R . For ϕ ∈ P L(Σ) and every d-dimensional cone σ ∈ Σ there exists a unique m ϕ,σ ∈ M such that for all n ∈ σ we have ϕ(n) = n(m ϕ,σ ).
Fixing for any σ ∈ Σ a d-dimensional cone σ ′ containing σ we put To any ϕ ∈ P L(Σ) we associate an invertible sheaf L ϕ on X as the subsheaf of rational functions on X generated over U σ by 1 mϕ,σ , considered as a rational function on X (L ϕ does not depend on the choice made above). The assignment ϕ → L ϕ gives an exact sequence 0 → M → P L(Σ) → Pic(X) → 0 (cf. [17], Corollary 2.5). Denote by θ : X × T → X the action of T on X and by p 1 : X × T → X the projection onto the first factor. The induced T -action on the sheaf of rational functions restricts to any subsheaf L ϕ , i.e., there is a canonical T -linearization [15], Ch. 1, § 3, for the notion of a T -linearization). In section four we will always consider L ϕ not merely as a line bundle on X but as a T -linearized line bundle with this T -linearization. In this sense P L(Σ) is isomorphic to the group Pic T (X) of isomorphism classes of T -linearized line bundles on X. Let Σ 1 ⊂ N be the set of primitive integral generators of the one-dimensional cones in Σ and put P L(Σ) + := {ϕ ∈ P L(Σ) R | ϕ(e) > 0 for all e ∈ Σ 1 }.

3.2
We shall introduce an adelic metric on the line bundle L ϕ as follows. For σ ∈ Σ and v ∈ Val(F ) define These subsets cover X(F v ) and we put for x ∈ K σ,v and any local section s of L ϕ at x The family · v = ( · x ) x∈X(Fv) is then a v-adic metric on L ϕ and L ϕ = (L ϕ , (where q v is the order of the residue field of F v for non-archimedean valuations and log(q v ) = 1 for archimedean valuations) we get an isomorphism The corresponding global function H Σ ( · , ϕ) : T (A) → C, is well defined since for almost all v the local component x v belongs to K T,v . The functions H Σ,v ( · , ϕ), ϕ ∈ P L(Σ), are related to the v-adic metric on L ϕ by the identity where s ϕ ∈ H 0 (T, L ϕ ) is the constant function 1. In particular, for every This gives an embedding M R → A T . For any archimedean place v and χ ∈ A T there is an

This gives decompositions
and where M 1 R,∞ is the minimal R-subspace of M R,∞ containing the image of U T under the map (8). ¿From now on we fix such a (non-canonical) splitting. By Dirichlet's unit theorem, the image of U T → M 1 R,∞ is a lattice of maximal rank. Its kernel is isomorphic to the character group of Cl d F , where Cl F is the ideal class group of F . For finite v we let dx v be the Haar measure on T (F v ) giving K T,v the volume one. For archimedean v we take on T (F v )/K T,v the pull-back of the Lebesgue measure on N R (normalized by the lattice N) and on K T,v the Haar measure with total mass one. The product measure gives an invariant measure dx v on T (F v ). On T (A) we get a Haar measure dx = v dx v .

3.4
We will denote by S 1 the unit circle. For a character χ : If χ is not trivial on K T,v thenĤ Σ,v (χ, ϕ) = 0 (assuming the convergence of the integral). We will show that these integrals do exists if Re(ϕ) is in P L(Σ) + .
Let v be an archimedean place of F . Any d-dimensional cone σ ∈ Σ is simplicial (since Σ is regular) and it is generated by the set σ ∩Σ 1 . Let χ be unramified, i.e., χ(x) = e −ix(m) with some m ∈ M R . Then we get To give the result for finite places we define rational functions R σ in variables u e , e ∈ Σ 1 , for any σ ∈ Σ by Although elementary, it is a very important observation that the polynomial Q Σ − 1 is a sum of monomials of degree not less than two (cf. [2], Prop. 2.2.3). Let χ be an unramified unitary character of T (F v ) and let Re(ϕ) be in P L(Σ) + . Then we can calculatê and this product converges for Re(s) > 1 (here π v denotes a local uniformizing element). By (10) and (11) we know that the global Fourier transform exists (i.e., the integral on the right converges absolutely) if Re(ϕ) is contained in is an absolutely convergent Euler product for Re(ϕ(e)) > 1/2 (for all e ∈ Σ 1 ) and hence is bounded for Re(ϕ) in any compact subset in 1 2 ϕ Σ + P L(Σ) + (by some constant depending only on this subset). converges absolutely and uniformly for (Re(ϕ), t) contained in any compact subset of (ϕ Σ + P L(Σ) + ) × T (A).
Proof. Let K be a compact subset of ϕ Σ + P L(Σ) + and let C v ⊂ T (F v ) (for every v ∈ Val(F )) be a compact subset, equal to K T,v for almost all v. Since any ϕ ∈ P L(Σ) C is a continuous piecewise linear function (with respect to a finite subdivision of N R into simplicial cones) there exists a constant c v ≥ 1 (depending on K and C v ) such that for Let S be a finite set of places containing Val ∞ (F ) and let By the preceding argument, there exists a c ′ > 0 such that for all ϕ ∈ K, x ∈ T (F ) and u ∈ U by the discussion above. From the explicit expression for the integral (cf. (12)) we derive the uniform convergence in ϕ on K. ✷

3.5
The aim is to apply Poisson's summation formula to the height zeta function. It remains to show thatĤ( · , ϕ) is absolutely integrable over A T . For χ ∈ A T and Re(ϕ) contained in 1 2 ϕ Σ + P L(Σ) + we put Lemma 3.5 Let K be a compact subset of P L(Σ) C such that for all ϕ ∈ K and e ∈ Σ 1 Re(ϕ(e)) > 1 2 .
Then there is a constant c = c(K) such that for all ϕ ∈ K, χ ∈ A T and m ∈ M R we have Proof. For K as above there exists a c ′ > 0 such that for all χ ∈ A T and m ∈ M R one has By the preceding lemma, we see thatĤ Hence we can apply Poisson's summation formula (together with (9)) and obtain where the Lebesgue measure dm on M R is normalized by M and with cl F the class number, R F the regulator and w F the number of roots of unity in F .
Theorem 3.6 For any ǫ > 0 there exists a δ > 0 and a constant c(ǫ) > 0 such that for all s with Re(s) > 1 − δ and all unramified Hecke characters which are non-trivial on For the trivial character χ = 1 one has 4 Twisted products 4.1 Let G, P, W = P \G etc. be as in section 2 and T, Σ, X = X Σ etc. be as in section 3. Let η : P → T be a homomorphism. Then P acts from the right on X × G by (x, g) · p := (xη(p), p −1 g).
Since π W : G → W is locally trivial, the quotient exists as a variety over F . Moreover, the projection X × G → G induces a morphism π : Y → W and Y becomes a locally trivial fiber bundle over W with fiber X (compare [13], I.5.16). Hence, by the properties of X (non-singular, projective), we see that Y is a non-singular projective variety over F ("projectivity" requires a short argument, cf. [23]). The quotient morphism X × G → Y will be denoted by π Y . Let ϕ ∈ P L(Σ) and let L ϕ be the invertible sheaf on X defined in section 3.1. Denote by L ϕ the corresponding G a -bundle over X, i.e., L ϕ = V(L ∨ ϕ ) = V(L −ϕ ) (with the notation of [11], II, Exercise 5.18). The canonical T -linearization The twisted product L ϕ × P G will then be a G a -bundle over Y and we define L Y ϕ to be the sheaf of local sections of L ϕ × P G over Y . Note that L Y ϕ (and even its isomorphism class in Pic(Y )) depends on the fixed T -linearization θ −ϕ . In fact, for ϕ ∈ P L(Σ) and

4.2
In the following proposition we collect all relevant facts about the geometry of twisted products which we will need in the sequel.
Suppose now that for ϕ ∈ P L(Σ) and By [17], Lemma 2.3, ✷ ϕ is a basis for H 0 (X, L ϕ ) (note the different sign conventions). It is easy to see that On the other hand, if (ϕ, λ) ∈ P L(Σ) ⊕X * (P ) is contained in the closure of P L(Σ) + × X * (P ) + then the trivial character corresponds to a global section of L ϕ . Hence Pulling back to Y and taking the d-th exterior power we get The canonical T -linearization of ω X (induced by the action of T on rational functions) We are going to introduce an adelic metric on the sheaves Let v be a place of F and let y ∈ (by (7)). Globally, for y ∈ Y o (F ), y = π Y (x, γ), x ∈ T (F ), γ ∈ G(F ), γ = p γ k γ and p γ , k γ as above, in P (A), K G , respectively, we get 4.4 Let ξ : P (A)/P (F ) → S 1 be an unramified character, i.e., ξ is trivial on P (A)∩K G . Using the Iwasawa decomposition we get a well defined function if g = pk as above. We denote by the corresponding Eisenstein series and we put E G P (λ, ξ) = E G P (λ, ξ, 1 G ). This series converges absolutely for Re(λ) contained in the cone ρ P + X * (P ) where the sum and integral on the right converge absolutely too.
We consider s = u + iv ∈ C such that u · ϕ is contained in the shifted cone ϕ Σ + P L(Σ) + and u · λ is contained in the cone 2ρ P + X * (P ) + . Then converges by Proposition 3.4 and is equal to (13)). Moreover,Ĥ Σ ( · , uϕ) is absolutely convergent on A T and therefore is bounded by some constant c (which is independent of η(p γ )). Thus we may calculate This shows the first assertion. Since converges, we can interchange the summation and integration and get The first thing to do is to show that Z Y o (L, s) can be continued meromorphically to a half-space beyond the abscissa of convergence and that there is no pole on this line with non-zero imaginary part. Then it remains to prove that this abscissa is at Re(s) = a(L) and to determine the order of the pole in s = a(L). We will see that this order is b(L).
The method which we will explain now consists in an iterated application of Cauchy's residue theorem. The proofs will be given in section 7.

5.2
Let E be a finite dimensional vector space over R and E C its complexification. Let V ⊂ E be a subspace and let l 1 , ..., l m ∈ E ∨ = Hom R (E, R) be linearly independent linear forms. Put is holomorphic in T B and there is a sufficient function c : V → R ≥0 such that for all compacts K ∈ T B , all z ∈ K and all v ∈ V we have the estimate (Cf. section 7.3 for a precise definition of a sufficient function. In particular, such a sufficient function is absolutely integrable over any subspace U ⊂ V .) In this case we call f distinguished with respect to the data (V ; l 1 , ..., l m ).
Let C be a connected component of B − ∪ m j=1 H j . By the conditions on g the integral (ν = dim V and dv is a fixed Lebesgue measure on V ) converges for every z ∈ T C andf C is a holomorphic function on T C .

Theorem 5.2
There is an open neighborhoodB containing C, and linear formsl 1 , ...,lm which vanish on V such that has a holomorphic continuation to TB. Moreover, for all j ∈ {1, ...,m} we have Ker(l j ) ∩ C = ∅.
We shall give the proof of this theorem in sections 7.3 and 7.4.

Put
and let ψ 0 : E 0 → P := E + 0 /π 0 (V ) be the canonical projection. We want to assume that π 0 (V ) ∩ E + 0 = {0}, so that Λ := ψ 0 (E + 0 ) is a strictly convex polyhedral cone. Let dy be the Lebesgue measure on E ∨ 0 normalized by the lattice ⊕ m j=1 Zl j . Let A ⊂ V be a lattice and let dv be the measure on V normalized by A. On V ∨ we have the Lebesgue measure dy ′ normalized by A ∨ and a section of the projection E ∨ 0 → V ∨ gives a measure dy ′′ on P ∨ with dy = dy ′ dy ′′ .
Define the X -function of the cone Λ by for all x ∈ P C with Re(x) contained in the interior of Λ (cf. section 7.1). Let B ⊂ E be as above and let f ∈ M(T B ) be a distinguished function with respect to (V ; l 1 , ..., l m ). Put (for z ∈ T B ). The functionf B + : T B + → C is holomorphic and has a meromorphic continuation to a neighborhood of 0 ∈ E C . In section 7.5 we will prove the following theorem.
¿From now on we will denote by K G ⊂ G(A) the maximal compact subgroup defined in section 8.2. such that for all (ϕ, λ) ∈ K, χ ∈ U T and m ∈ M R we have Proof. Write as in (12) For Re(ϕ) sufficiently small and e ∈ Σ ′ 1 we have is bounded for Re(ϕ) sufficiently small. If e ∈ Σ 1 − Σ ′ 1 then ϕ L (e) = 0. By the estimates of Rademacher (cf. Theorem 3.6), we have for χ e = 1 |L f (χ e , 1 + (ϕ + im)(e))| ≤ c e (1 + |m(e)| + (χ e ) ∞ ) ǫ for Re(ϕ(e)) > −δ and ϕ in a compact set (δ depends on ǫ, c e depends on this compact subset). If χ e = 1 (abusing notations we will denote from now on the trivial character by 1) then (ϕ + im)(e) (ϕ + im)(e) + 1 |L f (1, 1 + (ϕ + im)(e))| ≤ c e (1 + |m(e)|) ǫ Now we use Proposition 8.7 concerning estimates for Eisenstein series. This proposition tells us that there is for given ǫ > 0 an open neighborhood of 0 in X * (P ) R such that for Re(λ) contained in this neighborhood (For the definition of (· · ·) ∞ and the norms see section 8.5.) If we let λ vary in a compact subset in the tube domain over this neighborhood then there is a constant c 2 ≥ c 1 such that For Re(λ) sufficiently small and α ∈ ∆ P ′ we have Therefore, there are c 3 , c 4 > 0 such that for all such λ and m ∈ M R we have Putting everything together, we can conclude that there is a neighborhood B of 0 in P L(Σ) R ⊕ X * (P ) R such that for (ϕ, λ) in a compact subset K of the tube domain over B we have where · is a norm on M R,∞ . On the other hand, by Lemma 3.5, we have Now we may choose ǫ and c 5 such that for any system (σ v ) v|∞ of d-dimensional cones. This gives the claimed estimate. ✷

5.5
To begin with, we let . The connection with sections 5.2 and 5.3 is as follows: the set of linear forms l 1 , ..., l m is given as follows (We use the invariance of g under iM ′ R and Cauchy-Riemann differential equations to check that g is actually a function on is a meromorphic function on T B . Let E (0) be the common kernel of all maps (18,19). Note that there is an exact Let dy be the Lebesgue measure on E ∨ 0 normalized by the lattice generated by the linear forms (18,19). Denote by E + 0 the closed simplicial cone in E 0 defined by these linear forms, and by π 0 : E → E 0 the canonical projection. It is easily seen that π 0 (M ′′ R ) ∩ E + 0 = {0} (using the exact sequence above). Let be the canonical projection and put Λ = ψ 0 (E + 0 ), By the following theorem the function f ∈ M(T B ) is distinguished with respect to M ′′ R and the set of linear forms (18,19). Therefore, we can definef B + : has a holomorphic continuation to TB and g(0) = 0.
Proof. a) Define c 0 : M R,∞ → R ≥0 by Let dm 1 be the Lebesgue measure on M 1 R,∞ normalized by the image of U T . Then for for all s with Re(s) > 0. However, this is just with f, B + andf B + introduced in the preceding section. By Theorem 5.5,f B + extends to a meromorphic function on a tube domain over a neighborhood of 0 and in this tube domain the only singularities are the hyperplanes defined over R. Hence there is a δ > 0 such that Z Y o (L, s + a(L)) extends to a meromorphic function in the half-space Re(s) > −δ and the only possible pole is in s = 0 and its order is exactly b(L) (Theorems 5.3 and 5.5).
b) This result follows from a Tauberian theorem (cf. [8], Théorème III or [22], Problem 14.1 (in the constant stated there the factor 1 k 0 is missing)). ✷ 6 Non-vanishing of asymptotic constants 6.1 This section is devoted to the proof of the non-vanishing of g(0) claimed in Theorem 5.5. All notations are as in sections 5.2-5.5. The function g(ϕ, λ) which has been defined in 5.5 is given by whereφ,λ have been defined in 5.4. The function h L (ϕ, λ) was defined in 5.4: The uniform convergency of the integral above in any compact subset of T B (cf. Lemma 5.4) allows us to compute the limit first and then to integrate. We shall show that this limit vanishes if there are e ∈ Σ 1 − Σ ′ 1 with χ e = 1 or α ∈ ∆ P − ∆ P ′ with χ η •α = 1. Therefore, we may consider only Let η ′ : P ′ → T be the uniquely defined homomorphism such that for all α ∈ ∆ P ′ we have η ′ •α = η •α.
for the definition of c P ′ and c P ).
Proof. Recall that (cf. (12)) and that ζ Σ (χ,φ+im ′ ) is regular for ϕ in a tube domain over a neighborhood of 0 (Lemma 3.5). For e ∈ Σ ′ 1 we have ϕ L (e) > 0, hence we see that the function is holomorphic for Re(ϕ) in a neighborhood of 0. Let e ∈ Σ 1 − Σ ′ 1 . If χ e = 1 then the restriction of χ e to G m (A) 1 is non-trivial (by our construction of the embedding U T → A T , cf.

3.3), hence
ϕ → L f (χ e , 1 + ϕ(e)) is an entire function and ϕ(e)L f (χ e , 1 + ϕ(e)) tends to 0 as ϕ → 0. For α ∈ ∆ P ′ we have λ L , α > 0, hence λ L is contained in X * (P ′ ) + . Let α ∈ ∆ P − ∆ P ′ . If χ η •α = 1 then χ η •α restricted to G m (A) 1 is non-trivial and therefore vanishes as λ → 0 (cf. Proposition 8.3). We have shown that it suffices to take the sum over all χ ∈ U ′ T . To complete the proof, note that for χ ∈ U ′ T we have (cf. Proposition 2 By the absolute and uniform convergence of (cf. Lemma 3.5 and Theorem 3.6) and the convergence of we may change summation and integration in (21) and get for all ϕ ∈ P L(Σ) + where γ = p ′ γ k γ as above. Let I be the image of the homomorphism Note that I ⊂ T ′′ (A). Denote by K T ′ (resp. K T ′′ ) the maximal compact subgroup of T ′ (A) (resp. of T ′′ (A)). The linear forms when considered as functions on M ′′ , generate a sublattice of finite index in N ′′ = Hom(M ′′ , Z). This shows that there is a q > 0 such that the image of the q-th power homomorphism T ′′ (A) → T ′′ (A), t → t q , is contained in I. If v is any archimedean place of F the connected component of one in T ′′ (F v ) is therefore contained in I. Consequently,

and the left hand side is of finite index in T ′′ (A). Put
We observe that is absolutely integrable over A ′ T . Using Poisson's summation formula twice we get Now we collect all the terms together. Proof. Consider the embedding N ′′ R → N R and let Σ ′′ := {σ ∩ N ′′ R | σ ∈ Σ}. This is a complete fan in N ′′ R which consists of rational polyhedral cones, but which is not necessary a regular fan. We can obtain a regular fan by subdivision of the cones into regular ones (cf. [14], ch. I, §2, Theorem 11). This gives us a complete regular fanΣ ′′ such that any cone inΣ ′′ is contained in a cone of Σ ′′ . Denote byΣ ′′ 1 the set of primitive integral generators of the one-dimensional cones inΣ ′′ . Computing the integral as in section 3.4 we get (12)), where ζΣ′′(1,φ) is regular in a neighborhood of ϕ = 0 and positive for ϕ = 0. Letẽ ∈ N and σ ∈ Σ ′ be a cone containingẽ. Writeẽ = e∈σ∩Σ 1 t e · e (t e ∈ Z ≥0 ).
Then t e = 0 for all e ∈ σ ∩ Σ ′ 1 because ϕ L ∈ e∈Σ ′ 1 R >0 ϕ e (cf. 5.4). Hence, and this is a positive real number. ✷ In theorem 5.5 we claimed the non-vanishing of g(0). We are now in the position to prove Corollary 6.4 g(0) > 0.
Proof. By Lemma 6.2 it is enough to show that Then there exists a constant c > 0 such that for all t ∈ T ′′ (A) and j = 1, ..., ν we have Hence we can estimate We denote by χ Λ (v) the set-theoretic characteristic function of the cone Λ and by X Λ (v) the Laplace transform of the set-theoretic characteristic function of the dual cone where Re(v) ∈ Λ • (for these v the integral converges absolutely).
Consider a complete regular fan Σ on V , that is, a subdivision of the real space V into a finite set of convex rational simplicial cones, satisfying certain conditions (see [2], 1.2). Denote by Σ 1 the set of primitive generators of one dimensional cones in Σ. Denote by P L(Σ) R the vector space of real valued piecewise linear functions on V and by P L(Σ) C its complexification.
, Prop. 2.3.2, p. 614) For any compact K ⊂ P L(Σ) C with the property that Re(ϕ(v)) > 0 for all ϕ ∈ K and v = 0 there exists a constant κ(K) such that for all ϕ ∈ K and all y ∈ V ∨ the following inequality holds: The function ϕ is piecewise linear with respect to a complete fan of H ′ . Taking a subdivision, if necessary, we may assume it to be regular.

Proposition 7.2
The function X Λ (u) is absolutely integrable over any linear subspace U ⊂ H.
Proof. For h ∈ H we have is the Fourier transform of the function y ′ → e −ϕ(y ′ ) on H ′ ≃ H ∨ . The statement follows now from 7.1. ✷

7.3
The rest of this section is devoted to the proof of the meromorphic continuation of certain functions which are holomorphic in tube domains over convex finitely generated polyhedral cones. In section 5 we have already introduced the terminology and explained how this technical theorem is applied to height zeta functions. Let E be a finite dimensional vector space over R and E C its complexification. Let V ⊂ E be a subspace. We will call a function c : V → R ≥0 sufficient if it satisfies the following conditions: (i) For any subspace U ⊂ V and any v ∈ V the function U → R defined by u → c(v+u) is measurable on U and the integral is always finite (du is a Lebesgue measure on U). A meromorphic function f ∈ M(T B ) will be called distinguished with respect to the data (V ; l 1 , ..., l m ) if it satisfies the following conditions: (i) The function (ii) There exists a sufficient function c : V → R ≥0 such that for any compact K ⊂ T B there is a constant κ(K) ≥ 0 such that for all z ∈ K and all v ∈ V we have |g(z + iv)| ≤ κ(K)c(v).
Let C be a connected component of B − m j=1 H j and T C a tube domain over C. We will consider the following integral: Here we denoted by d = dim V and by dv a fixed Lebesgue measure on V . Proof. a) Let K ⊂ T C be a compact subset and let κ(K) ≥ 0 be a real number such that for all z ∈ K and all v ∈ V we have |g(z + iv)| ≤ κ(K)c(v). Since K is a compact and C doesn't intersect any of the hyperplanes H j there exist real numbers c j ≥ 0 for any j = 1, ..., m, such that for all z ∈ K and v ∈ V the following inequalities hold Therefore, for all x ∈ B 1 and j ∈ {1, ..., m 0 }, k ∈ {1, ...,ĵ, ..., m} we have Let C 1 be a connected component of As in (i) one shows that h C is a holomorphic function on T C . For x ∈ B 1 and λ ∈ [0, 1] we have If for some z = x + iy ∈ T C 1 (x ∈ C 1 ) and λ ∈ [0, 1] + iR, j ∈ {1, ..., m} we have l j (z + λv 0 ) = 0, then it follows that l j (x) + Re(λ)l j (v 0 ) = 0, and therefore, l j (x)l j (v 0 ) < 0 (since l j (x) has the same sign as l j (x 0 )). Consequently, j ∈ {1, ..., m 0 }. For z ∈ T C 1 and j ∈ {1, ..., m 0 } we put By our assumptions, we have 0 < Re(λ j (z)) < 1 2 . ¿From λ j (z) = λ j ′ (z), with j, j ′ ∈ {1, ..., m 0 } and j = j ′ it follows now that In particular, we have l j,j ′ (Re(z)) = 0. This is not possible, because z ∈ T C 1 . Assume now that x ∈ B 1 . We have, assuming that l k (v 0 ) = 0, that If l k (v 0 ) = 0 then we have l k (x + v 0 ) = l k (x). Fix a z ∈ T C 1 . Then there exist some numbers c 1 (z) ≥ 0, c 2 (z) ≥ 0 such that for all λ ∈ [0, 1], τ ∈ R with |τ | ≥ c 1 (z) we have For any z ∈ T C 1 we have therefore is a holomorphic function on T B 1 . Let K 1 ⊂ T B 1 (⊂ T B ) be a compact, and let This is a compact subset in T B . Put where κ(K(j)) is a constant such that |g(z + iv)| ≤ κ(K(j))c(v) for all z ∈ K(j) and all v ∈ V . For v 1 ∈ V 1 we put Then for all z ∈ K 1 and v ∈ V 1 we have Moreover, for any subspace U 1 ⊂ V 1 and all v 1 ∈ V 1 the function U 1 → R, u 1 → c j (u 1 +v 1 ) is measurable and we have This shows that f j is distinguished with respect to (V 1 ; (l j,k ) 1≤k≤m, k =j ). For z ∈ T B 1 we put If l k (v 0 ) = 0 we have (as above) for all x ∈ B 1 the following inequality Therefore, we conclude that the function Further, we have for z ∈ K 1 (with K 1 ⊂ T B 1 a compact) and v 1 ∈ V 1 the inequality where κ 0 (K 1 ) is some suitable constant and Again, for any subspace U 1 ⊂ V 1 and any u 1 ∈ V 1 we have that the map U 1 → R given by u 1 → c 0 (v 1 + u 1 ) is measurable, and that The Cauchy-Riemann equations imply that g 0 is invariant under Cv 0 , that is, for all z 1 , z 2 ∈ T B 1 with z 1 − z 2 ∈ Cv 0 we have g 0 (z 1 ) = g 0 (z 2 ). We see that f 0 is also invariant under Cv 0 (in this sense), as well as f 1 , ..., f m 0 (this can be seen from the explicit representation of these functions). For z ∈ T C 1 we have Moreover, for such z we havẽ where dv 1 dτ = dv (and Vol dτ ({λv 0 |λ ∈ [0, 1]}) = 1).
By our induction hypothesis, there exists an open and convex neighborhood B ′ of 0 in E and linear formsl 1 , ...,lm, which vanish on V , such that has a holomorphic continuation to T B ′ . (Strictly speaking, from the induction hypothesis it follows only that the linear formsl 1 , ...,lm ∈ E ∨ vanish on V 1 . But since the functions f 0 , ..., f m 0 "live" already on a tube domain in (E/Rv 0 ) C , it follows that the linear forms are also Rv 0 -invariant.) Now we notice thatf C (z) m j=1l j (z) is holomorphic on T C . LetB be the convex hull of B ′ ∪ C. Then we have thatf is holomorphic on TB (cf. [12], Theorem 2.5.10). Proof. By the proposition above, there exist linear formsl 1 , ...,lm such that V ⊂ ∩m j=1 Ker(l j ) andf C (z) m j=1l j (z) has a holomorphic continuation to a tube domain TB over a convex open neighborhoodB of 0 ∈ E containing C. Suppose that there exist an x 0 ∈ C and a j 0 ∈ {1, ...,m} such thatl j 0 (x 0 ) = 0. Then the functioñ is still holomorphic in TB′ withB ′ = (B − Ker(l j 0 )) ∪ C. It is easy to see thatB ′ is connected. The convex hull ofB ′ is equal toB. Therefore, already the functioñ is holomorphic on TB (cf. [12], loc. cit.).

7.5
As above, let E be a finite dimensional vector space over R and let l 1 , ..., l m be linearly independent linear forms on E. Put H j := Ker(l j ), for j = 1, ..., m, and let ψ : E 0 → P := E 0 /π 0 (V ) be the canonical projection. We want to assume that π 0 (V ) ∩ E + 0 = {0}, so that Λ := ψ(E + 0 ) is a strictly convex polyhedral cone. Let dy be the Haar measure on E ∨ 0 normalized by Vol dy (E ∨ 0 / ⊕ m j=1 Zl j ) = 1. Let A ⊂ V be a lattice, and let dv be a measure on V normalized by Vol dv (V /A) = 1. On V ∨ we have a measure dy ′ normalized by A ∨ and a section of the projection E ∨ 0 → V ∨ gives a measure dy ′′ on P ∨ with dy = dy ′ dy ′′ .
Let B ⊂ E be an open and convex neighborhood of 0, such that for all x ∈ B and j ∈ {1, ..., m} we have l j (x) > −1. Let f ∈ M(T B ) be a meromorphic function in the tube domain over B which is distinguished with respect to (V ; l 1 , ..., l m ). Put By 7.3, the functionf B + : T B + → C is holomorphic and it has a meromorphic continuation to a neighborhood of 0 ∈ E C . Put Proof. For j ∈ J := {1, ..., m} we define Let C be the set of connected components of V − m j=1 (H j,+ ∪ H j,− ). For a C ∈ C we put Denote by V C the complement to V C in V and let π C : The set C(v 1 ) is a convex open subset of V C . Let dv 1 , dv ′ be measures on V C (resp. on V C ), with dv 1 dv ′ = dv. For all s ∈ (0, 1] we have Here we denoted by d C := dim V C and by and by χ Cs the set-theoretic characteristic function of C s . We have put for any s ∈ (0, 1] and any The set is contained in T B and is compact. Further, there exist c ′ , c ′′ ≥ 0 such that for all s ∈ [0, 1] and all .
Therefore, for s ∈ (0, 1] and v ∈ V we have (The constant κ(K C ) and the function c : V → R ≥0 were introduced above.) The X -function corresponding to the cone E + 0 ⊂ E 0 (and the measure dy) is given by .
Since the map from V C to E 0 is injective and since π 0 (V C ) ∩ E + 0 = {0} we know that the function v 1 → X E + 0 (x 0 + iv 1 ) is absolutely integrable over V C (by 7.2). Therefore, For a fixed v ∈ V we consider the limit The estimate above shows that this limit is 0 if m − d C − #J C > 0. Therefore, we assume that m = d C + #J C . Then the map V C → R J−J C is an isomorphism. Since π 0 (V ) ∩ E + 0 = {0}, it follows that J C = J. There exists exactly one C ∈ C with J C = J and we denote it by C • . This C • contains 0 and for all sufficiently small s > 0 we have .
Using the theorem of dominated convergence (Lebesgue's theorem), we obtain ✷ 8 Some statements on Eisenstein series 8.1 Let G be a semi-simple simply connected algebraic group which is defined and split over F . Fix a Borel subgroup P 0 (defined over F ) and a Levi decomposition P 0 = S 0 U 0 , where S 0 is a maximal F -rational torus of G. Denote by g (resp. a 0 ) the Lie algebra of G (resp. S 0 ).
We are going to define a certain maximal compact subgroup K G ⊂ G(A). This maximal compact subgroup will have the advantage that the constant term of Eisenstein series, more precisely, certain intertwining operators, can be calculated explicitly, uniformly with respect to all places of F . In general, i.e., for an arbitrary maximal compact subgroup, there will be some places where such an explicit expression is not available. In principle, this should cause no problems. Any statement in this section should be valid for an arbitrary maximal compact subgroup.

8.2
Let Φ = Φ(G, S 0 ) be the root system of G with respect to S 0 . We denote by ∆ 0 the basis of simple roots determined by P 0 . For α ∈ Φ let be the corresponding root space.
Let ((H α ) α∈∆ 0 , (X α ) α∈Φ ) be the Chevalley basis of g. In particular, this means that This is a Q-structure for g and for any v ∈ Val(F ) the Lie algebra of where Φ + is the set of positive roots of Φ determined by ∆ 0 . Then k ⊕ p is a Cartan . In this case, K v contains exp(k) . In both cases K v is a maximal compact subgroup of G(F v ). Now let v be a finite place of F and let K v be the stabilizer of the lattice By [7], sec. 3, Example 2, K v is a maximal compact subgroup of G(F v ). In any case, the Iwasawa decomposition G(F v ) = P 0 (F v )K v holds (for non-archimedean v, cf. [7], loc. cit.). Then K G = v K v is a maximal compact subgroup of G(A) and G(A) = P 0 (A)K G .

8.3
As in section 2.3 we defined for any standard parabolic subgroup P ⊂ G The restriction of H P 0 to S 0 (A) is a homomorphism, its kernel will be denoted by S 0 (A) 1  Let (̟ α ) α∈∆ 0 be the basis of X * (S 0 ) which is determined by ̟ α , β = δ αβ for all α, β ∈ ∆ 0 . Let P ⊂ G be a standard parabolic subgroup. Then ̟ α for all α ∈ ∆ P lifts to a character of P and (̟ α ) α∈∆ P is a basis of X * (P ). Put Any χ ∈ U P is a character of P (A)/P (F )(P (A) ∩ K G ). Define converges absolutely and uniformly for Re(λ) contained in any compact subset of the open cone ρ P + X * (P ) + (cf. [10], Théorème III) and can be continued meromorphically to the whole of X * (P ) C .
For the Eisenstein series corresponding to P 0 a proof is given in [16], chapitre IV. In section 8.4 we will give an explicit expression for the Eisenstein series E G P , with P = P 0 as an iterated residue of E G P 0 which shows the claimed meromorphy on X * (P ) C . Let χ ∈ U 0 . The constant term of E G P 0 (λ, χ) along P = LU is by definition where the Haar measure on U(A) is normalized such that U(F )\U(A) gets volume one. It is an elementary calculation to show that for any parabolic subgroup P P 0 the constant term E G P 0 (λ, χ) P is orthogonal to all cusp forms in A 0 (L(F )U(A)\G(A)) (cf. [16], I.2.18, for the definition of this space). More precisely, for any parabolic subgroup P ⊇ P 0 the cuspidal component of E G P 0 (λ, χ) along P vanishes (cf. [16], I.3.5, for the definition of "cuspidal component").
Put S α = Ker(α) 0 ⊂ S 0 and G α = Z G (S α ). The Lie algebra of G α is a 0 ⊕ g −α ⊕ g α . There is a homomorphism ϕ α : SL 2,F → DG α (= derived group of G α ) such that dϕ α maps the matrices 0 1 0 0 , On A we take the measure dx that is described in Tate's thesis (then Vol(F \A) = 1). We have It is an exercise to compute this integral. The result is The Hecke L-functions are defined as follows. Let χ : G m (A)/G m (F ) → S 1 be an unramified character. For any finite place v we put
b) Let P ′ = L ′ U ′ be a standard parabolic subgroup containing P and suppose that χ •α = 1 for all α ∈ ∆ P ′ 0 . Then χ ∈ U P ′ and for all λ ∈ X * (P ′ ) C we have lim ϑ→0, ϑ∈X * (P ) + α∈∆ P −∆ P ′ ϑ, α E G P (ϑ + λ + ρ P , χ, g) = c P ′ c P E G P ′ (λ + ρ P ′ , χ, g). In fact, the cuspidal components of both sides along all non-minimal standard parabolic subgroups of L vanish. To compare the constant terms along P 0 ∩ L we can use (22) (for L instead of G and P 0 ∩ L instead of P 0 ) and the explicit expression of the functions c(w, ϑ + ρ 0 , 1) to get the identity stated above (note that w L ρ 0 + ρ 0 = 2ρ P ).
The explicit expression of the constant term of E G P 0 along P 0 in (22) and uniform estimates for L-functions as in [19], Theorem 5 (but for a larger strip), allow us to conclude that (32) is just the constant term of e s 2 f λ,χ (s, · ) along P 0 . Thus, by Proposition I.3.4 in [16], we have established (31). ¿From (31) it follows that |e s 2 f λ,χ (s, · )| is bounded by some constant in the strip −1 − a ≤ Re(s) ≤ 1 + b and this in turn implies (30). ✷.