Arithmetic Stratifications and Partial Eisenstein Series

Matthias Strauch

Mathematisches Institut der Westfälischen Wilhelms-Universität Münster (e-mail address: straucm@math.uni-muenster.de)

November 1999

Abstract. Let P \G and Q\H be generalized flag varieties over a number field F. In this paper we study certain locally trivial fibre bundles Y_{η} over P \G having Q\H as general fibre, and determine the arithmetic stratification of Y_{η} with respect to a line bundle. The arithmetic stratification is defined in terms of height zeta functions and the height zeta function of a stratum is of the form

∑_{γ∈P (F)\G(F)} e^{⟨sλ,H_{P} (γ)⟩}E_{Q}^{Qw^{−1}Q_{0}}(sμ, η(p_{γ})) ,

where E_{Q}^{Qw^{−1}Q_{0}} is a “partial Eisenstein series” associated to the Schubert cell Q\Qw^{−1}Q_{0}. The computation of the constant term of these gives estimates that allow one to determine the abcissa of convergence of the height zeta function of the stratum.

Résumé. Soient P \G et H\Q des variétés de drapeaux généralisées sur un corps de nombres F. Dans cet article nous étudions certaines fibrés localement triviales Y_{η} sur P \G ayant Q\H comme fibre générale, et nous déterminons la stratification arithmétique de Y_{η} relative à un faisceau inversible. La stratification arithmétique est définie par des fonctions zéta associées à un faisceau inversible métrisé et la fonction zéta d’une strate prend la forme

∑_{γ∈P (F)\G(F)} e^{⟨sλ,H_{P} (γ)⟩}E_{Q}^{Qw^{−1}Q_{0}}(sμ, η(p_{γ})) ,

E_{Q}^{Qw^{−1}Q_{0}} étant la série d’Eisenstein partielle associée à la cellule de Schubert Q\Qw^{−1}Q_{0}. Le calcul du terme constant de celles-ci fournit des estimations permettent de déterminer l’abcisse de convergence de la fonction zéta d’une strate.

Contents

Introduction 2 1 The fibre bundles: geometric-arithmetic preliminaries 4 2 Height zeta functions 9 3 Arithmetic stratification 18 References 23

Introduction

Consider a projective variety Y over a number field F and a metrized line bundle L = (L, (∥ ∥_{v})_{v}) on Y. For any subvariety U ⊂ Y there is a counting function

N_{U}(L, H) = #{y ∈ U(F) | H_{L}(y) ≤ H } .

Suppose N_{U}(L, H) is finite for all H. Then one would like to understand the asymptotic behaviour of the counting function N_{U}(L, H) as H tends to infinity. A useful tool when studying this question is the (formal) Dirichlet series

Z_{U}(L, s) = ∑_{y∈U(F)} \frac{1}{H_{L}(y)^{s}} ,

called a height zeta function. Whenever this series converges for all s ≫ 0 let σ_{U}(L) be its abcissa of convergence:

σ_{U}(L) = inf{s ∈ R | Z_{U}(L, s) converges} .

σ_{U}(L) depends only on the line bundle L underlying L and not on the metric. σ_{U}(L) = −∞ if and only if U(F) is a finite; otherwise

σ_{U}(L) = lim sup \frac{log N_{U}(L, H)}{log H} .

Sometimes one can go much further: namely, establish analytic continuation of the height zeta function. Then one even gets an asymptotic formula for the counting function, using a Tauberian Theorem (provided an assumption on the poles is satisfied). But before going into these questions, one encounters some features that should be considered beforehand.

A general phenomenon which may occur is this: there is a proper subvariety Y′ in Y, such that σ_{Y} (L) = σ_{Y′}(L) > σ_{Y −Y′}(L). Such a subvariety is called accumulating. In this case, counting rational points on Y′ is asymptotically the same as counting rational points on Y, and therefore one should study Y′ and Y − Y′ separately. It may happen that Y − Y′ contains again an accumulating subvariety. This leads us to the notion of the arithmetic stratification of Y with respect to L, introduced by V.V.Batyrev and Yu.I.Manin in [BM]. Suppose that Y (F) is dense in Y and that there is an open subset Y′ such that Z_{Y′}(F)(L, s) converges for all s ≫ 0. The arithmetic stratification of Y with respect to L is the descending sequence of Zariski open subvarieties

Y ⊃ Y_{0} ⊃ Y_{1} ⊃ Y_{2} …

where Y_{0} is the maximal Zariski open subset of Y for which the height zeta function Z_{Y_{0}}(L, s) converges for all s ≫ 0, and Y_{i+1} ⊂ Y_{i} is the maximal open subset of Y_{i} such that σ_{Y_{i+1}}(L) is strictly smaller than σ_{Y_{i}}(L). Note that the arithmetic stratification depends on the line bundle L. As an example, consider the variety obtained by blowing up the projective plane over F in a rational point. The projective line replacing this point will be an accumulating subvariety with respect to the anticanonical line bundle. For a further discussion cf. the introduction of [FMT] and [BM].

In this paper we study a class of smooth projective varieties, give a description of their Picard groups, and determine the arithmetic stratification with respect to an arbitrary line bundle. Here is a brief description of the varieties and the methods employed.

The varieties are locally trivial fibre bundles Y_{η} → W = P \G over a generalized flag variety P \G. The fibres are isomorphic to another flag variety Q\H, and Y_{η} is constructed via a homomorphism η : P ⟶ H which factors through a split torus. The Picard group is up to a finite cokernel given by the direct sum of character groups X^{∗}(Q) ⊕ X^{∗}(P). In P ic(Y_{η})_{R} there is the closed cone Λ(Y_{η})_{eff} generated by the line bundles having non-zero global sections, and we denote the interior of this cone by Λ(Y_{η})_{eff}^{∘}. From now on we assume that η satisfies assumption 1.4 below. Let L be a line bundle on Y corresponding to the pair (μ, λ) ∈ X^{∗}(Q) ⊕ X^{∗}(P). Suppose L lies in Λ(Y_{η})_{eff}^{∘} , and consider the arithmetic stratification Y_{η} ⊃ Y_{0} ⊃ Y_{1} ⊃ Y_{2}! … of Y_{η} with respect to L. It turns out that the strata Y_{i} − Y_{i+1} are always unions of twisted products Y^{w} = X^{w}×^{P} G over P \G. X^{w} = Q\Qw^{−1}Q_{0} is the Schubert cell in X corresponding to the Weyl group element w. The height zeta function of Y^{w} (with respect to a certain metric on L) is a double sum

∑_{γ∈P (F)\G(F)} e^{⟨sλ,H_{P} (γ)⟩}E_{Q}^{Qw^{−1}Q_{0}}(sμ, η(p_{γ})) .

E_{Q}^{Qw^{−1}Q_{0}} is what we call a “partial Eisenstein series”, because it is a subseries of a Langlands-Eisenstein series: the summation runs only over Q(F)\Q(F)w^{−1}Q_{0}(F). The partial Eisenstein series can be bounded from above and from below by its constant term E_{Q,Q_{0}}^{Qw^{−1}Q_{0}}. The constant term in turn is equal to

e^{⟨sη̌(wμ)−η̌(wρ_{0}−ρ_{0}),H_{P} (γ)⟩}E_{Q,Q_{0}}^{Qw^{−1}Q_{0}}(sμ, 1_{H}) .

Therefore, replacing the partial Eisenstein series by its constant term, one gets

E_{Q,Q_{0}}^{Qw^{−1}Q_{0}}(sμ, 1_{H}) · E_{P}^{G}(s(λ + η̌(wμ)) − η̌(wρ_{0} − ρ_{0}), 1_{G}) .

This suffices to determine the abcissa of convergence of the height zeta functions of the strata. The arithmetic stratification with respect to L can be described purely in terms of a finite number of explicitely computable constants involving the parameters μ, λ and the Weyl group elements.

The results in this paper are part of the authors Ph.D. thesis, whose second part concerning asymptotic formulas shall appear in [St]. It is a pleasure to thank Jens Franke for introducing me to the methods used in this paper and for his constant support while working on these questions.

1 The fibre bundles: geometric-arithmetic preliminaries

(1.1) Let G and H be two semi-simple linear algebraic groups over the number field F, and fix minimal F -rational parabolic subgroups P_{0} ⊂ G and Q_{0} ⊂ H with (F -rational) Levi decompositions P_{0} = L_{0}U_{0}, Q_{0} = M_{0}V_{0}. The quotients of G, respectively H, by parabolic subgroups P ⊃ P_{0}, Q ⊃ Q_{0} are (by definition) generalized flag varieties:

W = P \G and X = Q\H.

Consider a homomorphism η : P → T_{0}, where T_{0} is the maximal split torus in the center of M_{0}. P acts via η on X × G from the right: (x, g) · p = (xη(p), p^{−1}g), and there is a canonical morphism π from

Y = Y_{η} = (X × G)/P

to W, giving Y the structure of a locally trivial fibre bundle over W with general fibre X, cf. [J], 5.14.

(1.2) Let μ ∈ X^{∗}(Q) = Hom_{F-gps}(Q, G_{m,F}) be a character of Q and define an action of Q on H × A^{1} by q · (h, a) = (qh, μ(q)^{−1}a). The quotient of H × A^{1} by Q is a line bundle over X = Q\H, to be denoted by L_{μ} :

L_{μ} = Q\(H × A^{1}),

This gives an injection

X^{∗}(Q) ⟶ P ic(X)

with finite cokernel, cf. for instance [Sa], Prop. 6.10, Lemma 6.9. The same applies to X^{∗}(P) and P ic(W). The action of P on H × A^{1}, given by (h, a) · p = (hη(p), a), descends to an action of P on L_{μ}, and the quotient

L_{μ}^{Y} = (L_{μ} × G)/P ,

P acting on L_{μ} × G by (l, g) · p = (l · p, p^{−1}g), is a line bundle over Y. Using the exact sequence (6.10.2) in [Sa], it is not difficult to show that

(1.2.1) (μ, λ) ⟼ class of L_{μ}^{Y} ⊗ π^{∗}L_{λ}

is a monomorphism with finite cokernel

X^{∗}(Q) ⊕ X^{∗}(P) ⟶ P ic(Y) .

(1.3) The choice of Q_{0} determines a basis Δ_{0}^{H} of the root system of H with respect to T_{0}. The roots of T_{0} occuring in the Lie algebra of the unipotent radical of Q_{0} are defined to be positive and Δ_{0}^{H} is the basis of simple positive roots with respect to this set of positive roots. Let

W_{H} = N_{H(F)}(T_{0}(F))/Z_{H(F)}(T_{0}(F))

be the Weyl group of H (N_{H(F)} and Z_{H(F)} denoting the normalizer, respectively centralizer in H(F)), and let (., .) be a W_{H}-invariant positive definite form on X^{∗}(T_{0})_{R}. Define the set

X^{∗}(T_{0})_{R}^{+} = {x ∈ X^{∗}(T_{0})_{R} | for all β ∈ Δ_{0}^{H} : (x, β) > 0 } .

This is the positive Weyl chamber. Restriction of characters gives an injection and an isomorphism

X^{∗}(Q)_{R} ↪ X^{∗}(Q_{0})_{R} ˜ ⟶X^{∗}(T_{0})_{R} .

In the sequel we will always consider X^{∗}(Q)_{R} as a subspace of X^{∗}(Q_{0})_{R} which we will identify with X^{∗}(T_{0}). Let M denote the unique Levi component of Q which contains M_{0}. There is a subset Δ_{0}^{M} of Δ_{0}^{H} such that

X^{∗}(Q)_{R} = {x ∈ X^{∗}(T_{0})_{R} | for all β ∈ Δ_{0}^{M} : (x, β) = 0 } .

Denote by X^{∗}(Q)_{R}^{+} the interior (in X^{∗}(Q)_{R}) of the intersection of X^{∗}(Q)_{R} with the closure of X^{∗}(T_{0})_{R}^{+}. Let Δ_{Q}^{H} be the complement of Δ_{0}^{M} in Δ_{0}^{H}. Then we have

X^{∗}(Q)_{R}^{+} = {x ∈ X^{∗}(Q)_{R} | for all β ∈ Δ_{Q}^{H} : (x, β) > 0 } .

It is well known, that under the isomorphism X^{∗}(Q)_{R} → P ic(X)_{R} the closure of X^{∗}(Q)_{R}^{+} is mapped onto the cone of effective divisors Λ(X)_{eff}, i.e., the closed cone that is generated by those line bundles having non-zero global sections (cf. [J], II, 2.6).

The definitions depend on the once and for all chosen minimal parabolic subgroup and the fixed Levi component M_{0}. Define X^{∗}(P)_{R}^{+} by analogy (depending on P_{0} and L_{0}).

The following assumption will be used to describe Λ(Y)_{eff} and to determine the abcissa of convergence of the height zeta functions.

(1.4) Assumption. For all β ∈ Δ_{0}^{H} the element −β ∘ η of X^{∗}(P) lies in the closure of X^{∗}(P)_{R}^{+}.

Geometrically speaking, we consider those η which map the cone in X^{∗}(T_{0})_{R} generated by −Δ_{0}^{H} into the cone of effective divisors in X^{∗}(P)_{R}. This subset of Hom_{F-gps}(P, T_{0}) generates a sublattice of finite index. Moreover, it is easy to see that if one replaces η by a conjugate wηw^{−1}, w being an element of the Weyl group W_{H}, the variety Y_{wηw^{−1}} is isomorphic to Y_{η}. Sometimes such a conjugate of η satisfies (1.4). This is always the case if P is a maximal proper parabolic subgroup of G. Indeed, the group X^{∗}(P) is in this case of rank one, and if η is not trivial, the kernel of the induced homomorphism η̌: X^{∗}(T_{0})_{R} → X^{∗}(P)_{R} is a hyperplane, consisting of elements which are orthogonal to some χ ∈ X^{∗}(T_{0})_{R}, orthogonal with respect to the W-invariant positive definite form (., .) on X^{∗}(T_{0})_{R}. Replacing χ by −χ if necessary, we can assume that the half-space of ν ∈ X^{∗}(T_{0})_{R} with (ν, χ) < 0 is mapped to X^{∗}(P)_{R}^{+} by η̌. Consider the (closure of the) chamber that contains χ and the base Δ of the root system that corresp! onds to that chamber. For β ∈ Δ one has −β ∘ η ∈ \overline{X^{∗}(P)_{R}^{+}}, and this shows that a conjugate of η by an element of the Weyl group fulfills (1.4).

Identify W_{H} with its image in Aut(X^{∗}(T_{0})_{R}). The reflections along the elements of Δ_{0}^{H} generate W_{H}, and define the Bruhat ordering on W_{H} and a length function. Let w_{H} be the longest element of W. Define

Λ_{η} = \{(μ, λ) ∈ X^{∗}(Q)_{R} ⊕ X^{∗}(P)_{R} | μ ∈ \overline{X^{∗}(Q)_{R}^{+}} and η̌(w_{H}μ) + λ ∈ \overline{X^{∗}(P)_{R}^{+}} \} .

This is a closed cone in X^{∗}(Q)_{R} ⊕ X^{∗}(P)_{R}.

(1.5) Proposition. Suppose η fulfills assumption 1.4. Then the line bundle L_{μ}^{Y} ⊗ π^{∗}L_{λ} has non-zero global sections if and only if (μ, λ) lies in Λ_{η}.

Proof. Suppose H^{0}(Y, L_{μ}^{Y} ⊗ π^{∗}L_{λ}) ≠ 0. The pull-back of L_{μ}^{Y} ⊗ π^{∗}L_{λ} by the quotient map X × G ⟶ Y is isomorphic to the pull-back of L_{μ} by the projection onto the first factor X × G ⟶ X. It follows that L_{μ} has a non-zero global section, hence μ is an element of \overline{X^{∗}(Q)_{R}^{+}} (cf. [J], II, 2.6). Let T_{0} act on L_{μ} = Q\(H × A^{1}) by [h, a]t = [ht, a] and on H^{0}(X, L_{μ}) by (t · s)(x) = s(xt)t^{−1}. Because T_{0} is split, H^{0}(X, L_{μ}) decomposes as a direct sum of one-dimensional representations. For each χ ∈ X^{∗}(T_{0}) let m(χ) be the multiplicity of the corresponding weight space. Then one has

π_{∗}(L_{μ}^{Y} ⊗ π^{∗}L_{λ}) ≃ ⨁_{χ∈X^{∗}(Q)} L_{−χ∘η+λ}^{⊕m(χ)} .

Hence, there is a χ with m(χ) > 0 and −η̌(χ) + λ ∈ \overline{X^{∗}(P)_{R}^{+}}. This implies

η̌(w_{H}μ) + λ = −η̌(−w_{H}μ − χ) − η̌(χ) + λ ∈ \overline{X^{∗}(P)_{R}^{+}} ,

because −w_{H}μ − χ ∈ ∑_{β∈Δ_{0}^{H}} Z_{≥0}.

Now suppose that (μ, λ) lies in Λ_{η}. Then −w_{H}μ is a weight of the representation of T_{0} on H^{0}(X, L_{μ}). Therefore,

π_{∗}(L_{μ}^{Y} ⊗ π^{∗}L_{λ}) ≃ L_{(w_{H}μ)∘η+λ}^{⊕m(−w_{H}μ)} ⊕ ⨁_{χ≠−w_{H}μ} L_{−χ∘η+λ}^{⊕m(χ)}

has a non-zero global section, and so does L_{μ}^{Y} ⊗ π^{∗}L_{λ}. □

By Proposition 1.5, under the isomorphism X^{∗}(Q)_{R} ⊕ X^{∗}(P)_{R} ⟶ P ic(Y)_{R} induced by (1.2.1), the cone of effective divisors Λ(Y_{η})_{eff} is the image of Λ_{η}, and the interior Λ_{η}^{∘} of Λ_{η} is mapped onto Λ(Y_{η})_{eff}^{∘}.

(1.6) Next we discuss metrics on the line bundles introduced above. To do this, choose a maximal compact subgroup K_{G} = ∏_{v} K_{G,v} ⊂ G(A), such that

G(A) = P_{0}(A)K_{G} ,

where v runs over all places of F and A denotes the ring of adeles of F. For a place v and w ∈ W (F_{v}) choose k ∈ K_{G,v} which is mapped to w. If [g, a] ∈ L_{λ}(F_{v}) is an F_{v}-valued point of the line bundle L_{λ} over w, define its norm by

∥[g, a]∥_{w} = |λ(p)a|_{v} ,

where p ∈ P (F_{v}) is determined by g = pk and | · |_{v} denotes the local absolute value that is the Haar multiplier. The family ∥ · ∥_{v} = (∥ · ∥_{w})_{w∈W (F_{v})} is a v-adic metric on L_{λ} and L_{λ} = (L_{λ}, (∥ · ∥_{v})_{v}) is a metrization of L_{λ}, cf. [P], Lemme 6.2.3. Define a map

H_{P} = H_{P,K_{G}} : G(A) → Hom_{R}(X^{∗}(P) ⊗ R, R)

by ⟨λ, H_{P} (pk)⟩ = log(∏_{v} |λ(p_{v})|_{v}), for λ ∈ X^{∗}(P), p = (p_{v})_{v} ∈ P (A), k ∈ K_{G} and extend linearly. An easy computation shows (cf. [FMT], p. 428) that for the (exponential) height function we have

(1.6.1) H_{L_{λ}}(w) = e^{−⟨λ,H_{P} (γ)⟩} ,

whenever γ ∈ G(F) maps to w ∈ W (F) = P (F)\G(F).

Similarly, we fix a maximal compact subgroup K_{H} = ∏_{v} K_{H,v} ⊂ H(A) such that H(A) = Q_{0}(A)K_{H}, and provide all line bundles L_{μ}, μ ∈ X^{∗}(Q), with the metrics as defined above. Furthermore, K_{H} is supposed to induce maximal compact subgroups of all the standard Levi subgroups. More precisely, we assume that

for each parabolic subgroup Q′ ⊃ Q_{0} of H with Levi decomposition Q′ = M′V′, M′ ⊃ M_{0}, one has

• Q′(A) ∩ K_{H} = (M′(A) ∩ K_{H})(V′(A) ∩ K_{H}) ,

• M′(A) ∩ K_{H} is a maximal compact subgroup of M′(A) .

By [MW], I.1.4., such K_{H} exist. This implies in particular that K_{H} contains the maximal compact subgroup of T_{0}(A).

Finally, we define metrics on the line bundles L_{μ}^{Y} = (L_{μ} × G)/P. Let ∥ · ∥_{v} be the v-adic metric on L_{μ} defined by means of K_{H,v}. Recall the action L_{μ} × P → L_{μ} = Q\(H × A^{1}), given by l · p = l · η(p) = [hη(p), a] if l = [h, a]. For an element [l, g] ∈ L_{μ}(F_{v}) mapping to y = [x, g] ∈ Y (F_{v}) define

∥ [l, g] ∥_{y} = ∥ l · η(p) ∥_{xη(p)} ,

where g = pk, p ∈ P (F_{v}), k ∈ K_{H,v}. It follows from the remark above, that this is welldefined (because P (F_{v}) ∩ K_{G,v} is mapped into K_{H,v}). The family ∥ · ∥_{v} = (∥ · ∥_{y})_{y∈Y (F_{v})} is a v-adic metric on L_{μ}^{Y} and L_{μ}^{Y} = (L_{μ}^{Y} , (∥ · ∥_{v})_{v}) is a metrization of L_{μ}^{Y}. For γ ∈ G(F) and δ ∈ H(F) mapping to x ∈ X(F), the pair (x, γ) maps to an element y ∈ Y (F), and

(1.6.2) H_{L_{μ}^{Y}} (y) = e^{−⟨μ,H_{Q}(δη(p))⟩} ,

with g = pk, p ∈ P (A), k ∈ K_{G}.

2 Height zeta functions

Denote by s a variable (taking values in R). By (1.6.1) the height zeta function of the generalized flag variety X = Q\H corresponding to the metrized line bundle L_{μ} is equal to (the value of) an Langlands-Eisenstein series, whenever the series converges:

Z_{X}(L_{μ}, s) = ∑_{x∈X(F)} H_{L_{μ}}(x)^{−s} = ∑_{δ∈Q(F)\H(F)} e^{⟨sμ,H_{Q}(δ)⟩} = E_{Q}^{H}(sμ, 1_{H}) .

This crucial observation is due to J.Franke. In [FMT] all analytic properties of these Eisenstein series that are needed for applications concerning rational points on X are proved. In this paper we only need results on the domain of convergence, and so we found it appropriate to prove these more accessible statements here again instead of deducing them from much deeper theorems. Hence our first task is to determine the domain of convergence of these Eisenstein series. Let ρ_{Q} be half the sum of the roots of T_{0} that occur in the Lie algebra of V (= unipotent radical of Q), counted with multiplicity. For convenience, in the definition of E_{Q}^{H}, we shifted the coordinate on X^{∗}(Q) ⊗ R by −ρ_{Q}, compared with the usual convention.

The following elementary lemma will turn out to be very useful.

(2.1) Lemma. Let F ⊂ H(A) and C ⊂ X^{∗}(Q)_{R} be relatively compact subsets. Then there is a constant c > 1 such that for all v ∈ F, μ ∈ C and h ∈ H(A)

c^{−1}e^{⟨μ,H_{Q}(h)⟩} ≤ e^{⟨μ,H_{Q}(hv)⟩} ≤ ce^{⟨μ,H_{Q}(h)⟩} .

Proof. Put h = qk, hv = q_{1}k_{1}, q, q_{1} ∈ Q(A), k, k_{1} ∈ K_{H}. Note that q^{−1}q_{1} = kvk_{1}^{−1} ∈ K_{H} \overline{F}K_{H} ∩ Q(A) and the latter is a compact subset of Q(A). Hence there is c > 1 such that for all v_{1} ∈ K_{H}FK_{H} and μ ∈ C

c^{−1} ≤ e^{⟨μ,H_{Q}(v_{1})⟩} ≤ c .

Hence we have for any triple (μ, v, h) ∈ C × F × H(A)

\begin{aligned}c^{−1}e^{⟨μ,H_{Q}(h)⟩} & ≤ & e^{⟨μ,H_{Q}(q^{−1}q_{1})⟩}e^{⟨μ,H_{Q}(q)⟩} & = & e^{⟨μ,H_{Q}(hv)⟩} , \\ e^{⟨μ,H_{Q}(hv)⟩} & = & e^{⟨μ,H_{Q}(q^{−1}q_{1})⟩}e^{⟨μ,H_{Q}(q)⟩} & ≤ & ce^{⟨μ,H_{Q}(hv)⟩} . \end{aligned} □

(2.2) The most important tool for us to investigate the Eisenstein series are Siegel domains. To introduce these, consider the homomorphism

\begin{aligned}β̄ = (β)_{β∈Δ_{0}^{H}} : & T_{0} & → & G_{m,F}^{d} , \\ & t & ↦ & (β(t))_{β∈Δ_{0}^{H}} . \end{aligned}

It is surjective with finite kernel (d = dim(T)). Embed R_{>0} into A^{∗} by mapping x ∈ R to the idèle (x_{v})_{v} with archimedean components x_{v} = x and non-archimedean components x_{v} = 1. This way we consider R_{>0}^{d} as a subgroup of G_{m,F}^{d} (A). Let A_{M_{0}} the connected component of the preimage of R_{>0}^{d} under β̄, which contains the trivial element. For all w ∈ W we have wA_{M_{0}}w^{−1} = A_{M_{0}}. One has M_{0}(A) = A_{M_{0}}M_{0}(A)^{1} with

M_{0}(A)^{1} = ⋂_{χ∈X^{∗}(M_{0})} ker(|χ| : M_{0}(A) → R_{>0}) ,

where |χ|((m_{v})_{v}) := ∏_{v} |χ(m_{v})|_{v}. For t_{0} ∈ A_{M_{0}} we define

A_{M_{0}}(t_{0}) = {t ∈ A_{M_{0}} | for all β ∈ Δ_{0}^{H} : β(t) > β(t_{0}) } .

Recall the Levi decomposition Q = MV, M being the unique Levi component containing M_{0}. For a compact subset Ω ⊂ Q_{0}(A) put

S_{Q,Ω} = ΩA_{M}A_{M_{0}}(t_{0})K_{H} ,

where A_{M} = A_{M_{0}} ∩ Z(M(A)), Z(M(A)) denoting the center of M(A). If Ω is sufficiently big and t_{0} is sufficiently small (i.e., β(t_{0}) is sufficiently small for all β ∈ Δ_{0}^{H}) then the following holds, c.f. [MW], I.2.1:

H(A) = Q(F)S_{Q,Ω} .

In this case S_{Q,Ω} is called a Siegel domain. With these preparations we can determine the abcissa of convergence of the Eisenstein series.

(2.3) Proposition. a) The series

E_{Q}^{H}(μ + 2ρ_{Q}, h) = ∑_{δ∈Q(F)\H(F)} e^{⟨μ+2ρ_{Q},H_{Q}(δh)⟩} ,

μ ∈ X^{∗}(Q) ⊗ C, h ∈ H(A), converges uniformly for (ℜ(μ), h) in any compact subset of X^{∗}(Q)_{R}^{+} × H(A).

b) For any μ ∈ X^{∗}(Q)_{R} − X^{∗}(Q)_{R}^{+} and any h ∈ H(A) the series E_{Q}^{H}(μ + 2ρ_{Q}, h) does not converge.

Proof. a) (This part is extracted from [Go].) For each β ∈ Δ_{0}^{H} there is a maximal proper parabolic subgroup Q_{β} containing Q_{0} and a generator μ_{β} of X^{∗}(Q_{β}) lying in X^{∗}(Q_{β})_{R}^{+} such that

X^{∗}(Q_{0})_{R}^{+} = ∑_{β∈Δ_{0}^{H}} R_{>0}μ_{β} .

Attached to μ_{β} there is an ample line bundle L_{μ_{β}} on Q_{β}\H. Hence, there exists c_{β} > 0 such that for all δ ∈ H(F)

e^{⟨μ_{β},H_{Q_{β}} (δ)⟩} = H_{L_{μ_{β}}} (Q_{β}(F)δ)^{−1} ≤ c_{β} .

Now let U ⊂ H(A) be an open and relatively compact subset. By Lemma (2.1) there is a constant c > 0 such that for all β ∈ Δ_{Q}^{H}, δ ∈ H(F) and h ∈ U

e^{⟨μ_{β},H_{Q_{0}}(δh)⟩} ≤ c .

Therefore, H(F)U is contained in V_{0}(A)M_{0}(A)^{1}A_{M_{0}}^{−}(t_{0}^{−})K_{H} for some t_{0}^{−} ∈ A_{M_{0}}, where

A_{M_{0}}^{−}(t_{0}^{−}) = {t ∈ A_{M_{0}} | for all β ∈ Δ_{0}^{H} : μ_{β}(t) < μ_{β}(t_{0}^{−})} .

Next we use the fact that H(A) = Q(F)S_{Q,Ω} with some Siegel domain S_{Q,Ω}. By the definition of S_{Q,Ω} we see that there is a fundamental domain for the action of Q(F) on H(F)U which is contained in

Ω′(A_{M} ∩ A_{M_{0}}^{−}(t_{0}^{−}))K_{H}

for some compact subset Ω′ ⊂ Q_{0}(A) which contains Ω. By Lemma 2.1, for h in a compact subset of H(A), we can bound E_{Q}^{H}(μ + 2ρ_{Q}, h) by the integral

∫_{Q(F)\H(F)U} e^{⟨μ+2ρ_{Q},H_{Q}(h_{1})⟩}dh_{1} ,

with some Haar measure dh_{1} on H(A). Now, dh_{1} = e^{⟨−2ρ_{Q},H_{Q}(m)⟩}dvdmdk with Haar measures dv on V (A), dm on M(A), dk on K_{H} (recall Q = MV and H(A) = Q(A)K_{H}). Therefore, the integral above can be bounded by

∫_{A_{M} ∩A_{M_{0}}^{−}(t_{0}^{−})} e^{⟨μ,H_{Q}(a)⟩}da ,

with some Haar measure da on A_{M}. Writing μ = ∑_{β∈Δ_{Q}^{H}} s_{β}μ_{β} with s_{β} ∈ R_{>0}, we see that this integral is equal to

(constant depending on t_{0}^{−} and da) · ∏_{β∈Δ_{Q}^{H}} \frac{1}{s_{β}} .

Hence we have shown that E_{Q}^{H}(μ + 2ρ_{Q}, h) converges uniformly for (μ, h) in compact subsets of X^{∗}(Q)_{R}^{+} × H(A).

b) To prove the second assertion, it is enough to show that E_{Q}^{H}(μ + 2ρ_{Q}, 1_{H}) tends to infinity as μ approaches the boundary of X^{∗}(Q)_{R}^{+}. Let Q′ ⊃ Q be a parabolic subgroup with Levi component M′ ⊃ M such that Q ∩ M′ is a maximal proper parabolic subgroup of M′. Decompose X^{∗}(Q)_{R} as follows:

X^{∗}(Q)_{R} = X^{∗}(Q′)_{R} ⊕ X^{∗}(T_{M}/T_{M′})_{R} ,

where T_{M} (resp. T_{M′}) is the maximal split torus in the center of M (resp. M′). With respect to this decomposition we have 2ρ_{Q} = 2ρ_{Q′} + 2ρ_{Q∩M′}^{M′}, where 2ρ_{Q∩M′}^{M′} is the sum of roots of T_{0} (counted with multiplicity) which occur in the Lie algebra of the unipotent radical of Q ∩ M′. X^{∗}(T_{M}/T_{M′})_{R} is a one dimensional R-vector space and the image of X^{∗}(Q)_{R}^{+} under the projection onto this space is the open half-line generated by ρ_{Q∩M′}^{M′}. Now let μ ∈ X^{∗}(Q)_{R}^{+} tend to the boundary of X^{∗}(Q)_{R}^{+}. Call μ′ the image of μ under the projection onto X^{∗}(T_{M}/T_{M′})_{R}, and assume that μ′ tends to zero. Next we note that

E_{Q}^{H}(μ + 2ρ_{Q}, 1_{H}) = ∑_{δ∈Q′(F)\H(F)} e^{⟨μ−μ′+2ρ_{Q′},H_{Q′}(δ)⟩}E_{Q}^{Q′}(μ′ + 2ρ_{Q∩M′}^{M′}, δ) .

Hence it suffices to show that E_{Q∩M′}^{M′}(μ′ +2ρ_{Q∩M′}^{M′}, 1_{H}) tends to infinity as μ′ tends to zero. In other words, it suffices to consider the case when Q is a maximal proper parabolic subgroup. Let Q^{−} be the opposite parabolic subgroup with Levi component V^{−}. The map from H to Q\H maps V^{−} isomorphically onto an open subset of Q\H. On V^{−}(A) the function h ↦ e^{⟨2ρ_{Q},H_{Q}(h)⟩} is bounded from above but assumes every sufficiently small positive value. Let F ⊂ V^{−}(A) be an open and relatively compact subset such that V^{−}(F)F = V^{−}(A). Then, with the notations from part a) of the proof, there is some open and relatively compact subset ω ⊂ Q_{0}(A) such that a fundamental domain for the action of Q(F) on H(F)FK_{H} contains

ω(A_{M} ∩ A_{M_{0}}^{−}(t_{0}^{−}))K_{H} .

Here we used the fact that A_{M} is one-dimensional. By Lemma 2.1, E_{Q}^{H}(μ + 2ρ_{Q}, 1_{H}) is bounded from below by the integral

∫_{Q(F)\H(F)FK_{H}} e^{⟨μ+2ρ_{Q},H_{Q}(h)⟩}dh ,

with some Haar measure dh on H(A), and this in turn can be bounded from below by

∫_{A_{M} ∩A_{M_{0}}^{−}(t_{0}^{−})} e^{⟨μ,H_{Q}(a)⟩}da ,

with some appropriate Haar measure da on A_{M}. This integral tends to infinity as μ tends to zero, and this completes the proof. □

(2.4) Of course, the Proposition 2.3 applies equally to the Eisenstein series

E_{P}^{G}(λ, G) = ∑_{γ∈P (F)\G(F)} e^{⟨λ,H_{P} (γg)⟩} .

For μ ∈ X^{∗}(Q)_{R}^{+} denote by a_{μ} the abcissa of convergence of E_{Q}^{H}(sμ, h). By Prop. (2.3) we have

a_{μ} = inf{a | aμ ∈ 2ρ_{Q} + X^{∗}(Q)_{R}^{+} } .

For μ ∉ X^{∗}(Q)_{R}^{+} we put a_{μ} = ∞. The height zeta function of the variety Y with respect to the metrized line bundle L_{μ}^{Y} ⊗ π^{∗}L_{λ} has, by (1.6.1) and (1.6.2), the following formal expression:

\begin{aligned}∑_{y∈Y (F)} H_{L_{μ}^{Y} ⊗π^{∗}L_{λ}}(y)^{−s} & = ∑_{γ∈P (F)\G(F)} e^{⟨sλ,H_{P} (γ)⟩} ∑_{δ∈Q(F)\H(F)} e^{⟨sμ,H_{Q}(δη(p_{γ}))⟩} \\ & = ∑_{γ∈P (F)\G(F)} e^{⟨sλ,H_{P} (γ)⟩}E_{Q}^{H}(sμ, η(p_{γ})) \end{aligned}

where γ = p_{γ}k_{γ} with p_{γ} ∈ P (A), k_{γ} ∈ K_{G}. Even if L_{μ}^{Y} ⊗ π^{∗}L_{λ} lies in the interior of the cone of effective divisors Λ(Y)_{eff}, this series will in general not converge for any s. The reason for this is the occurrence of accumulating subvarieties. A posteriori it turns out that the right approach to find these is to decompose the fibre X into locally closed subvarieties, the images of the Bruhat cells:

X = ∐_{w∈W^{M}} X^{w} ,

where W^{M} ⊂ W consists of the elements of minimal length in the classes wW_{M}, w running through W, and W_{M} is the Weyl group of M with respect to T_{0}. X^{w} is the image of the generalized Bruhat cell Qw^{−1}Q_{0} ⊂ H. Denote by w_{H}^{M} the element of minimal length in w_{H}W_{M}. Then X^{w_{H}^{M}} is the open stratum, which we denote by X^{∘}. The partial Eisenstein series are defined by summation over the rational points of X^{w}:

E_{Q}^{Qw^{−1}Q_{0}}(μ, h) = ∑_{δ∈Q(F)\Q(F)w^{−1}Q_{0}(F)} e^{⟨μ,H_{Q}(δh)⟩} .

The behaviour of the partial Eisenstein series on Siegel domains is of importance for us. To understand how a partial Eisenstein series decreases when one goes to infinity on such a Siegel domain, we will study its constant term. The constant term Θ_{Q_{0}} of a function Θ : H(A) → C, which is left-invariant under V_{0}(F) and lies in L_{1,loc}(H(A)), is defined by

Θ_{Q_{0}}(h) = ∫_{V_{0}(F)\V_{0}(A)} Θ(vh)dv ,

where the measure dv on V_{0}(F)\V_{0}(A) is the quotient of the Haar measures and is normalized by ∫_{V_{0}(F)\V_{0}(A)} dv = 1. The following statement is the key to determine the abcissa of convergence of the height zeta functions of the strata. We put ρ_{0} = ρ_{Q_{0}}.

(2.5) Proposition. Let F ⊂ H(A) and C ⊂ 2ρ_{Q} + X^{∗}(Q)_{R}^{+} be relatively compact subsets (C relatively compact in X^{∗}(Q)_{R}), and fix t_{0} ∈ A_{M_{0}}. a) There is a constant c > 1, such that for all μ ∈ C, w ∈ W^{M}, t ∈ A_{M_{0}}(t_{0}) and v ∈ F

c^{−1}E_{Q,Q_{0}}^{Qw^{−1}Q_{0}}(μ, 1_{H}) ≤ e^{−⟨w(μ−ρ_{0})+ρ_{0},H_{Q_{0}}(t)⟩}E_{Q}^{Qw^{−1}Q_{0}}(μ, tv) ≤ cE_{Q,Q_{0}}^{Qw^{−1}Q_{0}}(μ, 1_{H}) .

b) Now let w = w_{H}^{M}. Then there is a constant c > 1, such that for all μ ∈ 2ρ_{Q} + X^{∗}(Q)_{R}^{+}

E_{Q}^{H}(sμ, 1_{H}) ≤ cE_{Q}^{Qw^{−1}Q_{0}}(sμ, 1_{H}) .

Proof. a) Let V_{0} be the unipotent radical of Q_{0}, and let F_{0} ⊂ V_{0}(A) be a compact subset such that V_{0}(F)F_{0} = V_{0}(A). Because

{t^{−1}vt | t ∈ A_{M_{0}}(t_{0}), v ∈ F_{0}}

is relatively compact, Lemma 2.1 ensures the existence of a c > 1 such that for all h ∈ H(A), t ∈ A_{M_{0}}, v ∈ F, v_{0} ∈ F_{0} one has

c^{−1}e^{⟨μ,H_{Q}(ht(t^{−1}v_{0}t))⟩} ≤ e^{⟨μ,H_{Q}(htv)⟩} ≤ ce^{⟨μ,H_{Q}(ht(t^{−1}v_{0}t))⟩} .

Therefore

c^{−1} ∫_{V_{0}(F)\V_{0}(A)} E_{Q}^{Qw^{−1}Q_{0}}(μ, v_{0}t)dv_{0} ≤ E_{Q}^{Qw^{−1}Q_{0}}(μ, tv) ≤ c ∫_{V_{0}(F)\V_{0}(A)} E^{Qw^{−1}Q_{0}}(μ, v_{0}t)dv_{0} .

Moreover,

\begin{aligned}∫_{V_{0}(F)\V_{0}(A)} E_{Q}^{Qw^{−1}Q_{0}}(μ, v_{0}t)dv_{0} & = & ∫_{(V_{0}∩wV_{0}w^{−1})(A)\V_{0}(A)} e^{⟨μ,H_{Q_{0}}(w^{−1}v_{0}t))⟩}dv_{0} \\ & = & e^{⟨wμ+ρ_{0}−wρ_{0},H_{Q}(t)⟩}E_{Q,Q_{0}}^{Qw^{−1}Q_{0}}(μ, 1_{H}) . \end{aligned}

And this proves assertion a).

b) Because H(F) is dense in H and Qw^{−1}Q_{0} is open (and dense) in H, there are elements δ_{1}, …, δ_{n} ∈ H(F) such that

H(F) = ∪_{i=1}^{n}Q(F)w^{−1}Q_{0}(F)δ_{i} .

Therefore,

(2.5.1) E_{Q}^{H}(μ, 1_{H}) ≤ ∑_{i=1}^{n} E_{Q}^{Qw^{−1}Q_{0}}(μ, δ_{i}) .

By Lemma 2.1, there exists c_{1} > 1 such that for all μ ∈ C, δ ∈ H(F) and i = 1, …, n

e^{⟨μ,H_{Q}(δδ_{i})⟩} ≤ c_{1}e^{⟨μ,H_{Q}(δ))⟩} .

The sum on the right of (2.5.1) can be bounded by nc_{1}E_{Q}^{Qw^{−1}Q_{0}}(sμ, 1_{H}). □

(2.6) The following lemma explains the significance of assumption 1.4 : the geometric assertion that η̌ maps the cone in X^{∗}(T_{0})_{R} generated by −Δ_{0}^{H} into the cone X^{∗}(P)_{R}^{+} ⊂ X^{∗}(P)_{R}, has as analytic consequence that there exists a fixed Siegel domain such that η maps the “P -part” of each rational element γ ∈ G(F) into this fixed Siegel domain. To be more precise, put

T_{0}(A)^{1} = M_{0}(A)^{1} ∩ T_{0}(A) ,

cf. (2.2). By the product formula we have T_{0}(F) ⊂ T_{0}(A)^{1}, and the quotient T_{0}(F)\T_{0}(A)^{1} is compact. Let Ω_{0} ⊂ T_{0}(A)^{1} be a compact subset such that T_{0}(A)^{1} = T (F)Ω_{0}.

(2.7) Lemma. There is t_{0} ∈ A_{M_{0}}, such that for all γ ∈ G(F) one has

η(p_{γ}) ∈ T_{0}(A)^{1}A_{M_{0}}(t_{0}) ,

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

Proof. For β ∈ Δ_{0}^{H} we have

e^{⟨β,H_{Q_{0}}(η(p_{γ}))⟩} = e^{⟨β∘η,H_{P} (γ)⟩} = H_{L_{−β∘η}}(P (F)γ) .

By hypothesis, the line bundle L_{−β∘η} lies in the cone of effective divisors, hence the height function associated to L_{−β∘η} is bounded from below. □

(2.8) Before proving our result about the abcissa of convergence we have to introduce the following constants. For (μ, λ) ∈ X^{∗}(Q)_{R} ⊕ X^{∗}(P)_{R} and w ∈ W^{M} put

ǎ_{μ,λ}^{w} = inf {a | for all a′ ≥ a : a′(λ + η̌(wμ)) − η̌(wρ_{0} − ρ_{0}) ∈ 2ρ_{P} + X^{∗}(P)_{R}^{+} } ,
a_{μ,λ}^{w} = max {a_{μ}, ǎ_{μ,λ}^{w}} .

In general, ǎ_{μ,λ}^{w} may not be finite (by convention, the infimum of the empty set is ∞). For w = w_{H}^{M} put a_{μ,λ}^{∘} = a_{μ,λ}^{w}. The stratification of X gives rise to a stratification of Y because all strata X^{w} = Q\Qw^{−1}Q_{0} are stable under the action of T_{0} from the right:

Y = ∐_{w∈W^{M}} Y^{w} with Y^{w} = (X^{w} × G)/P .

The open stratum, corresponding to w = w_{H}^{M} will be denoted by Y^{∘}. The corresponding height zeta functions take the following form:

Z_{Y^{w}}(L_{μ}^{Y} ⊗ π^{∗}L_{λ}, s) = ∑_{γ∈P (F)\G(F)} e^{⟨sλ,H_{P} (γ)⟩}E_{Q}^{Qw^{−1}Q_{0}}(sμ, η(p_{γ})) ,

with the notations from Lemma 2.7.

(2.9) Proposition. a) If λ+η̌(wμ) is not contained in X^{+}(P)_{R}^{+}, then Z_{Y^{w}}(L_{μ}^{Y} ⊗π^{∗}L_{λ}, s) does not converge for any s > 0.

b) If λ + η̌(wμ) is in X^{∗}(P)_{R}^{+}, then the abcissa of convergence of Z_{Y^{w}}(L_{μ}^{Y} ⊗ π^{∗}L_{λ}, s) is in the interval [ǎ_{μ,λ}^{w}, a_{μ,λ}^{w}].

c) Suppose that L_{μ}^{Y} ⊗ π^{∗}L_{λ} lies in Λ(Y)_{eff}^{∘}. Then the abcissa of convergence of Z_{Y^{∘}}(L_{μ}^{Y} ⊗ π^{∗}L_{λ}, s) is a_{μ,λ}^{∘}.

Proof. a) Let us fix t_{0} such that the assertion of Lemma 2.7 holds. Then we can write η(p_{γ}) = ϑ_{γ}o_{γ}a_{γ} with ϑ_{γ} ∈ T (F), o_{γ} ∈ Ω_{0} and a_{γ} ∈ A_{M_{0}}(t_{0}). Since all partial Eisenstein series are left-invariant under T (F) one has

E_{Q}^{Qw^{−1}Q_{0}}(sμ, η(p_{γ})) = E_{Q}^{Qw^{−1}Q_{0}}(sμ, a_{γ}o_{γ}) .

For each real number a > a_{μ} there is by Proposition 2.5 a c_{1} > 0 such that for all w ∈ W^{M} and all s ∈ (a_{μ}, a]

c_{1}E_{Q,Q_{0}}^{Qw^{−1}Q_{0}}(sμ, 1_{H}) ≤ e^{−⟨w(sμ−ρ_{0})+ρ_{0},H_{Q_{0}}(a_{γ})⟩}E_{Q}^{Qw^{−1}Q_{0}}(sμ, η(p_{γ})) ≤ c_{1}^{−1}E_{Q,Q_{0}}^{Qw^{−1}Q_{0}}(sμ, 1_{H}) ,

where E_{Q,Q_{0}}^{Qw^{−1}Q_{0}} is the constant term of the partial Eisenstein series:

E_{Q,Q_{0}}^{Qw^{−1}Q_{0}}(sμ, 1_{H}) = ∫_{V_{0}(F)\V_{0}(A)} E_{Q}^{Qw^{−1}Q_{0}}(sμ, v_{0})dv_{0} .

For s ∈ (a_{μ}, a] this function is bounded from below by a positive number c_{2}. For such s we have then

\begin{matrix}Z_{Y^{w}}(LY_{μ} ⊗ π^{∗}L_{λ}, s) ≥ & c_{1}E_{Q,Q_{0}}^{Qw^{−1}Q_{0}}(sμ, 1_{H}) ∑_{γ∈P (F)\G(F)} e^{⟨sλ,H_{P} (a_{γ})⟩}e^{⟨swμ−wρ_{0}+ρ_{0},H_{Q_{0}}(η(p_{γ}))⟩} \\ ≥ & c_{1}c_{2}E_{P}^{G}(s(λ + η̌(wμ)) − η̌(wρ_{0} − ρ_{0}), 1_{G}) . \end{matrix}

It is well-known that ρ_{0} − wρ_{0} lies in the closed cone generated by the positive simple roots, and by assumption 1.4 we conclude that η̌(wρ_{0} − ρ_{0}) lies in the closure of X^{∗}(P)_{R}^{+}. Therefore, if λ + η̌(wμ) is not contained in X^{∗}(P)_{R}^{+}, the same is true for s(λ + η̌(wμ)) − η̌(wρ_{0} −ρ_{0}), for all s > 0. In this case, Z_{Y^{w}}(L_{λ} ⊗π^{∗}L_{μ}, s) does not converge for any s > 0.

b) Suppose now that λ + η̌(wμ) is an element of X^{∗}(P)_{R}^{+}. The above estimate shows that the abcissa of convergence of Z_{Y^{w}}(L_{λ} ⊗ π^{∗}L_{μ}, s) is greater or equal to ǎ_{μ,λ}^{w}. On the other hand, using the estimate above, we get that

Z_{Y^{w}}(L_{μ}^{Y} ⊗ π^{∗}L_{λ}, s) ≤ c_{1}^{−1}E_{Q,Q_{0}}^{Qw^{−1}Q_{0}}(sμ, 1_{H})E_{P}^{G}(s(λ + η̌(wμ)) − η̌(wρ_{0} − ρ_{0}), 1_{G}) .

Hence, this function converges for s > max{a_{μ}, ǎ_{μ,λ}^{w}} = a_{μ,λ}^{w}, by Proposition 2.3 a).

c) Let w = w_{H}^{M}. By Proposition 2.5 b) and the estimate above, we have for s ∈ (a_{μ}, a]

E_{Q,Q_{0}}^{Qw^{−1}Q_{0}}(sμ, 1_{H}) ≥ c_{1}E_{Q}^{Qw^{−1}Q_{0}}(sμ, 1_{H}) ≥ c_{3}E_{Q}^{H}(sμ, 1_{H}) ,

for some c_{3} > 0. This shows that Z_{Y^{∘}}(L_{μ}^{Y} ⊗ π^{∗}L_{λ}, s) converges only for s > a_{μ,λ}^{∘}. By part b) we conclude that the abcissa of convergence is exactly a_{μ,λ}^{∘}. □

3 Arithmetic stratification

In this section we determine the arithmetic stratification of Y = (Q\H × G)/P with respect to line bundles L_{μ}^{Y} ⊗ π^{∗}L_{λ} which lie in the interior of the cone of effective divisors Λ(Y)_{eff}^{∘}. Otherwise, there is no open non-empty subset for which the corresponding height zeta function converges for all s ≫ 0. First we need a lemma.

(3.1) Lemma. If X^{w′} ⊂ \overline{X^{w}}, then a_{μ,λ}^{w′} ≥ a_{μ,λ}^{w}.

Proof. The assertion follows if we can show that for all a > a_{μ} one has

a(λ + η̌(wμ)) − η̌(wρ_{0} − ρ_{0}) − a(λ + η̌(w′μ)) + η̌(wρ_{0} − ρ_{0}) ∈ \overline{X^{∗}(P)_{R}^{+}} .

Assumption 1.4 on η ensures us that this is true if we have

aw′μ − w′ρ_{0} − awμ + wρ_{0} ∈ ∑_{β∈Δ_{0}^{H}} R_{≥0}β

for all a > a_{μ}. Note that

aw′μ − w′ρ_{0} − awμ + wρ_{0} = w′(aμ − 2ρ_{Q}) − w(aμ − 2ρ_{Q}) + w′(2ρ_{Q} − ρ_{0}) − w(2ρ_{Q} − ρ_{0}) .

Now let w_{M} be the longest element in W_{M}. Then we have w_{H} = w_{H}^{M}w_{M} and 2ρ_{Q} − ρ_{0} = w_{M}ρ_{0}. Therefore,

aw′μ − w′ρ_{0} − awμ + wρ_{0} = w′(aμ − 2ρ_{Q}) − w(aμ − 2ρ_{Q}) + w′w_{M}ρ_{0} − ww_{M}ρ_{0} .

By [J], II, 13.8 (4), the assumption X^{w′} ⊂ \overline{X^{w}} implies that w′ ≤ w with respect to the Bruhat ordering on W_{H}. It follows from the description of the Bruhat ordering given in [J], II, Proposition 13.7, together with

l(w′w_{M}) = l(w′) + l(w_{M}), l(ww_{M}) = l(w) + l(w_{M})

that moreover

w′w_{M} ≤ ww_{M} .

Hence we are done, if we can show that whenever w_{1} ≤ w_{2} and x ∈ \overline{X^{∗}(Q_{0})_{R}^{+}} one has

w_{1}x − w_{2}x ∈ ∑_{β∈Δ_{0}^{H}} R_{≥0}β .

For a root β of H with respect to T_{0} we denote by s_{β} the reflection along β. By [BGG], Proposition 2.8, there exist positive roots β_{1}, …, β_{k} such that with s_{i} = s_{β_{i}}

\begin{matrix}w_{2} = s_{k} · s_{k−1} · . . . · s_{1} · w_{1} , \\ for all i = 1, ..., k : l(s_{i} · s_{i−1} · . . . · s_{1} · w_{1}) = l(s_{i−1} · . . . · s_{1} · w_{1}) + 1 . \end{matrix}

Because

w_{1}x − w_{2}x = ∑_{i=1}^{k} s_{i−1} · . . . · s_{1} · w_{1}x − s_{i} · s_{i−1} · . . . · s_{1} · w_{1}x ,

we can assume that w_{2} = s_{β}w_{1} with a positive root β and l(w_{2}) = l(w_{1}) + 1. By [Bou], ch. VI, §1, Proposition 17, this implies w_{2}^{−1}β < 0, i.e. w_{1}^{−1}β > 0. Now suppose x lies in the closure of X^{∗}(Q_{0})_{R}^{+}. Then one has

w_{1}x − w_{2}x = w_{1}x − (w_{1}x − 2\frac{(w_{1}x, β)}{(β, β)}β) = 2\frac{(x, w_{1}^{−1}β)}{(β, β)}β ,

and this proves the assertion. □

To describe the arithmetic stratification, consider those a_{μ,λ}^{w} which are finite and order them:

{a_{μ,λ}^{w}| w ∈ W^{M}, a_{μ,λ}^{w} < ∞} = {a_{0}, a_{1}, …, a_{r_{μ,λ}} } ,

with a_{0} > a_{1} > … > a_{r_{μ,λ}} and put formally a_{−1} = ∞.

(3.2) Theorem. Let Y = Y_{η} be the fibre bundle over W = P \G defined in (1.1), and suppose η fulfills the assumption of (1.4).

a) If L_{μ}^{Y} ⊗ π^{∗}L_{λ} is not an element of Λ(Y)_{eff}^{∘}, then there is no open non-empty subset such that the corresponding height zeta function converges for s ≫ 0.

b) Suppose L_{μ}^{Y} ⊗ π^{∗}L_{λ} lies in Λ(Y)_{eff}^{∘}. For i = 0, 1, …, r = r_{μ,λ} put

Y_{i} = Y − ⋃_{w∈W^{M} , a_{μ,λ}^{w}=a_{i−1}} \overline{Y^{w}} .

Then the arithmetic stratification of Y with respect to L_{μ}^{Y} ⊗ π^{∗}L_{λ} is

Y ⊃ Y_{0} ⊃ Y_{1} ⊃ … ⊃ Y_{r} .

Proof. First we prove the second assertion. Lemma 3.1 implies that

Y_{i} = ⋃_{w∈W^{M} , a_{μ,λ}^{w}≤a_{i}} Y^{w} .

Hence we conclude that Y_{i+1} is properly contained in Y_{i} (i = 0, ..., r−1). Let σ_{Y_{i}}(L) be the abcissa of convergence of Z_{Y_{i}}(L, s) with L = L_{μ}^{Y} ⊗ π^{∗}L_{λ}. We shall show that σ_{Y_{i}}(L) = a_{i}. Obviously,

Z_{Y_{i}}(L, s) = ∑_{w∈W^{M} , a_{μ,λ}^{w}≤a_{i}} Z_{Y^{w}}(L, s) .

By Proposition 2.9 b), Z_{Y^{w}}(L, s) converges for s > a_{μ,λ}^{w}. This shows that Z_{Y_{i}}(L, s) converges for s > a_{i}. On the other hand, there is a w ∈ W^{M} such that a_{μ,λ}^{w} = a_{i} and in order for Z_{Y^{w}}(L, s) to converge it is necessary that s > ǎ_{μ,λ}^{w}. Because the closure of X^{∘} is X, we have a_{μ,λ}^{∘} = a_{r}, by Lemma 3.1, and Z_{Y^{∘}}(L, s) is a summand of Z_{Y_{i}}(L, s). By Proposition 2.5 c), we know that for Z_{Y^{∘}}(L, s) to converge it is necessary that s > a_{μ}. Hence, for Z_{Y_{i}}(L, s) to converge it is necessary that s > max{ǎ_{μ,λ}^{w}, a_{μ}} = a_{μ,λ}^{w} = a_{i} and we conclude that the abcissa of convergence of Z_{Y_{i}}(L, s) is a_{i}.

Fix i ∈ { −1, …, r } and consider an open subset Y′ ⊂ Y_{i} which contains Y_{i+1} properly (Y_{−1} = Y, Y_{r+1} = ∅). The case i = −1 can only occur if Z_{Y} (L, s) does not converge for any s > 0. We have to show that Z_{Y′}(L, s) converges only for s > a_{i} (and then this sum converges for all s > a_{i}). By hypothesis, Y′ properly contains the open subset Y_{i+1}, hence there is a w ∈ W^{M} such that a_{μ,λ}^{w} = a_{i} und Y′ ∩ Y^{w} ≠ ∅. Y′ ∩ Y^{w} is an open and dense subset of Y^{w}. Therefore, there are open and dense subsets X′ ⊂ X^{w}, G′ ⊂ G, such that the image of X′ × G′ under the canonical projection X × G → Y is contained in Y′ ∩ Y^{w}. Moreover, we may and we will assume that P G′ ⊂ G′. Let H′ ⊂ H be the preimage of X′ under! the projection H → X. For h ∈ H(A) put

E_{Q}^{H′}(sμ, h) = ∑_{δ∈Q(F)\H′(F)} e^{⟨sμ,H_{Q}(δt)⟩} .

Then we have

(3.2.1) Z_{Y′}(L, s) ≥ ∑_{γ∈P (F)\G′(F)} e^{⟨sλ,H_{P} (γ)⟩}E_{Q}^{H′}(sμ, η(p_{γ})) .

For each ϑ ∈ T_{0}(F) the set H′ϑ is open in Qw^{−1}Q_{0}, hence H̃ = ∪_{ϑ∈T_{0}(F)}H′ϑ is an open subset of Qw^{−1}Q_{0} and there exist ϑ_{1}, …, ϑ_{m} ∈ T_{0}(F) such that H̃ = ∪_{1≤j≤m}H′ϑ_{j}. For h ∈ H(A) we define as above

EH̃ ; Q(sμ, h) = ∑_{δ∈Q(F)\H̃(F)} e^{⟨sμ,H_{Q}(δt)⟩} .

Fix a > a_{μ}. By Lemma 2.1 there is a c_{1} > 0 such that for all s ∈ (a_{μ}, a] and t ∈ T_{0}(A)

(3.2.2) E_{Q}^{H′}(sμ, t) ≥ c_{1}EH̃ ; Q(sμ, t) .

Note that for all ϑ ∈ T_{0}(F) and h ∈ H(A) one has EH̃ ; Q(sμ, ϑh) = EH̃ ; Q(sμ, h). For each η ∈ Q_{0}(F) the set H̃η is open in Qw^{−1}Q_{0}. Therefore, there are η_{1}, …, η_{n} ∈ Q_{0}(F) such that Qw^{−1}Q_{0} = ∪_{1≤k≤n}H̃η_{k}. Write η(p_{γ}) = ϑ_{γ}a_{γ}o_{γ} with ϑ_{γ} ∈ T_{0}(F), o_{γ} ∈ Ω_{0} and a_{γ} ∈ A_{M_{0}}(t_{0}), where t_{0} is as in Lemma 2.7. The set {a_{γ}^{−1}η_{j}a_{γ}o_{γ} | γ ∈ P (F)\G′(F), j = 1, … n } is relatively compact in H(A). So we can find c_{2} > 0 such that for all γ ∈ P (F)\G′(F) and j = 1, …, n we have

EH̃ ; Q(sμ, η_{j}a_{γ}o_{γ}) ≥ c_{2}EH̃ ; Q(sμ, a_{γ}o_{γ}) = c_{2}EH̃ ; Q(sμ, η(p_{γ})) .

Replacing nc_{2} by c_{2} we even get the following estimate

(3.2.3) EH̃ ; Q(sμ, η(p_{γ})) ≥ c_{2}E_{Q}^{Qw^{−1}Q_{0}}(sμ, η(p_{γ})) .

Putting (3.2.1), (3.2.2) and (3.2.3) together and using Proposition 2.5 we have

\begin{aligned}Z_{Y′}(L, s) & ≥ c_{1}c_{2} ∑_{γ∈P (F)\G′(F)} e^{⟨sλ,H_{P} (γ)⟩}E_{Q}^{Qw^{−1}Q_{0}}(sμ, η(p_{γ})) \\ & ≥ c_{3} ∑_{γ∈P (F)\G′(F)} e^{⟨s(λ+η̌(wμ))−η̌(wρ_{0}−ρ_{0}),H_{P} (γ)⟩} , \end{aligned}

with a suitable c_{3} > 0. Because G′ is dense in G there is c_{4} > 0 such that

∑_{γ∈P (F)\G′(F)} e^{⟨s(λ+η̌(wμ))−η̌(wρ_{0}−ρ_{0}),H_{P} (γ)⟩} ≥ c_{4}E_{P}^{G}(s(λ + η̌(wμ)) − η̌(wρ_{0} − ρ_{0})) .

(The argument is the same as in the proof of Prop. 2.5 b).) This shows that the abcissa of convergence of Z_{Y′}(L, s) is not less than ǎ_{μ,λ}^{w}. Because Y′ is dense in Y, the intersection Y′ ∩ Y^{∘} is not empty. By the same reasoning as above we find c_{5} > 0 such that

Z_{Y′}(L, s) ≥ c_{5} ∑_{γ∈P (F)\G′(F)} e^{⟨sλ,H_{P} (γ)⟩}E_{Q}^{Q(w_{H}^{M})^{−1}Q_{0}}(sμ, η(p_{γ})) .

Again by Proposition 2.5 b), E_{Q}^{Q(w_{H}^{M})^{−1}Q_{0}}(sμ, ·) converges only for s > a_{μ}, so we can conclude that Z_{Y′}(L, s) converges only for s > max{ǎ_{μ,λ}^{w}, a_{μ}} = a_{μ,λ}^{w} = a_{i}, and this proves the second statement.

To prove the first assertion, observe that we have just seen that the height zeta function of any open non-empty subset can be bounded from below by the height zeta function of Y^{∘}. But this series does not converge for any s > 0 if L_{μ}^{Y} ⊗ π^{∗}L_{λ} is not in the interior of the positive cone, by proposition 2.9. □

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