HEIGHT ZETA FUNCTIONS OF EQUIVARIANT COMPACTIFICATIONS OF UNIPOTENT GROUPS 5 1 0 JOSEPH SHALIKA AND YURI TSCHINKEL 2 n a J 0 Abstract. We prove Manin’s conjecture for bi-equivariant com- 1 pactifications of unipotent groups. ] T N Contents . h Introduction 1 t a 2. Nilpotent Lie algebras and unipotent groups 7 m 3. Coadjoint orbits 11 [ 4. Integral structures 15 1 5. Representations: basics 17 v 6. Universal enveloping algebra 23 9 7. Geometry 27 9 3 8. Height zeta function 29 2 9. Analytic properties of the height zeta function 32 0 . 10. Appendix: Elliptic operators 42 1 0 References 44 5 1 : v i Introduction X r Let F be a number field and X an algebraic variety over F; we write a X(F) for the set of its F-rational points. The height of an F-rational point x = (x : ... : x ) ∈ Pn(F) of a projective space is given by 0 n H(x) := max|x | , j v j v Y where the product is over the set of all valuations of F and |·| is the v standard v-adic absolute value. Let G be a linear algebraic group over F and ρ : G → PGL n+1 a projective rational representation of G. Assume that there exists a point e ∈ Pn with trivial stabilizer (under the action of ρ(G)). We are 1 2 JOSEPH SHALIKAANDYURITSCHINKEL interested in the asymptotics of N(B) := {γ ∈ G(F)|H(ρ(γ)·e) ≤ B}, B → ∞. An alternative geometric description of this problem is as follows: Con- sider the Zariski closure X ⊂ Pn of the orbit {ρ(γ)·e|γ ∈ G(F)}. Then X is an equivariant compactification of G, embedded by a G- linearized (ample) line bundle L. Choosing a particular height in the ambient projective space amounts to choosing an adelic metrization L := (L,k ·k ) of L (see Section 8 for the definitions). In this setup, v the problem is to understand (0.1) N(L,B) := {γ ∈ G(F)|H (γ) ≤ B}, B → ∞, L where H is the height defined by L. L In this paper we consider smooth projective bi-equivariant compact- ifications of a unipotent group G over F. This means that G is con- tained in X as a Zariski open subset and that the natural left and right actions of G on itself extend to left and right actions of G on X. Alternatively, one may think of X as an equivariant compactification of the homogeneous space G×G/G. The main result is the determination of the asymptotic (0.1) for ar- bitrary bi-equivariant compactifications X as above and L = −K , X the anticanonical line bundle equipped with a smooth adelic metriza- tion, proving Manin’s conjecture [12] and its refinement by Peyre [22] for this class of varieties. This generalizes the theorem for equivariant compactifications of the Heisenberg group proved in [28]. It turns out that the geometric language is more adequate for the de- scription of the asymptotic behavior. More precisely, denote by Pic(X) the Picard group of X, this is a free abelian group generated by the classesoftheirreducibleboundarycomponentsD ,α ∈ A(wewillgen- α erally identify divisors and their classes in Pic(X)). Our main result is a proof of Manin’s conjecture: Theorem 1. Let X be a smooth projective bi-equivariant compactifi- cation of G, with boundary X \G = ∪ D α∈A α a normal crossings divisor consisting of geometrically irreducible com- ponents. Then τ(−K ) N(−K ,B) = X Blog(B)b−1(1+o(1)), as B → ∞, X (b−1)! HEIGHT ZETA FUNCTIONS 3 where b = rkPic(X) = #A is the number of boundary components and τ(−K ) is the Tamagawa number defined by Peyre [22]. X We now give an outline of the proof. In Section 2 we recall some basic structural results concerning nilpotent algebras and unipotent groups. In Section 3 we discuss coadjoint orbits and their parametriza- tion and in Section 4 integral structures. In Section 5 we collect facts regarding unitary representations of unipotent groups over the adeles. In Section 6 we study the action of the universal enveloping algebra in representation spaces. All of the above material is standard and can be found in the books [7], [10] and the papers [17], [21]. In Section 7 we turn to equivariant compactifications of unipotent groups and describe the relevant geometric invariants and construc- tions. In Section 8 we introduce the height pairing H = H : Pic(X) ×G(A) → C, v C v Y generalizing the usual heights, and the height zeta function (1.1) Z(s;g) := H(s;γg)−1. γ∈G(F) X By the projectivity of X, the series converges to a function which is continuous and bounded in g and holomorphic in s for ℜ(s) contained in some cone Λ ⊂ Pic(X) . Our goal is establish its analytic prop- R erties, and in particular, to obtain a meromorphic continuation of the 1-parameter height zeta function Z(sL) = H (sL,γ)−1, L γ∈G(F) X the restriction of the multiparameter zeta function Z(s;g) to the com- plex line through L and the identity g = e ∈ G(A ). F To describe the polar set, we use the classes D as a basis of Pic(X). α In this basis, the pseudo-effective cone Λ (X) ⊂ Pic(X) consists of eff R classes (l ) ∈ Pic(X) with l ≥ 0 for all α. Let α R α −K = κ = κ D ∈ Pic(X) , X α α R α∈A X be the anticanonical class. We know (see Proposition 7.3) that κ ≥ 2, α forallα ∈ A. Conjecturally, analyticpropertiesofheight zeta functions Z(sL) depend on the location of L = (l ) ∈ Pic(X) with respect to α the anticanonical class and the cone Λ (X) (see [12], [22] and [3]). eff Precisely, define 4 JOSEPH SHALIKAANDYURITSCHINKEL • a(L) := inf{a | aL+K ∈ Λ (X)} = max (κ /l ); X eff α α α • b(L) := #{α|κ = a(L)l }; α α • C(L) := {α|κ 6= a(L)l }; α α • c(L) := l−1. α∈/C(L) α Then, conjecturally, Q c(L)τ(L) h(s) (1.2) Z(sL) = + , (s−a(L))b(L) (s−a(L))b(L)−1 where h(s) is a holomorphic function (for ℜ(s) > a(L)−δ, some δ > 0) and τ(L) is a positive real number. Given this, Tauberian theorems imply c(L)τ(L) N(L,B) = Ba(L)log(B)b(L)−1(1+o(1)), a(L)(b(L)−1)! as B → ∞, for certain constants τ(L) defined in [3]. Here we es- tablish this for L = −K , via a spectral expansion of Z(s;g) from X Equation (1.1). Thebi-equivariance ofX implies thatH isinvariantunder theaction on both sides ofa compact open subgroup K ofthe finite adeles G(A ). fin Furthermore, we assume that H is smooth for archimedean v. We v observe that Z(s;g) ∈ L2(G(F)\G(A))K and we proceed to analyze its spectral decomposition. We get a formal identity (1.3) Z(s;g) = Z (s;g), ̺ ̺ X where the summation is over all irreducible unitary representations (̺,H ) of G(A) occurring in the right regular representation of G(A) in ̺ L2(G(F)\G(A)). These are parametrized by F-rational orbits O = O ̺ under the coadjoint action of G on the dual of its Lie algebra g∗. The relevant orbits are integral - there exists a lattice in g∗(F) such that Z (s;g) = 0 unless the intersection of O with this lattice is nonempty. ̺ The pole of highest order is contributed by the trivial representation and integrality insures that this representation is “isolated”. Let ̺ be a representation as above. Then ̺ arises from some π = IndG(ψ), M where M ⊂ G is an F-rational subgroup and ψ is a certain character of M(A)(seeProposition5.5). Inparticular, forthetrivialrepresentation, HEIGHT ZETA FUNCTIONS 5 M = G and ψ is the trivial character. Further, there exists a finite set of places S = S such that dim ̺ = 1 for v ∈/ S and consequently ̺ v (1.4) Z (s;g′) = ZS(s;g′)·Z (s;g′), ̺ S where ZS(s;g′) := H (s;m g′)−1ψ(m g′)dm , v v v v v v v∈/SZM(Fv) Y (with an appropriately normalized Haar measure dm on M(F )) and v v the function Z is the projection of Z to ⊗ ̺ . S v∈S v The first key result is the explicit computation of height integrals: H (s;m g′)−1ψ(m g′)dm v v v v v v ZM(Fv) foralmostallv (seeSection9). Thishasbeendonein[5]forequivariant compactifications of additive groupsGn; thesame approach works here a as well. We regard the height integrals as geometric versions of Igusa’s integrals (see [6]). For the trivial representation and v ∈/ S, we have (1.5) q −1 H(s;g )−1dg = q−dimX D0(k ) v , ZG(Fv) v v v A⊆A A v α∈A qvsα−κα+1 −1! X Y where DA := ∩α∈ADα, DA0 := DA \∪A′)ADA′ and q is the cardinality of the residue field k at v. Restricting to the v v linethrough−K , we findthattheresulting Euler product ZS(−sK ) X X isregularizedbyaproductof(truncated)Dedekindzetafunctions, thus is holomorphic for ℜ(s) > 1, admits a meromorphic continuation to ℜ(s) > 1−δ, forsomeδ > 0, andhasanisolatedpoleoforderrk Pic(X) at s = 1, with the expected leading coefficient τ(−K ). Similarly, we X identify the poles of ZS for nontrivial representations: again, they are regularized by products of (truncated) Dedekind zeta functions and thus admit a meromorphic continuation to the same domain, with at most an isolated pole at s = 1; but the order of the pole at s = 1 is strictly smaller than rk Pic(X). Next we need to estimate dim ̺ and the local integrals for nonar- v chimedeanv ∈ S (seeSections5.7and9). Thenweturntoarchimedean places. Using integration by parts, we prove in Lemma 9.7 that for all 6 JOSEPH SHALIKAANDYURITSCHINKEL ǫ > 0 and all (left or right) G-invariant differential operators ∂ there exist constants c = c(ǫ,∂) and N = N(∂) such that (1.6) |∂H (s;g )−1| dg ≤ c·kskN, v v v v ZG(Fv) for all s with ℜ(s ) > κ −1+ǫ, for all α ∈ A. α α Letv bereal. Itisknownthat̺ admitsastandardmodel(π ,L2(Rr)), v v where 2r = dim O. More precisely, there exists an isometry j : (π ,L2(Rr)) → (̺ ,H ), v v v an analog of the Θ-distribution. Moreover, the universal enveloping algebra U(g) surjects onto the Weyl algebra of differential operators with polynomial coefficients acting on the smooth vectors C∞(Rr) ⊂ L2(Rr). In particular, we can find an operator ∆ acting as the r- dimensional harmonic oscillator r ∂2 ( −a x2), ∂x2 j j j=1 j Y with a > 0. We choose an orthonormal basis of L2(Rr) consisting of j ∆-eigenfunctions {ω˜ } (which are well known) and analyze λ H (s;g )−1ω (g )dg , v v λ v v ZG(Fv) where ω = j(ω˜ ). Using integration by parts and (1.6) we find that λ λ for all n ∈ N there exist constants c = c(n,∆) and N ∈ N such that this integral is bounded by (1.7) c·λ−n ·kskN, forswithℜ(s ) > κ −1+ǫ, forallα. Thisestimatesufficestoconclude α α that for each ̺ the function Z is holomorphic in a neighborhood of S̺ κ; indeed it will be majorized by λ−n, λ X the spectral zeta function of a compact manifold, which converges for sufficiently large n ≥ 0 (see Section 9 and the Appendix). Now the issue is to prove the convergence of the sum in (1.3). Using any element ∂ ∈ U(g) acting in H by a scalar λ(∂) 6= 0 (for example, ̺ any element in the center of U(g)) we can improve the bound (1.7) to c·λ−n1 ·λ(∂)−n2 ·kskN HEIGHT ZETA FUNCTIONS 7 (for any n ,n ∈ N and some constants c = c(n ,n ,∆,∂) and N = 1 2 1 2 N(∆,∂). However, we have to insure the uniformity of such estimates over the set of all ̺. This relies on a parametrization of coadjoint orbits. There is a finite set Σ of “packets” of coadjoint orbits, each parametrized by a locally closed subvariety Z ⊂ g∗, and for each σ a σ finite set of F-rational polynomials {P } on g∗ such that each P is σ,j σ,j invariant under the coadjoint action and nonvanishing on the stratum Z . Consequenty, the corresponding derivatives σ ∂ ∈ U(g) σ,j act in H by multiplication by the scalar ̺ λ (ℓ) = P (2πiℓ), ℓ ∈ O. ̺,j σ,j Recall that ℓ varies over a lattice; applying several times ∂ = ∂ σ j σ,j we obtain the uniform convergence of the right hand side in (1.3). Q The last technical point is to prove that both expressions (1.1) and (1.3)forZ(−sK ;g)definecontinuousfunctionsonG(F)\G(A). Then X (1.3) gives the desired meromorphic continuation of Z(−sK ;e). X The techniques described above should allow the treatment of arbi- trary height functions; here we restricted to the anticanonical height H as in the original conjecture of Manin [12], to avoid some tech- −KX nical issues with L-primitive fibrations (see [3] and [5]). Acknowledgements. The second author was partially supported by NSF grants 0739380, 0901777, and 1160859. He is very grateful to the referees for comments and suggestions that helped to improve the exposition. 2. Nilpotent Lie algebras and unipotent groups In this section we recall basic properties of nilpotent Lie algebras and unipotent groups. We work over a field F of characteristic zero. 2.1. Nilpotent algebras. Let g = (g,[,]) be an n-dimensional Lie algebra over F: an affine space over F of dimension n together with a bracket [·,·] satisfying the Jacobi identity. Denote by z the center of g g. For a subset h ⊂ g we denote by n (h) := {X ∈ g|[X,h] ⊂ h} g 8 JOSEPH SHALIKAANDYURITSCHINKEL its normalizer and by z (h) := {X ∈ g|[X,Y] = 0,∀Y ∈ h} g its centralizer. Let g ⊂ g ⊂ ... ⊂ g ⊂ g 1 2 k be a sequence of subalgebras. A weak Malcev basis through this se- quence is a basis (X ,...,X ) of g such that 1 n • for all j ∈ 1,...,k there exists ann such that g = hX ,...,X i; j j 1 nj • for all i = 1,...,n the F-vector space hX ,...,X i is a Lie subal- 1 i gebra. Assume that all g above are ideals. A strong Malcev basis through j this sequence is a weak Malcev basis such that • for all i = 1,...,n the F-vector space hX ,...,X i is an ideal. 1 i The ascending central series of g is defined as g := 0; 0 g := {x ∈ g|[x,g] ⊆ g }. j j−1 From now on we will assume that g is nilpotent, that is, there exists an n such that g = g. n Example 2.2. Some common examples are: • the Heisenberg algebra h := hX,Y,Zi, [X,Y] = Z; 3 • the upper-triangular algebra n ⊂ gl ; n n • the algebra k = hX ,X ,X ,Yi: [X ,X ] = 0,[Y,X ] = X . 4 1 2 3 i j i i−1 Lemma 2.3. If g is nilpotent then for any ascending sequence of al- gebras (resp. ideals) there exists a weak (resp. strong) Malcev basis passing through it. Proof. Indeed, for any subalgebra h ( n (h), g and for any X ∈ n (h) \ h the vector space h ⊕ FX is a subalgebra. g (cid:3) Same argument works for ideals. There is no canonical choice of a Malcev basis through a given sub- algebra. Lemma 2.4. (Kirillov’s lemma) Let g be a noncommutative nilpotent Lie algebra with 1-dimensional center z (g) = hZi. Then there exist g X,Y ∈ g such that • [X,Y] = Z; • g = z (Y)⊕FX. g HEIGHT ZETA FUNCTIONS 9 Proof. Choose some Y ∈ g \g . Then g := z (Y) is a subalgebra of 2 1 0 g codimension one and there is an X in its complement as required. (cid:3) Notation 2.5. We refer to the quadruple (Z,Y,X,g ) in Lemma 2.4 as 0 a reducing quadruple. 2.6. Polarizations. Denote by g∗ the dual Lie algebra. Each ℓ ∈ g∗ determines a skew-symmetric bilinear form B : g×g → F ℓ (X,Y) → ℓ([X,Y]). For any subalgebra h ⊂ g denote by r (h) := h∩h⊥ℓ = {h ∈ h|ℓ([h,h′]) = 0, ∀h′ ∈ h} ℓ its radical with respect to B . Clearly, the maximum dimension of an ℓ isotropic subspace in g is 1 d = dimr + (dimh−dimr ). ℓ ℓ 2 Definition 2.7. A subalgebra m ⊂ g is called polarizing for ℓ if ℓ • m is isotropic for B , that is, B (m,m′) = 0 for all m,m′ ∈ m ; ℓ ℓ ℓ ℓ • dimm is the maximal possible dimension d for isotropic sub- ℓ spaces. Such subalgebras exist, and all have the same dimen- sion. Example 2.8. For the Heisenberg algebra h and any ℓ with ℓ(Z) 6= 0 3 a polarizing subalgebra is the ideal m = hZ,Yi. ℓ Remark 2.9. A polarizing algebra m is not necessarily an ideal. An ℓ ℓ ∈ g∗ can have many polarizing subalgebras. In general, there does not exist a finite set of subalgebras such that for each ℓ ∈ g∗ one of the subalgebras in this set is polarizing for ℓ. Acanonicalconstructionofapolarizingalgebra(byVergne[31])goes as follows: fix a strong Malcev basis (X ,...,X ) for g. Put 1 n n m := r (g ), ℓ ℓ j j=1 X where g := hX ,...,X i and r (g ) is the radical of g with respect to j 1 j ℓ j j B . ℓ Alternatively, a polarizing subalgebra may be constructed inductively: Case 1. If z := z ∩Ker(ℓ) 6= 0 ℓ g 10 JOSEPH SHALIKAANDYURITSCHINKEL consider the projection pr : g → g := g/z 0 ℓ andwriteℓ fortheinducedlinearformong . Ifm ⊂ g isapolarizing 0 0 ℓ0 0 algebra for ℓ the preimage pr−1(m ) is a polarizing algebra for ℓ. 0 ℓ0 Case 2. Otherwise, z(g) = hZi and ℓ(Z) 6= 0. Then there exists a Y ∈ g \g such that codimz (Y) = 1 (by Lemma 2.4). Let ℓ be the 2 1 g Y restriction of ℓ to z (Y) and m a polarizing algebra for ℓ in z (Y). g Y Y g Then m = m . ℓ Y Proposition 2.10. Let Z ⊂ g∗ be an algebraic variety, defined over F. There exists a Zariski open subset Z0 ⊂ Z, a positive integer k ≤ dimg and an F-morphism pol : Z0 → Gr(k,g) such that for every point ℓ in Z0 the image pol(ℓ) in the Grassmannian of k-dimensional subspaces in g corresponds to a polarizing subalgebra for ℓ. Proof. Consider g∗ over the function field of Z and apply Vergne’s construction to the generic point. Alternatively, consider the subvariety of all subalgebras m ⊂ g over the function field F(Z) of dimension k such that ℓ([m,m]) = 0, with ℓ ∈ g∗(F(Z)). Take the maximal k such that this variety has an F(Z)- rational point. This point defines an F(Z)-rational point in Gr(k,g). Specializing, we get polarizations on some open subset Z0 ⊂ Z. (cid:3) 2.11. Unipotent groups. Let V be a finite dimensional vector space over F and N ⊂ GL(V) the subgroup of all upper-triangular unipo- tent matrices. Denote by n the F-vector space of all upper-triangular nilpotent matrices. The (standard) maps exp : n → N log : N → n are biregular F-morphisms (polynomial maps) between algebraic vari- eties. Let G be a (connected) unipotent linear algebraic group defined over F. Then there exists an F-rational representation ρ : G → GL(V), F for some V, realizing G as a closed subgroup of N. We fix this repre- sentation. Then g := log(G) ⊂ n