Height zeta functions of equivariant compactifications of semi-direct products of algebraic groups
aa r X i v : . [ m a t h . N T ] A p r HEIGHT ZETA FUNCTIONS OF EQUIVARIANTCOMPACTIFICATIONS OF SEMI-DIRECT PRODUCTS OFALGEBRAIC GROUPS
SHO TANIMOTO AND YURI TSCHINKEL
Abstract.
We apply the theory of height zeta functions to study the asymp-totic distribution of rational points of bounded height on projective equivariantcompactifications of semi-direct products.
Contents
Introduction 11. Geometry 22. Height zeta functions 63. Harmonic analysis 104. The projective plane 175. Geometrization 23References 36
Introduction
Let X be a smooth projective variety over a number field F and L a very ampleline bundle on X . An adelic metrization L = ( L, k·k ) on L induces a height function H L : X ( F ) → R > , let N ( X ◦ , L , B ) := { x ∈ X ◦ ( F ) | H L ( x ) ≤ B } , X ◦ ⊂ X, be the associated counting function for a subvariety X ◦ . Manin’s program, initiatedin [FMT89] and significantly developed over the last 10 years, relates the asymp-totic of the counting function N ( X ◦ , L , B ), as B → ∞ , for a suitable Zariski open X ◦ ⊂ X , to global geometric invariants of the underlying variety X . By generalprinciples of diophantine geometry, such a connection can be expected for varietieswith sufficiently positive anticanonical line bundle − K X , e.g., for Fano varieties.Manin’s conjecture asserts that(0.1) N ( X ◦ , −K X , B ) = c · B log( B ) r − , where r is the rank of the Picard group Pic( X ) of X , at least over a finite extensionof the ground field. The constant c admits a conceptual interpretation, its mainingredient is a Tamagawa-type number introduced by Peyre [Pey95]. Date : August 29, 2018.
For recent surveys highlighting different aspects of this program, see, e.g., [Tsc09],[CL10], [Bro07], [Bro09].Several approaches to this problem have evolved: • passage to (universal) torsors combined with lattice point counts; • variants of the circle method; • ergodic theory and mixing; • height zeta functions and spectral theory on adelic groups.The universal torsor approach has been particularly successful in the treatmentof del Pezzo surfaces, especially the singular ones. This method works best over Q ;applying it to surfaces over more general number fields often presents insurmount-able difficulties, see, e.g., [dlBF04]. Here we will explain the basic principles of themethod of height zeta functions of equivariant compactifications of linear algebraicgroups and apply it to semi-direct products; this method is insensitive to the groundfield. The spectral expansion of the height zeta function involves 1-dimensional aswell as infinite-dimensional representations, see Section 3 for details on the spectraltheory. We show that the main term appearing in the spectral analysis, namely, theterm corresponding to 1-dimensional representations, matches precisely the predic-tions of Manin’s conjecture, i.e., has the form (0.1). The analogous result for theuniversal torsor approach can be found in [Pey98] and for the circle method appliedto universal torsors in [Pey01].Furthermore, using the tools developed in Section 3, we provide new examplesof rational surfaces satisfying Manin’s conjecture. Acknowledgments.
The second author was partially supported by NSF grantsDMS-0739380 and 0901777. 1.
Geometry
In this section, we collect some general geometric facts concerning equivariantcompactifications of solvable linear algebraic groups. Here we work over an alge-braically closed field of characteristic 0.Let G be a linear algebraic group. In dimension 1, the only examples are theadditive group G a and the multiplicative group G m . Let X ( G ) × := Hom( G, G m ) , the group of algebraic characters of G . In the situations we consider, this is atorsion-free Z -module of finite rank (see [Bor91, Section 8], for conditions insuringthis property).Let X be a projective equivariant compactification of G . After applying equi-variant resolution of singularities, if necessary, we may assume that X is smoothand that the boundary X \ G = D = ∪ ι D ι , is a divisor with normal crossings. Here D ι are irreducible components of D . LetPic G ( X ) be the group of equivalence classes of G -linearized line bundles on X .Generally, we will identify divisors, associated line bundles, and their classes inPic( X ), resp. Pic G ( X ). Proposition 1.1.
Let X be a smooth and proper equivariant compactification of aconnected solvable linear algebraic group G . Then, EIGHT ZETA FUNCTIONS 3 (1) we have an exact sequence → X ( G ) × → Pic( X ) G → Pic( X ) → , (2) Pic G ( X ) = ⊕ ι ∈I Z D ι , and (3) the closed cone of pseudo-effective divisors of X is spanned by the boundarycomponents: Λ eff ( X ) = X ι ∈I R ≥ D ι . Proof.
The first claim follows from the proof of [MFK94, Proposition 1.5]. Thecrucial point is to show that the Picard group of G is trivial. As an algebraicvariety, a connected solvable group is a product of an algebraic torus and an affinespace. The second assertion holds since every finite-dimensional representation ofa solvable group has a fixed vector. For the last statement, see [HT99, Theorem2.5]. (cid:3) Proposition 1.2.
Let X be a smooth and proper equivariant compactification forthe left action of a linear algebraic group. Then the right invariant top degreedifferential form ω satisfies − div( ω ) = X ι ∈I d ι D ι , where d ι > . The same result holds for the right action and the left invariant form.Proof. See [HT99, Theorem 2.7]. (cid:3)
Proposition 1.3.
Let X be a smooth and proper equivariant compactification ofa connected linear algebraic group. Let f : X → Y be a birational morphism to anormal projective variety Y . Then Y is an equivariant compactification of G suchthat the contraction map f is a G -morphism.Proof. Choose an embedding
Y ֒ → P N , and let L be the pull back of O (1) on X .Since Y is normal, Zariski’s main theorem implies that the image of the completelinear series | L | is isomorphic to Y . This linear series carries a G -linearization[MFK94, Corollary 1.6]. Now we apply the same argument as in the proof of[HT99, Corollary 2.4] to this linear series, and our assertion follows. (cid:3) The simplest solvable groups are G a and G m , as well as their products. Newexamples arise as semi-direct products. For example, let ϕ d : G m → G m = GL ,a a d and put G d := G a ⋊ ϕ d G m , where the group law is given by( x, a ) · ( y, b ) = ( x + ϕ d ( a ) y, ab ) . It is easy to see that G d ≃ G − d .One of the central themes in birational geometry is the problem of classification ofalgebraic varieties. The classification of G -varieties, i.e., varieties with G -actions, isalready a formidable task. The theory of toric varieties, i.e., equivariant compactifi-cations of G = G nm , is very rich, and provides a testing ground for many conjectures SHO TANIMOTO AND YURI TSCHINKEL in algebraic and arithmetic geometry. See [HT99] for first steps towards a classifica-tion of equivariant compactifications of G = G na , as well as [Sha09], [AS09], [Arz10]for further results in this direction.Much less is known concerning equivariant compactifications of other solvablegroups; indeed, classifying equivariant compactifications of G d is already an inter-esting open question. We now collect several results illustrating specific phenomenaconnected with noncommutativity of G d and with the necessity to distinguish ac-tions on the left, on the right, or on both sides. These play a role in the analysis ofheight zeta functions in following sections. First of all, we have Lemma 1.4.
Let X be a biequivariant compactification of a semi-direct product G ⋊ H of linear algebraic groups. Then X is a one-sided (left- or right-) equivariantcompactification of G × H .Proof. Fix one section s : H → G ⋊ H . Define a left action by( g, h ) · x = g · x · s ( h ) − , for any g ∈ G , h ∈ H , and x ∈ X . (cid:3) In particular, there is no need to invoke noncommutative harmonic analysis inthe treatment of height zeta functions of biequivariant compactifications of generalsolvable groups since such groups are semi-direct products of tori with unipotentgroups and the lemma reduces the problem to a one-sided action of the direct product. Height zeta functions of direct products of additive groups and tori canbe treated by combining the methods of [BT98] and [BT96a] with [CLT02], seeTheorem 2.1. However, Manin’s conjectures are still open for one-sided actions ofunipotent groups, even for the Heisenberg group.The next observation is that the projective plane P is an equivariant compact-ification of G d , for any d . Indeed, the embedding( x, a ) ( a : x : 1) ∈ P defines a left-sided equivariant compactification, with boundary a union of two lines.In contrast, we have Proposition 1.5. If d = 1 , , or − , then P is not a biequivariant compactificationof G d .Proof. Assume otherwise. Let D and D be the two irreducible boundary com-ponents. Since O ( K P ) ∼ = O ( − D and D are lines orone of them is a line and the other a conic. Let ω be a right invariant top degreedifferential form. Then ω/ϕ d ( a ) is a left invariant differential form. If one of D and D is a conic, then the divisor of ω takes the form − div( ω ) = − div( ω/ϕ d ( a )) = D + D , but this is a contradiction. If D and D are lines, then without loss of generality,we can assume that − div( ω ) = 2 D + D and − div( ω/ϕ d ( a )) = D + 2 D . However, div( a ) is a multiple of D − D , which is also a contradiction. (cid:3) Combining this result with Proposition 1.3, we conclude that a del Pezzo surfaceis not a biequivariant compactification of G d , for d = 1 , , or , −
1. Another sampleresult in this direction is:
EIGHT ZETA FUNCTIONS 5
Proposition 1.6.
Let S be the singular quartic del Pezzo surface of type A + A defined by x + x x + x x = x x − x = 0 Then S is a one-sided equivariant compactification of G , but not a biequivariantcompactification of G d if d = 0 .Proof. For the first assertion, see [DL10, Section 5]. Assume that S is a biequivari-ant compactification of G d . Let π : e S → S be its minimal desingularization. Then e S is also a biequivariant compactification of G d . It has three ( − L , L ,and L , which are the strict transforms of { x = x = x = 0 } , { x + x = x = x = 0 } , and { x = x = x = 0 } , respectively, and has four ( − R , R , R , and R . The nonzero intersectionnumbers are given by: L .R = L .R = R .R = R .R = R .L = L .R = 1 . Since the cone of curves is generated by the components of the boundary, thesenegative curves must be in the boundary because each generates an extremal ray.Since the Picard group of e S has rank six, the number of boundary components isseven. Thus, the boundary is equal to the union of these negative curves.Let f : e S → P be the birational morphism which contracts L , L , L , R ,and R . This induces a biequivariant compactification on P . The birational map f ◦ π − : S P is given by S ∋ ( x : x : x : x : x ) ( x : x : x ) ∈ P .The images of R and R are { y = 0 } and { y = 0 } and we denote them by D and D , respectively. The images of L and L are (0 : 0 : 1) and (0 : 1 : − P must fix (0 : 0 : 1), (0 : 1 : − D ∩ D = (0 : 1 : 0). Thus, the group action must fix the line D , and thisfact implies that all left and right invariant vector fields vanish along D . It followsthat − div( ω ) = − div( ω/ϕ d ( a )) = 2 D + D , which contradicts d = 0. (cid:3) Example 1.7.
Let l ≥ d ≥
0. The Hirzebruch surface F l = P P (( O ⊕ O ( l )) ∗ ) is abiequivariant compactification of G d . Indeed, we may take the embedding G d ֒ → F l ( x, a ) (( a : 1) , [1 ⊕ xσ l ]) , where σ is a section of the line bundle O (1) on P such thatdiv( σ ) = (1 : 0) . Let π : F l → P be the P -fibration. The right action is given by(( x : x ) , [ y ⊕ y σ l ]) (( ax : x ) , [ y ⊕ ( y + ( x /x ) d xy ) σ l ]) , on π − ( U = P \ { (1 : 0) } ) and(( x : x ) , [ y ⊕ y σ l ]) (( ax : x ) , [ a l y ⊕ ( y + ( x /x ) l − d xy ) σ l ]) , on U = π − ( P \ { (0 : 1) } ). Similarly, one defines the left action. The boundaryconsists of three components: two fibers f = π − ((0 : 1)), f = π − ((0 : 1)) andthe special section D characterized by D = − l . SHO TANIMOTO AND YURI TSCHINKEL
Example 1.8.
Consider the right actions in Examples 1.7. When l > d >
0, theseactions fix the fiber f and act multiplicatively, i.e., with two fixed points, on thefiber f . Let X be the blow up of two points (or more) on f and of one fixed point P on f \ D . Then X is an equivariant compactification of G d which is neither atoric variety nor a G a -variety. Indeed, there are no equivariant compactificationsof G m on F l fixing f , so X cannot be toric. Also, if X were a G a -variety, wewould obtain an induced G a -action on F l fixing f and P . However, the boundaryconsists of two irreducible components and must contain f , D , and P because D is a negative curve. This is a contradiction.For l = 2 and d = 1, blowing up two points on f we obtain a quintic del Pezzosurface with an A singularity. Manin’s conjecture for this surface is proved in[Der07].In Section 5, we prove Manin’s conjecture for X with l ≥ Height zeta functions
Let F be a number field, o F its ring of integers, and Val F the set of equivalenceclasses of valuations of F . For v ∈ Val F let F v be the completions of F with respectto v , for nonarchimedean v , let o v be the corresponding ring of integers and m v themaximal ideal. Let A = A F be the adele ring of F .Let X be a smooth and projective right-sided equivariant compactification of asplit connected solvable linear algebraic group G over F , i.e., the toric part T of G is isomorphic to G nm . Moreover, we assume that the boundary D = ∪ ι ∈I D ι consistsof geometrically irreducible components meeting transversely. We are interested inthe asymptotic distribution of rational points of bounded height on X ◦ = G ⊂ X ,with respect to adelically metrized ample line bundles L = ( L, ( k · k A )) on X . Wenow recall the method of height zeta functions; see [Tsc09, Section 6] for moredetails and examples. Step 1.
Define an adelic height pairing H : Pic G ( X ) C × G ( A F ) → C , whose restriction to H : Pic G ( X ) × G ( F ) → R ≥ , descends to a height system on Pic( X ) (see [Pey98, Definition 2.5.2]). This meansthat the restriction of H to an L ∈ Pic G ( X ) defines a Weil height corresponding tosome adelic metrization of L ∈ Pic G ( X ), and that it does not depend on the choiceof a G -linearization on L . Such a pairing appeared in [BT95] in the context of toricvarieties, the extension to general solvable groups is straightforward.Concretely, by Proposition 1.1, we know that Pic G ( X ) is generated by boundarycomponents D ι , for ι ∈ I . The v -adic analytic manifold X ( F v ) admits a “partitionof unity”, i.e., a decomposition into charts X I,v , labeled by I ⊆ I , such that ineach chart the local height function takes the form H v ( s , x v ) = φ ( x v ) · Y ι ∈ I | x ι,v | s ι v , where for each ι ∈ I , x ι is the local coordinate of D ι in this chart, s = X ι ∈I s ι D ι , EIGHT ZETA FUNCTIONS 7 and φ is a bounded function, equal to 1 for almost all v . Note that, locally, theheight function H ι,v ( x v ) := | x ι,v | v is simply the v -adic distance to the boundary component D ι . To visualize X I,v (foralmost all v ) consider the partition induced by X ( F v ) = X ( o v ) ρ −→ ⊔ I ⊂I X ◦ I ( F q ) , where X I := ∪ ι ∈ I D I , X ◦ I := X I \ ∪ I ′ ) I X I ′ , is the stratification of the boundary and ρ is the reduction map; by convention X ∅ = G . Then X I,v is the preimage of X ◦ I ( F q ) in X ( F v ), and in particular, X ∅ ,v = G ( o v ), for almost all v .Since the action of G lifts to integral models of G , X , and L , the nonarchimedeanlocal height pairings are invariant with respect to a compact subgroup K v ⊂ G ( F v ),which is G ( o v ), for almost all v . Step 2.
The height zeta function Z ( s , g ) := X γ ∈ G ( F ) H ( s , γg ) − , converges absolutely to a holomorphic function, for ℜ ( s ) sufficiently large, anddefines a continuous function in L ( G ( F ) \ G ( A F )) ∩ L ( G ( F ) \ G ( A F )). Formally,we have the spectral expansion(2.1) Z ( s , g ) = X π Z π ( s , g ) , where the “sum” is over irreducible unitary representations occurring in the rightregular representation of G ( A F ) in L ( G ( F ) \ G ( A F )). The invariance of the globalheight pairing under the action of a compact subgroup K ⊂ G ( A F ), on the side ofthe action, insures that Z π are in L ( G ( F ) \ G ( A F )) K . Step 3.
Ideally, we would like to obtain a meromorphic continuation of Z to atube domain T Ω = Ω + i Pic( X ) R ⊂ Pic( X ) C , where Ω ⊂ Pic( X ) R is an open neighborhood of the anticanonical class − K X . It isexpected that Z is holomorphic for ℜ ( s ) ∈ − K X + Λ ◦ eff ( X )and that the polar set of the shifted height zeta function Z ( s − K X , g ) is the sameas that of(2.2) X Λ eff ( X ) ( s ) := Z Λ ∗ eff ( X ) e −h s , y i d y, the Laplace transform of the set-theoretic characteristic function of the dual coneΛ eff ( X ) ∗ ⊂ Pic( X ) ∗ R . Here the Lebesgue measure d y is normalized by the duallattice Pic( X ) ∗ ⊂ Pic( X ) ∗ R . In particular, for κ = − K X = X ι κ ι D ι , SHO TANIMOTO AND YURI TSCHINKEL the restriction of the height zeta function Z ( s , id) to the one-parameter zeta function Z ( sκ, id) should be holomorphic for ℜ ( s ) >
1, admit a meromorphic continuationto ℜ ( s ) > − ǫ , for some ǫ >
0, with a unique pole at s = 1, of order r = rk Pic( X ).Furthermore, is is desirable to have some growth estimates in vertical strips. Inthis case, a Tauberian theorem implies Manin’s conjecture (0.1) for the countingfunction; the quality of the error term depends on the growth rate in vertical strips.Finally, the leading constant at the pole of Z ( sκ, id) is essentially the Tamagawa-type number defined by Peyre. We will refer to this by saying that the height zetafunction Z satisfies Manin’s conjecture; a precise definition of this class of functionscan be found in [CLT01, Section 3.1].This strategy has worked well and lead to a proof of Manin’s conjecture for thefollowing varieties: • toric varieties [BT95], [BT98], [BT96a]; • equivariant compactifications of additive groups G na [CLT02]; • equivariant compactifications of unipotent groups [ST04], [ST]; • wonderful compactifications of semi-simple groups of adjoint type [STBT07].Moreover, applications of Langlands’ theory of Eisenstein series allowed to proveManin’s conjecture for flag varieties [FMT89], their twisted products [Str01], andhorospherical varieties [ST99], [CLT01].The analysis of the spectral expansion (2.1) is easier when every automorphicrepresentation π is 1-dimensional, i.e., when G is abelian: G = G na or G = T , analgebraic torus. In these cases, (2.1) is simply the Fourier expansion of the heightzeta function and we have, at least formally,(2.3) Z ( s , id) = Z b H ( s , χ )d χ, where(2.4) ˆ H ( s , χ ) = Z G ( A F ) H ( s , g ) − ¯ χ ( g )d g, is the Fourier transform of the height function, χ is a character of G ( F ) \ G ( A F ), andd χ an appropriate measure on the space of automorphic characters. For G = G na ,the space of automorphic characters is G ( F ) itself, for G an algebraic torus it is(noncanonically) X ( G ) × R × U G , where U G is a discrete group.The v -adic integration technique developed by Denef and Loeser (see [DL98],[DL99], and [DL01]) allows to compute local Fourier transforms of height functions,in particular, for the trivial character χ = 1 and almost all v we obtain b H v ( s ,
1) = Z G ( F v ) H ( s , g ) − d g = τ v ( G ) − X I ⊂I X ◦ I ( F q ) q dim( X ) Y ι ∈ I q − q s ι − κ ι +1 − ! , where X I are strata of the stratification described in Step 1 and τ v ( G ) is the localTamagawa number of G , τ v ( G ) = G ( F q ) q dim( G ) . Such height integrals are geometric versions of Igusa’s integrals; a comprehensivetheory in the analytic and adelic setting can be found in [CLT10].The computation of Fourier transforms at nontrivial characters requires a finerpartition of X ( F v ) which takes into account possible zeroes of the phase of the EIGHT ZETA FUNCTIONS 9 character in G ( F v ); see [CLT02, Section 10] for the the additive case and [BT95,Section 2] for the toric case. The result is that in the neighborhood of κ = X ι κ ι D ι ∈ Pic G ( X ) , the Fourier transform is regularized as follows b H ( s , χ ) = (Q v / ∈ S ( χ ) Q ι ∈I ( χ ) ζ F,v ( s ι − κ ι + 1) Q v ∈ S ( χ ) φ v ( s , χ ) G = G na , Q v / ∈ S ( χ ) Q ι ∈I L F,v ( s ι − κ ι + 1 + im ( χ ) , χ u ) Q v ∈ S ( χ ) φ v ( s , χ ) G = T, where • I ( χ ) ( I ; • S ( χ ) is a finite set of places, which, in general, depends on χ ; • ζ F,v is a local factor of the Dedekind zeta function of F and L F,v a localfactor of a Hecke L -function; • m ( χ ) is the “coordinate” of the automorphic character χ of G = T under theembedding X ( G ) × R ֒ → Pic G ( X ) R in the exact sequence (1) in Proposition 1.1and χ u is the “discrete” component of χ ; • and φ v ( s , χ ) is a function which is holomorphic and bounded.In particular, each b H ( s , χ ) admits a meromorphic continuation as desired and wecan control the poles of each term. Moreover, at archimedean places we may useintegration by parts with respect to vector fields in the universal enveloping algebraof the corresponding real of complex group to derive bounds in terms of the “phase”of the occurring oscillatory integrals, i.e., in terms of “coordinates” of χ .So far, we have not used the fact that X is an equivariant compactification of G .Only at this stage do we see that the K -invariance of the height is an important,in fact, crucial, property that allows to establish uniform convergence of the rightside of the expansion (2.1); it insures that b H ( s , χ ) = 0 , for all χ which are nontrivial on K . For G = G na this means that the trivial repre-sentation is isolated and that the integral on the right side of Equation (2.3) is infact a sum over a lattice of integral points in G ( F ). Note that Manin’s conjecture fails for nonequivariant compactifications of the affine space, there are counterex-amples already in dimension three [BT96b]. The analytic method described abovefails precisely because we cannot insure the convergence on the Fourier expansion.A similar effect occurs in the noncommutative setting; one-sided actions do not guarantee bi- K -invariance of the height, in contrast with the abelian case. Ana-lytically, this translates into subtle convergence issues of the spectral expansion, inparticular, for infinite-dimensional representation. Theorem 2.1.
Let G be an extension of an algebraic torus T by a unipotent group N such that [ G, G ] = N over a number field F . Let X be an equivariant compacti-fication of G over F and Z ( s , g ) = X γ ∈ G ( F ) H ( s , γg ) − , the height zeta function with respect to an adelic height pairing as in Step 1. Let Z ( s , g ) = Z Z χ ( s , g ) d χ, be the integral over all 1-dimensional automorphic representations of G ( A F ) occur-ring in the spectral expansion (2.1) . Then Z satisfies Manin’s conjecture.Proof. Let 1 → N → G → T → , the defining extension. One-dimensional automorphic representations of G ( A F ) areprecisely those which are trivial on N ( A F ), i.e., these are automorphic charactersof T . The K -invariance of the height (on one side) insures that only unramifiedcharacters, i.e., K T -invariant characters contribute to the spectral expansion of Z .Let M = X ( G ) × be the group of algebraic characters. We have Z ( s , id) = Z M R ×U T Z G ( F ) \ G ( A F ) Z ( s , g ) ¯ χ ( g ) d g d χ = Z M R ×U T Z G ( A F ) H ( s , g ) − ¯ χ ( g ) d g d χ = Z M R F ( s + im ( χ )) d m, where F ( s ) := X χ ∈U T b H ( s , χ u ) . Computations of local Fourier transforms explained above show that F can beregularized as follows: F ( s ) = Y ι ∈I ζ F ( s ι − κ ι + 1) · F ∞ ( s ) , where F ∞ is holomorphic for ℜ ( s ι ) − κ ι > − ǫ , for some ǫ >
0, with growth controlin vertical strips. Now we have placed ourselves into the situation considered in[CLT01, Section 3]: Theorem 3.1.14 establishes analytic properties of integrals Z M R Q ι ∈I ( s ι − κ ι + im ι ) · F ∞ ( s + im ) d m, where the image of ι : M R ֒ → R I intersects the simplicial cone R I≥ only in theorigin. The main result is that the analytic properties of such integrals match thoseof the X -function (2.2) of the image cone under the projection0 / / M R ι / / R I π / / R I− dim( M ) / / / / X ( G ) × R / / Pic G ( X ) R / / Pic( X ) R / / R I≥ under π isprecisely Λ eff ( X ) ⊂ Pic( X ) R . (cid:3) Harmonic analysis
In this section we study the local and adelic representation theory of G := G a ⋊ ϕ G m , EIGHT ZETA FUNCTIONS 11 an extension of T := G m by N := G a via a homomorphism ϕ : G m → GL . Thegroup law given by ( x, a ) · ( y, b ) = ( x + ϕ ( a ) y, ab ) . We fix the standard Haar measuresd x = Y v d x v and d a × = Y v d a × v , on N ( A F ) and T ( A F ). Note that G ( A F ) is not unimodular; d g := d x d a × is a rightinvariant measure on G ( A F ) and d g/ϕ ( a ) is a left invariant measure.Let ̺ be the right regular unitary representation of G ( A F ) on the Hilbert space: H := L ( G ( F ) \ G ( A F ) , d g ) . We now discuss the decomposition of H into irreducible representations. Let ψ = Y v ψ v : A F → S , be the standard automorphic character and ψ n the character defined by x → ψ ( nx ) , for n ∈ F × . Let W := ker( ϕ : F × → F × ) , and π n := Ind G ( A F ) N ( A F ) × W ( ψ n ) , for n ∈ F × . The following proposition we learned from J. Shalika [Sha]. Proposition 3.1.
Irreducible automorphic representations, i.e., irreducible unitaryrepresentations occurring in H = L ( G ( F ) \ G ( A F )) , are parametrized as follows: H = L ( T ( F ) \ T ( A F )) ⊕ dM n ∈ ( F × /ϕ ( F × )) π n , Remark 3.2.
Up to unitary equivalence, the representation π n does not dependon the choice of a representative n ∈ F × /ϕ ( F × ). Proof.
Define H := { φ ∈ H | φ (( x, g ) = φ ( g ) } , and let H be the orthogonal complement of H . It is straightforward to prove that H ∼ = L ( T ( F ) \ T ( A F )) . The following two lemmas prove that H ∼ = dM n ∈ F × /ϕ ( F × ) π n . (cid:3) Lemma 3.3.
For any φ ∈ L ( G ( F ) \ G ( A F )) ∩ H , the projection of φ onto H isgiven by φ ( g ) := Z N ( F ) \ N ( A F ) φ (( x, g ) dx. Proof.
It is easy to check that φ ∈ H . Also, for any φ ′ ∈ H , we have Z G ( F ) \ G ( A F ) ( φ − φ ) φ ′ d g = 0 . (cid:3) Lemma 3.4.
We have H ∼ = dM n ∈ F × /ϕ ( F × ) π n . Proof.
First we note that the underlying Hilbert space of π n is L ( W \ T ( A F )), andthat the group action is given by( x, a ) · f ( b ) = ψ n ( ϕ ( b ) x ) f ( ab ) , where f is a square-integrable function on T ( A F ). For φ ∈ C ∞ c ( G ( F ) \ G ( A F )) ∩H ,define f n,φ ( a ) := Z N ( F ) \ N ( A F ) φ ( x, a ) ψ n ( x ) d x. Then, k φ k L = Z T ( F ) \ T ( A F ) Z N ( F ) \ N ( A F ) | φ ( x, a )) | d x d a × = Z T ( F ) \ T ( A F ) X α ∈ F (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z N ( F ) \ N ( A F ) φ ( x, a ) ψ ( αx )d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d a × = Z T ( F ) \ T ( A F ) X α ∈ F × (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z N ( F ) \ N ( A F ) φ ( x, a ) ψ ( αx )d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d a × = Z T ( F ) \ T ( A F ) X α ∈ F × X n ∈ F × /ϕ ( F × ) W (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z N ( F ) \ N ( A F ) φ ( x, a ) ψ ( nϕ ( α ) x )d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d a × = Z T ( F ) \ T ( A F ) X α ∈ F × X n ∈ F × /ϕ ( F × ) W (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z N ( F ) \ N ( A F ) φ ( x, αa ) ψ ( nx )d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d a × = X n ∈ F × /ϕ ( F × ) W Z T ( A F ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z N ( F ) \ N ( A F ) φ ( x, a ) ψ n ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d a × = X n ∈ F × /ϕ ( F × ) k f n,φ k L . The second equality is the Plancherel theorem for N ( F ) \ N ( A F ). Third equalityfollows from the previous lemma. The fourth equality follows from the left G ( F )-invariance of φ . Thus, we obtain an unitary operator: I : H → dM n ∈ F × /ϕ ( F × ) π n . Compatibility with the group action is straightforward, so I is actually a morphismof unitary representations. We construct the inverse map of I explicitly. For f ∈ C ∞ c ( W \ T ( A F )), define φ n,f ( x, a ) := 1 W X α ∈ F × ψ n ( ϕ ( α ) x ) f ( αa ) . EIGHT ZETA FUNCTIONS 13
The orthogonality of characters implies that R N ( F ) \ N ( A F ) φ n,f ( x, a ) · φ n,f ( x, a ) d x = R N ( F ) \ N ( A F ) ( P α ∈ F × ψ n ( ϕ ( α ) x ) f ( αa )) · ( P α ∈ F × ψ n ( ϕ ( α ) x ) f ( αa )) d x = P α ∈ F × | f ( αa ) | . Substituting, we obtain k φ n,f k = Z T ( F ) \ T ( A F ) Z N ( F ) \ N ( A F ) | φ n,f ( x, a ) | d x d a × = 1 W Z T ( F ) \ T ( A F ) X α ∈ F × | f ( αa ) | d a × = k f k n . Lemma 3.3 implies that that φ f ∈ H and we obtain a morphismΘ : dM n ∈ F × /ϕ ( F × ) π n → H . Now we only need to check that Θ I = id and I Θ = id . The first follows from thePoisson formula: For any φ ∈ C ∞ c ( G ( F ) \ G ( A F )) ∩ H ,Θ Iφ = X n ∈ F × /ϕ ( F × ) W X α ∈ F × ψ n ( ϕ ( α ) x ) Z N ( F ) \ N ( A F ) φ ( y, αa ) ψ n ( y )d y = X n ∈ F × /ϕ ( F × ) W X α ∈ F × Z N ( F ) \ N ( A F ) φ ( ϕ ( α ) y, αa ) ψ n ( ϕ ( α )( y − x ))d y = X n ∈ F × /ϕ ( F × ) W X α ∈ F × Z N ( F ) \ N ( A F ) φ (( y + x, a )) ψ n ( ϕ ( α ) y )d y = X α ∈ F Z N ( F ) \ N ( A F ) φ (( y, x, a )) ψ ( αy )d y = φ ( x, a ) . The other identity, I Θ = id is checked by a similar computation. (cid:3) To simplify notation, we now restrict to F = Q . For our applications in Sec-tions 4 and 5, we need to know an explicit orthonormal basis for the unique infinite-dimensional representation π = L ( A × Q ) of G = G . For any n ≥
1, define compactsubgroups of G ( Z p ) G ( p n Z p ) := { ( x, a ) | x ∈ p n Z p , a ∈ p n Z p } . Let v p : Q p → Z be the discrete valuation on Q p . Lemma 3.5.
Let K p = G ( p n Z p ) . • When n = 0 , an orthonormal basis for L ( Q × p ) K p is given by { p j Z × p | j ≥ } . • When n ≥ , an orthonormal basis for L ( Q × p ) K p is given by { λ p ( · /p j ) p j Z × p | j ≥ − n, λ p ∈ M p } , where M p is the set of multiplicative characters on Z × p / (1 + p n Z p ) . Moreover, let K fin = Q p K p where K p = G ( p n p Z p ) and n p = 0 for almost all p .Let S be the set of primes with n p = 0 and N = Q p p n p . Then an orthonormalbasis for L ( A × Q , fin ) K fin is given by {⊗ p ∈ S λ p ( a p · p − v p ( a p ) ) mN Z × p ( a p ) ⊗ ′ p/ ∈ S m Z × p ( a p ) | m ∈ N , λ p ∈ M p } . Proof.
For the first assertion, let f ∈ L ( Q × p ) K p where K p = G ( Z p ). Since it is K p -invariant, we have f ( b p · a p ) = f ( a p ) , for any b ∈ Z × p . Hence f takes the form of f = ∞ X j = −∞ c j p j Z × p where c j = f ( p j ) and P ∞ j = −∞ | c j | < + ∞ . On the other hand we have ψ p ( a p · x p ) f ( a p ) = f ( a p )for any x p ∈ Z p . This implies that f ( p j ) = 0 for any j <
0. Thus the first assertionfollows. The second assertion is treated similarly. The last assertion follows fromthe first and the second assertions. (cid:3)
We denote these vectors by v m,λ where m ∈ N and λ ∈ M := Q p ∈ S M p . Notethat M is a finite set. Also we define θ m,λ,t ( g ) := Θ( v m,λ ⊗ | · | it ∞ )( g )= X α ∈ Q × ψ ( αx ) v m,λ ( αa fin ) | αa ∞ | it ∞ . The following proposition is a combination of Lemma 3.5 and the standardFourier analysis on the real line:
Proposition 3.6.
Let f ∈ H K . Suppose that (1) I ( f ) is integrable, i.e., I ( f ) ∈ L ( A × ) K ∩ L ( A × ) , (2) the Fourier transform of f is also integrable i.e. Z + ∞−∞ | ( f, θ m,λ,t ) | d t < + ∞ , for any m ∈ N and λ ∈ M .Then we have f ( g ) = X λ ∈ M ∞ X m =1 π Z + ∞−∞ ( f, θ m,λ,t ) θ m,λ,t ( g ) d t a.e., where ( f, θ m,λ,t ) = Z G ( Q ) \ G ( A Q ) f ( g ) θ m,λ,t ( g ) d g. EIGHT ZETA FUNCTIONS 15
Proof.
For simplicity, we assume that n p = 0 for all primes p . Let I ( f ) = h ∈ L ( A × ) K ∩ L ( A × ). It follows from the proof of Lemma 3.4 that f ( g ) = Θ( h )( g ) = X α ∈ Q × ψ ( αx ) h ( αa ) . Note that this infinite sum exists in both L and L sense. It is easy to check that Z A × h ( a ) v m ( a fin ) | a ∞ | − it ∞ d a × = ( f, θ m,t ) . Write h = X m v m ⊗ h m , where h m ∈ L ( R > , d a ×∞ ). The first and the second assumptions imply that h m and the Fourier transform of b h m both are integrable. Hence the inverse formula ofFourier transformation on the real line implies that h ( a ) = X m π Z + ∞−∞ ( f, θ m,t ) v m ( a fin ) | a ∞ | it ∞ d t a.e. . Apply Θ to both sides, and our assertion follows. (cid:3)
We recall some results regarding Igusa integrals with rapidly oscillating phase,studied in [CLT09]:
Proposition 3.7.
Let p be a finite place of Q and d, e ∈ Z . Let Φ : Q p × C → C , be a function such that for each ( x, y ) ∈ Q p , Φ(( x, y ) , s ) is holomorphic in s =( s , s ) ∈ C . Assume that the function ( x, y ) Φ( x, y, s ) belongs to a boundedsubset of the space of smooth compactly supported functions when ℜ ( s ) belongs to afixed compact subset of R . Let Λ be the interior of a closed convex cone generatedby (1 , , (0 , , ( d, e ) . Then, for any α ∈ Q × p , η α ( s ) = Z Q p | x | s p | y | s p ψ p ( αx d y e )Φ( x, y, s )d x × p d y × p , is holomorphic on T Λ . The same argument holds for the infinite place when Φ is asmooth function with compact supports.Proof. For the infinite place, use integration by parts and apply the convexityprinciple. For finite places, assume that d, e are both negative. Let δ ( x, y ) = 1 if | x | p = | y | p = 1 and 0 else. Then we have η α ( s ) = X n,m ∈ Z Z Q p | x | s p | y | s p ψ p ( αx d y e )Φ( x, y, s ) δ ( p − n x, p − m y ) d x × p d y × p = X n,m ∈ Z p − ( ns + ms ) · η α,n,m ( s ) , where η α,n,m ( s ) = Z | x | p = | y | p =1 ψ p ( αp nd + me x d y e )Φ( p n x, p m y, s ) d x × p d y × p . Fix a compact subset of C and assume that ℜ ( s ) is in that compact set. Theassumptions in our proposition mean that the support of Φ( · , s ) is contained in afixed compact set in Q p , so there exists an integer N such that η α,n,m ( s ) = 0 if n < N or m < N . Moreover our assumptions imply that there exists a positivereal number δ such that Φ( · , s ) is constant on any ball of radius δ in Q p . Thisimplies that if 1 /p n < δ , then for any u ∈ Z × p , η α,n,m ( s ) = Z ψ p ( αp nd + me x d y e u d )Φ( p n xu, p m y, s ) d x × p d y × p = Z ψ p ( αp nd + me x d y e u d )Φ( p n x, p m y, s ) d x × p d y × p = Z Z Z × p ψ p ( αp nd + me x d y e u d ) d u × Φ( p n x, p m y, s ) d x × p d y × p , and the last integral is zero if n is sufficiently large because of [CLT09, Lemma2.3.5]. Thus we conclude that there exists an integer N such that η α,n,m ( s ) = 0 if n > N or m > N . Hence we obtained that η α ( s ) = X N ≤ n,m ≤ N p − ( ns + ms ) · η α,n,m ( s ) , and this is holomorphic everywhere.The case of d < e = 0 is treated similarly.Next assume that d < e >
0. Then again we have a constant c such that η α,n,m ( s ) = 0 if 1 /p n < δ and n | d | − me > c . We may assume that c is sufficientlylarge so that the first condition is unnecessary. Then we have | η α ( s ) | ≤ X N ≤ n X m p − ne ( e ℜ ( s )+ | d |ℜ ( s )) · p ( n | d |− me ) e ℜ ( s ) · | η α,n,m ( s ) |≤ X N ≤ n p − ne ( e ℜ ( s )+ | d |ℜ ( s )) · p ce ℜ ( s ) − p − ℜ ( s e Thus η α ( s ) is holomorphic on T Λ . (cid:3) From the proof of Proposition 3.7, we can claim more for finite places:
Proposition 3.8.
Let ǫ > be any small positive real number. Fix a compactsubset K of Λ , and assume that ℜ ( s ) is in K . Define: κ ( K ) := max n , − ℜ ( s ) | d | , − ℜ ( s ) | e | o if d < e < n , − ℜ ( s ) | d | o if d < e ≥ . Then we have | η α ( s ) | ≪ / | α | κ ( K )+ ǫp as | α | p → .Proof. Let | α | p = p − k , and assume that both d, e are negative. By changing vari-ables, if necessary, we may assume that N in the proof of Proposition 3.7 is zero.If k is sufficiently large, then one can prove that there exists a constant c such that η α,n,m ( s ) = 0 if n | d | + m | e | > k + c . Also it is easy to see that | p − ( ns + ms ) | ≤ p ( n | d | + m | e | ) κ ( K ) . EIGHT ZETA FUNCTIONS 17
Hence we can conclude that | η α ( s ) | ≪ k / | α | κ ( K ) p ≪ / | α | κ ( K )+ ǫp . The case of d < e = 0 is treated similarly.Assume that d < e >
0. Then we have a constant c such that η α,n,m ( s ) = 0if n | d | − me > k + c . Thus we can conclude that | η α ( s ) | ≤ X m ≥ X n ≥ p − ( n ℜ ( s )+ m ℜ ( s )) | η α,n,m ( s ) |≪ X m ≥ p − m ℜ ( s ) ( me + k ) p ( me + k ) κ ( K,s ) ≪ k / | α | κ ( K,s ) p X m ≥ ( m + 1) p − m ( ℜ ( s ) − eκ ( K,s )) ≪ / | α | κ ( K,s )+ ǫp . where κ ( K, s ) = max (cid:26) , − ℜ ( s ) | d | : ( ℜ ( s ) , ℜ ( s )) ∈ K (cid:27) . Thus we can conclude that | η α ( s ) | ≪ / | α | κ ( K,s )+ ǫp ≪ / | α | κ ( K )+ ǫp . (cid:3) The projective plane
In this section, we implement the program described in Section 2 for the simplestequivariant compactifications of G = G = G a ⋊ G m , namely, the projective plane P , for a one-sided , right, action of G given by G ∋ ( x, a ) [ x : x : x ] = ( a : a − x : 1) ∈ P . The boundary consists of two lines, D and D given by the vanishing of x and x . We will use the following identities:div( a ) = D − D , div( x ) = D + D − D , div( ω ) = − D , where D is given by the vanishing of x and ω is the right invariant top degreeform. The height functions are given by H D ,p ( a, x ) = max {| a | p , | a − x | p , }| a | p , H D ,p ( a, x ) = max {| a | p , | a − x | p , } , H D , ∞ ( a, x ) = p | a | + | a − x | + 1 | a | , H D ,p ( a, x ) = p | a | + | a − x | + 1 , H D = Y p H D ,p × H D , ∞ , H D = Y p H D ,p × H D , ∞ , and the height pairing by H ( s , g ) = H s D ( g ) H s D ( g ) , for s = s D + s D and g ∈ G ( A ). The height zeta function takes the form Z ( s , g ) = X γ ∈ G ( Q ) H ( s , γg ) − . The proof of Northcott’s theorem shows that the Dirichlet series Z ( s , g ) convergesabsolutely and normally to a holomorphic function, for ℜ ( s ) is sufficiently large,which is continuous in g ∈ G ( A ). Moreover, if ℜ ( s ) is sufficiently large, then Z ( s , g ) ∈ L ( G ( Q ) \ G ( A )) ∩ L ( G ( Q ) \ G ( A )) . According to Proposition 3.1, we have the following decomposition: L ( G ( Q ) \ G ( A )) = L ( G m ( Q ) \ G m ( A )) ⊕ π, and we can write Z ( s , g ) = Z ( s , g ) + Z ( s , g ) . The analysis of Z ( s , id) is a special case of our considerations in Section 2, in par-ticular Theorem 2.1 (for further details, see [BT98] and [CLT10]). The conclusionhere is that there exist a δ > h which is holomorphic on the tubedomain T > − δ such that Z ( s , id) = h ( s + s )( s + s − . The analysis of Z ( s , id), i.e., of the contribution from the unique infinite-dimensionalrepresentation occurring in L ( G ( Q ) \ G ( A )), is the main part of this section. Define K = Y p K p · K ∞ = Y p G ( Z p ) · { (0 , ± } . Since the height functions are K -invariant, Z ( s , g ) ∈ π K ≃ L ( A × ) K . Lemma 3.5 provides a choice of an orthonormal basis for L ( A × fin ). Combining withthe Fourier expansion at the archimedean place, we obtain the following spectralexpansion of Z : Lemma 4.1.
Assume that ℜ ( s ) is sufficiently large. Then Z ( s , g ) = X m ≥ π Z ∞−∞ ( Z ( s , g ) , θ m,t ( g )) θ m,t ( g )d t, where θ m,t ( g ) = Θ( v m ⊗ | · | it )( g ) .Proof. See Lemma 5.3. (cid:3)
It is easy to see that( Z ( s , g ) , θ m,t ( g )) = Z G ( Q ) \ G ( A ) Z ( s , g ) θ m,t ( g )d g = Z G ( A ) H ( s , g ) − ¯ θ m,t ( g )d g = X α ∈ Q × Z G ( A ) H ( s , g ) − ¯ ψ ( αx ) v m ( αa fin ) | αa ∞ | − it d g = X α ∈ Q × Y p H ′ p ( s , m, α ) · H ′∞ ( s , t, α ) , EIGHT ZETA FUNCTIONS 19 where H ′ p ( s , m, α ) = Z G ( Q p ) H p ( s , g p ) − ¯ ψ p ( αx p ) m Z × p ( αa p )d g p , H ′∞ ( s , t, α ) = Z G ( R ) H ∞ ( s , g ∞ ) − ¯ ψ ∞ ( αx ∞ ) | αa ∞ | − it d g ∞ . Note that θ m,t (id) = 2 | m | it . Hence we can conclude that Z ( s , id) = X α ∈ Q × ∞ X m =1 π Z ∞−∞ Y p H ′ p ( s , m, α ) · H ′∞ ( s , t, α ) | m | it d t = X α ∈ Q × ∞ X m =1 π Z ∞−∞ Y p Z G ( Q p ) H p ( s , g p ) − ¯ ψ p ( αx p ) m Z × p ( αa p ) | αa | − itp d g p · H ′∞ ( s , t, α )d t = X α ∈ Q × π Z ∞−∞ Y p b H p ( s , α, t ) · b H ∞ ( s , α, t )d t, where b H p ( s , α, t ) = Z G ( Q p ) H p ( s , g p ) − ¯ ψ p ( αx p ) Z p ( αa p ) | a p | − itp d g p , b H ∞ ( s , α, t ) = Z G ( R ) H ∞ ( s , g ∞ ) − ¯ ψ ∞ ( αx ∞ ) | a ∞ | − it d g ∞ . It is clear that b H p ( s , α, t ) = b H p (( s − it, s + it ) , α, , so we only need to study b H p ( s , α ) = b H p ( s , α, P over Spec( Z ), and for anyprime p , we have the reduction map modulo p : ρ : G ( Q p ) ⊂ P ( Q p ) = P ( Z p ) → P ( F p )This is a continuous map from G ( Q p ) to P ( F p ). Consider the following open sets: U φ = ρ − ( P \ ( D ∪ D )) = {| a | p = 1 , | x | p ≤ } U D = ρ − ( D \ ( D ∩ D )) = {| a | p < , | a − x | p ≤ } U D = ρ − ( D \ ( D ∩ D )) = {| a | p > , | a − x | p ≤ } U D ,D = ρ − ( D ∩ D ) = {| a − x | p > , | a − x | p > } . The height functions have a partial left invariance, i.e., they are invariant underthe left action of the compact subgroup { (0 , b ) | b ∈ Z × p } . This implies that b H p ( s , α ) = Z G ( Q p ) H p ( s , g ) − Z Z × p ¯ ψ p ( αbx )d b × Z p ( αa )d g. We record the following useful lemma (see, e.g., [CLT02, Lemma 10.3]):
Lemma 4.2. Z Z × p ¯ ψ p ( bx )d b × = | x | p ≤ , − p − if | x | p = p, . Lemma 4.3.
Assume that | α | p = 1 . Then b H p ( s , α ) = ζ p ( s + 1) ζ p (2 s + s ) ζ p ( s + s ) . Proof.
We apply Lemma 4.2 and obtain b H p ( s , α ) = Z U ∅ + Z U D + Z U D ,D = 1 + p − ( s +1) − p − ( s +1) + p − (2 s + s ) − p − ( s + s ) (1 − p − ( s +1) )(1 − p − (2 s + s ) )= ζ p ( s + 1) ζ p (2 s + s ) ζ p ( s + s ) . (cid:3) Lemma 4.4.
Assume that | α | p > . Let | α | p = p k . Then b H p ( s , α ) = p − k ( s +1) b H p ( s , . Proof.
Use the lemma, and we obtain that b H p ( s , α ) = Z U D + Z U D ,D = p − k ( s +1) − p − ( s +1) + p − k ( s +1) p − (2 s + s ) − p − ( s + s ) (1 − p − ( s +1) )(1 − p − (2 s + s ) )= p − k ( s +1) b H p ( s , . (cid:3) Lemma 4.5.
Assume that | α | p < . Let | α | p = p − k . Then b H p ( s , α ) is holomorphicon the tube domain T Λ over the cone Λ = { s > − , s + s > , s + s > } . Moreover, for any compact subset of Λ , there exists a constant C > such that | b H p ( s , α ) | ≤ Ck max { , p − k ℜ ( s − } for any s with real part in this compact set.Proof. It is easy to see that b H p ( s , α ) = Z U ∅ + Z U D + Z U D + Z U D ,D = 1 + p − ( s +1) − p − ( s +1) + Z U D + Z U D ,D . EIGHT ZETA FUNCTIONS 21 On U D , we choose (1 : x : x ) as coordinates, then we have Z U D = Z | x | p ≤ , | x | p < | x | s − p Z Z × p ¯ ψ p ( αb x x )d b × Z p ( αx − )d x d x × = Z p − k ≤| x | p < | x | s − p d x × Hence this integral is holomorphic everywhere, and we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z U D (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < k max { , p − k Re( s − } . On U D ,D , we choose ( x : 1 : x ) as coordinates and obtain11 − p − Z U D ,D = Z | x | p , | x | p < | x | s +1 p | x | s − p Z Z × p ¯ ψ p ( αb x x )d b × d x × d x × = − p − − p − p ( k +1)( s +1) p − (1 − [ − k ] )(2 s + s ) − p − (2 s + s ) + Z | x | s +1 p | x | s − p d x × d x × , where the last integral is over {| x | p < , | x | p < , | αx | p ≤ | x | p } , and is equal to X j,l ≥ , j − l ≤ k p − l ( s +1) − j ( s − = X j ≤ [ k ] p − j ( s − p − ( s +1) − p − ( s +1) + X j> [ k ] p − j ( s − p − (2 j − k )( s +1) − p − ( s D +1) = X j ≤ [ k ] p − j ( s − p − ( s +1) − p − ( s +1) + p − ( [ k ] +1)(2 s + s ) − p − (2 s + s ) p k ( s +1) − p − ( s +1) . From this we can see that R U D ,D is holomorphic on T Λ and that for any compactsubset of Λ, we can find a constant C > (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z U D ,D (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < Ck max { , p − k ℜ ( s − } , for ℜ ( s ) in this compact. (cid:3) Next, we study the local integral at the real place. Again, b H ∞ ( s , α, t ) = b H ∞ (( s − it, s + it ) , α, , and we start with b H ∞ ( s , α ) = b H ∞ ( s , α, . Lemma 4.6.
The function s b H ∞ ( s , α ) , is holomorphic on the tube domain T Λ ′ over Λ ′ = { s > − , s > , s + s > } . Moreover, for any r ∈ N and any compact subset of Λ ′ r = { s > − r, s > } , there exists a constant C > such that | b H ∞ ( s , α ) | < C | α | r ∞ , for any s in the tube domain over this compact.Proof. Let U ∅ = X ( R ) \ ( D ∪ D ), U D i be a small tubular neighborhood of D i minus D ∩ D , and U D ,D be a small neighborhood of D ∩ D . Then { U ∅ , U D , U D , U D ,D } is an open covering of X ( R ), and consider the partitionof unity for this covering; θ ∅ , θ D , θ D , θ D ,D . Then we have b H ∞ ( s , α ) = Z U ∅ H − ∞ ¯ ψ ∞ ( αx ∞ ) θ ∅ d g ∞ + Z U D + Z U D + Z U D ,D . On U D ,D , we choose ( x : 1 : x ) as analytic coordinates and obtain Z U D ,D = Z R | x | s +1 | x | s − ¯ ψ ( α x x )Φ( s , x , x )d x × dx × , where Φ is a smooth bounded function with compact support. Such oscillatoryintegrals have been studied in [CLT09], in our case the integral is holomorphicif Re( s ) > − s + s ) >
0. Assume that ℜ ( s ) is sufficiently large.Integration by parts implies that Z U D ,D = 1 α r Z R | x | s +1 − r | x | s − r ¯ ψ ( α x x )Φ ′ ( s , x , x ) dx × d x × , and this integral is holomorphic if ℜ ( s ) > − r and ℜ ( s ) > − r . Thus, oursecond assertion follows. The other integrals are studied similarly. (cid:3) Lemma 4.7.
For any compact set K ⊂ Λ ′ , there exists a constant C > such that | b H ∞ ( s , α, t ) | < C | α | (1 + t ) , for any s ∈ T K .Proof. Consider a left invariant differential operator ∂ a = a∂/∂a . Integrating byparts we obtain that b H ∞ ( s , α, t ) = − t Z G ( R ) ∂ a H ∞ ( s , g ∞ ) − ¯ ψ ∞ ( αx ∞ ) | a ∞ | − it d g ∞ , According to [CLT02], ∂ a H ∞ ( s , g ∞ ) − = H ∞ ( s , g ∞ ) − × (a bounded smooth function) , so we can apply the discussion of the previous proposition. (cid:3) Lemma 4.8.
The Euler product Y p b H p ( s , α, t ) · b H ∞ ( s , α, t ) , is holomorphic on the tube domain T Ω over Ω = { s > , s > , s + s > } . EIGHT ZETA FUNCTIONS 23
Moreover, let α = βγ , where gcd( β, γ ) = 1 . Then for any ǫ > and any compactset K ⊂ Ω ′ = { s > , s > , s + s > } , there exists a constant C > such that | Y p b H p ( s , α, t ) · b H ∞ ( s , α, t ) | < C · max { , p | β | −ℜ ( s − }| β | − ǫ | γ | ℜ ( s − , for all s ∈ T K . Theorem 4.9.
There exists δ > such that Z ( s + s , id) is holomorphic on T > − δ .Proof. Let δ > { s > δ, s > − δ } . It follows from the previous proposition that for any ǫ > K ⊂ Λ, there exists a constant
C > | Y p b H p ( s , α, t ) · b H ∞ ( s , α, t ) | < C (1 + t ) | β | − ǫ − δ | γ | δ . From this inequality, we can conclude that the integral Z ∞−∞ Y p b H p ( s , α, t ) · b H ∞ ( s , α, t )d t, converges uniformly and absolutely to a holomorphic function on T K . Furthermore,we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∞−∞ Y p b H p ( s , α, t ) · b H ∞ ( s , α, t )d t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < C ′ | b | − ǫ − δ | c | δ . For sufficiently small ǫ > δ >
0, the sum X α ∈ Q × π Z ∞−∞ Y p b H p ( s , α, t ) · b H ∞ ( s , α, t )d t, converges absolutely and uniformly to a function in s + s . This concludes theproof of our theorem. (cid:3) Geometrization
In this section we geometrize the method described in Section 4. Our maintheorem is:
Theorem 5.1.
Let X be a smooth projective equivariant compactification of G = G over Q , under the right action. Assume that the boundary divisor has strictnormal crossings. Let a, x ∈ Q ( X ) be rational functions, where ( x, a ) are the stan-dard coordinates on G ⊂ X . Let E be the Zariski closure of { x = 0 } ⊂ G . Assumethat: • the union of the boundary and E is a divisor with strict normal crossings, • div( a ) is a reduced divisor, and • for any pole D ι of a , one has − ord D ι ( x ) > . Then Manin’s conjecture holds for X . The remainder of this section is devoted to a proof of this fact. Blowing up thezero-dimensional subschemeSupp(div ( a )) ∩ Supp(div ∞ ( a )) , if necessary, we may assume thatSupp(div ( a )) ∩ Supp(div ∞ ( a )) = ∅ . The height functions are invariant under the right action of some compact subgroup K p ⊂ G ( Z p ). Moreover, we can assume that K p = G ( p n p Z p ), for some n p ∈ Z ≥ .Let S be the set of bad places for X ; note that n p = 0 for all p / ∈ S . For simplicity,we assume that the height function at the infinite place is invariant under the actionof K ∞ = { (0 , ± } . Lemma 5.2.
We have Z ( s , g ) ∈ L ( G ( Q ) \ G ( A )) K ∩ L ( G ( Q ) \ G ( A )) . Proof.
First it is easy to see that Z G ( Q ) \ G ( A ) | Z ( s , g ) | d g ≤ Z G ( Q ) \ G ( A ) X γ ∈ G ( Q ) | H ( s , γg ) | − d g = Z G ( A ) H ( ℜ ( s ) , g ) − d g, and the last integral is bounded when ℜ ( s ) is sufficiently large. (See [CLT10,Proposition 4.3.4].) Hence it follows that Z ( s , g ) is integrable. To conclude that Z ( s , g ) is square-integrable, we prove that Z ( s , g ) ∈ L ∞ for ℜ ( s ) sufficiently large.Let u, v be sufficiently large positive real numbers. Assume that ℜ ( s ) is in a fixedcompact subset of Pic G ( X ) ⊗ R and sufficiently large. Then we have H ( ℜ ( s ) , g ) − ≪ H ( a ) − u · H ( x ) − v where H ( a ) = Y p max {| a p | p , | a p | − p } · q | a ∞ | ∞ + | a ∞ | − ∞ H ( x ) = Y p max { , | x p | p } · p | x ∞ | ∞ . Since Z ( s , g ) is G ( Q )-periodic, we may assume that | a p | p = 1 where g p = ( x p , a p ).Then we obtain that X γ ∈ G ( Q ) H ( ℜ ( s ) , γg ) − ≪ X α ∈ Q × X β ∈ Q H ( αa ) − u · H ( αx + β ) − v ≤ X α ∈ Q × H , fin ( α ) − u Z ( αx ) , where Z ( x ) = X β ∈ Q H ( x + β ) − v . EIGHT ZETA FUNCTIONS 25
It is known that Z is a bounded function for sufficiently large v , (see [CLT02]) sowe can conclude that Z ( s , g ) is also a bounded function because X α ∈ Q × H , fin ( α ) − u < + ∞ , for sufficiently large u . (cid:3) By Proposition 3.1, the height zeta function decomposes as Z ( s , id) = Z ( s , id) + Z ( s , id) . Analytic properties of Z ( s , id) were established in Section 2. It remains to showthat Z ( s , id) is holomorphic on a tube domain over an open neighborhood of theshifted effective cone − K X + Λ eff ( X ). To conclude this, we use the spectral decom-position of Z : Lemma 5.3.
We have Z ( s , id) = X λ ∈ M ∞ X m =1 π Z + ∞−∞ ( Z ( s , g ) , θ m,λ,t ) θ m,λ,t (id) d t. Proof.
To apply Proposition 3.6, we need to check that Z satisfies the assumptionsof Proposition 3.6. The proof of Lemma 3.4 implies that I ( Z ) = Z N ( Q ) \ N ( A ) Z ( s , g ) ψ ( x ) d x. Thus we have Z T ( A ) | I ( Z ) | d a × = Z T ( A ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z N ( Q ) \ N ( A ) Z ( s , g ) ψ ( x ) d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d a × ≤ X α ∈ Q × Z T ( A ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z N ( A ) H ( s , g ) − ψ ( αx ) d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d a × = X α ∈ Q × Y p Z T ( Q p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z N ( Q p ) H p ( s , g p ) − ψ p ( αx p ) d x p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d a × p × Z T ( R ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z N ( R ) H ∞ ( s , g ∞ ) − ψ ∞ ( αx ) d x ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d a ×∞ . Assume that p / ∈ S . Since the height function is right K p -invariant, we obtain thatfor any y p ∈ Z p , Z N ( Q p ) H p ( s , g p ) − ψ p ( αx p ) d x p = Z N ( Q p ) H p ( s , ( x p + a p y p , a p )) − ψ p ( αx p ) d x p = Z N ( Q p ) H p ( s , g p ) − ψ p ( αx p ) Z Z p ψ p ( αa p y p ) d y p d x p = 0 if | αa p | p > Z T ( Q p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z N ( Q p ) H p ( s , g p ) − ψ p ( αx p ) d x p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d a × p ≤ Z G ( Q p ) H p ( ℜ ( s ) , g p ) − Z p ( αa p ) d g p . Similarly, for p ∈ S , we can conclude that Z T ( Q p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z N ( Q p ) H p ( s , g p ) − ψ p ( αx p ) d x p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d a × p ≤ Z G ( Q p ) H p ( ℜ ( s ) , g p ) − N Z p ( αa p ) d g p . Then the convergence of the following sum X α ∈ Q × Y p Z G ( Q p ) H − p N Z p ( αa p ) d g p · Z T ( R ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z N ( R ) H − ∞ ψ ∞ ( αx ) d x ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d a ×∞ , can be verified from the detailed study of the local integrals which we will conductlater. See proofs of Lemmas 5.6, 5.9, and 5.10.Next we need to check that Z + ∞−∞ | ( Z ( s , g ) , θ m,λ,t ) | d t < + ∞ . It is easy to see that( Z ( s , g ) , θ m,λ,t ) = Z G ( Q ) \ G ( A Q ) Z ( s , g ) θ m,λ,t d g = Z G ( A Q ) H ( s , g ) − θ m,λ,t d g = X α ∈ Q × Z G ( A Q ) H ( s , g ) − ψ ( αx ) v m,λ ( αa fin ) | αa ∞ | − it ∞ d g = X α ∈ Q × Y p H ′ p ( s , m, λ, α ) · H ′∞ ( s , t, α ) , where H ′ p ( s , m, λ, α ) is given by= Z G ( Q p ) H p ( s , g p ) − ψ p ( αx p ) m Z × p ( αa p ) d g p , p / ∈ S = Z G ( Q p ) H p ( s , g p ) − ψ p ( αx p ) λ p ( αa p /p v p ( αa p ) ) mN Z × p ( αa p ) d g p , p ∈ S and H ′∞ ( s , t, α ) = Z G ( R ) H ∞ ( s , g ∞ ) − ψ ∞ ( αx ∞ ) | αa ∞ | − it ∞ d g ∞ . The integrability follows from the proof of Lemma 5.9. Thus we can apply Proposi-tion 3.6, and the identity in our statement follows from the continuity of Z ( s , g ). (cid:3) We obtained that Z ( s , id) = X λ ∈ M ∞ X m =1 π Z + ∞−∞ ( Z ( s , g ) , θ m,λ,t ) θ m,λ,t (id) d t = X λ ∈ M , λ ( − ∞ X m =1 π Z + ∞−∞ ( Z ( s , g ) , θ m,λ,t ) Y p ∈ S λ p (cid:16) mN · p − v p ( m/N ) (cid:17) (cid:12)(cid:12)(cid:12) mN (cid:12)(cid:12)(cid:12) it ∞ d t. EIGHT ZETA FUNCTIONS 27
We will use the following notation: λ S ( αa p ) := Y q ∈ S λ q ( p v p ( αa p ) ) , p / ∈ Sλ S,p ( αa p ) := λ p (cid:18) αa p p v p ( αa p ) (cid:19) Y q ∈ S \ p λ q ( p v p ( αa p ) ) , p ∈ S. Proposition 5.4. If ℜ ( s ) is sufficiently large, then Z ( s , id ) = X λ ∈ M , λ ( − X α ∈ Q × π Z + ∞−∞ Y p b H p ( s , λ, t, α ) · b H ∞ ( s , t, α ) d t, where b H p ( s , λ, t, α ) is given by Z G ( Q p ) H p ( s , g p ) − ψ p ( αx p ) λ S ( αa p ) Z p ( αa p ) | a p | − itp d g p , p / ∈ S Z G ( Q p ) H p ( s , g p ) − ψ p ( αx p ) λ S,p ( αa p ) N Z p ( αa p ) | a p | − itp d g p , p ∈ S and b H ∞ ( s , t, α ) = Z G ( R ) H ∞ ( s , g ∞ ) − ψ ∞ ( αx ∞ ) | a ∞ | − it ∞ d g ∞ Proof.
For simplicity, we assume that S = ∅ . We have seen that Z ( s , id) = ∞ X m =1 X α ∈ Q × π Z + ∞−∞ Y p H ′ p ( s , m, α ) · H ′∞ ( s , t, α ) | m | it ∞ d t. On the other hand, it is easy to see that b H p ( s , t, α ) = ∞ X j =0 Z G ( Q p ) H p ( s , g p ) − ψ p ( αx p ) p j Z × p ( αa p ) (cid:12)(cid:12)(cid:12)(cid:12) p j α (cid:12)(cid:12)(cid:12)(cid:12) − itp d g p . Hence we have the formal identity: Y p b H p ( s , t, α ) · b H ∞ ( s , t, α ) = ∞ X m =1 Y p H ′ p ( s , m, α ) · H ′∞ ( s , t, α ) | m | it ∞ , and our assertion follows from this. To justify the above identity, we need to addressconvergence issues; this will be discussed below (see the proof of Lemma 5.6). (cid:3) Thus we need to study the local integrals in Proposition 5.4. We introduce somenotation: I = { ι ∈ I | D ι ⊂ Supp(div ( a )) }I = { ι ∈ I | D ι ⊂ Supp(div ∞ ( a )) }I = { ι ∈ I | D ι Supp(div( a )) } . Note that I = I ⊔ I ⊔ I and I = ∅ . Also D ι ⊂ Supp(div ∞ ( x )) for any ι ∈ I because D = ∪ ι ∈I D ι = Supp(div( a )) ∪ Supp(div ∞ ( x )) . Let − div( ω ) = X ι ∈I d ι D ι , where ω = d x d a/a is the top degree right invariant form on G . Note that ω definesa measure | ω | on an analytic manifold G ( Q v ), and for any finite place p , | ω | = (cid:18) − p (cid:19) d g p , where d g p is the standard Haar measure defined in Section 3. Lemma 5.5.
Consider an open convex cone Ω in Pic G ( X ) R , defined by the follow-ing relations: s ι − d ι + 1 > ι ∈ I s ι − d ι + 1 + e ι > ι ∈ I s ι − d ι + 1 > ι ∈ I where e ι = | ord D ι ( x ) | . Then b H p ( s , λ, t, α ) and b H ∞ ( s , t, α ) are holomorphic on T Ω .Proof. First we prove our assertion for b H ∞ . We can assume that b H v ( s , t ) = b H v ( s − it m ( a ) , , where m ( a ) ∈ X ∗ ( G ) ⊂ Pic G ( X ) is the character associated to the rational function a (by choosing an appropriate height function). It suffices to discuss the case when t = 0. Choose a finite covering { U η } of X ( R ) by open subsets and local coordinates y η , z η on U η such that the union of the boundary divisor D and E is locally definedby y η = 0 or y η · z η = 0. Choose a partition of unity { θ η } ; the local integral takesthe form b H ∞ ( s , α ) = X η Z G ( R ) H ∞ ( s , g ∞ ) − ψ ∞ ( αx ∞ ) θ η d g ∞ . Each integral is a oscillatory integral in the variables y η , z η . For example, assumethat U η meets D ι , D ι ′ , where ι, ι ′ ∈ I . Then Z G ( R ) H ∞ ( s , g ∞ ) − ψ ∞ ( αx ∞ ) θ η d g ∞ = Z R | y η | s ι − d ι | z η | s ι ′ − d ι ′ ψ ∞ (cid:18) αfy e ι η z e ι ′ η (cid:19) Φ( s , y η , z η )d y η d z η , where Φ is a smooth function with compact support and f is a nonvanishing analyticfunction. Shrinking U η and changing variables, if necessary, we may assume that f is a constant. Proposition 3.7 implies that this integral is holomorphic everywhere.The other integrals can be studied similarly.Next we consider finite places. Let p be a prime of good reduction. SinceSupp(div ( a )) ∩ Supp(div ∞ ( a )) = ∅ , the smooth function Z p ( αa p ) extends to a smooth function h on X ( Q p ). Let U = { h = 1 } . Then b H p ( s , λ, α ) = Z U H p ( s , g p ) − ψ p ( αx p ) λ S ( αa p )d g p . Now the proof of [CLT10, Lemma 4.4.1] implies that this is holomorphic on T Ω because U ∩ ( ∪ ι ∈I D ι ( Q p )) = ∅ . Places of bad reduction are treated similarly. (cid:3) EIGHT ZETA FUNCTIONS 29
Lemma 5.6.
Let | α | p = p k > . Then, for any compact set in Ω and for any δ > , there exists a constant C > such that | b H p ( s , λ, t, α ) | < C | α | − min ι ∈I {ℜ ( s ι ) − d ι +1 − δ } p , for ℜ ( s ) in that compact set.Proof. First assume that p is a good reduction place. Let ρ : X ( Z p ) → X ( F p )be the reduction map modulo p where X is a smooth integral model of X overSpec( Z p ). Note that ρ ( {| a | p < } ) ⊂ ∪ ι ∈I D ι ( F p ) , where D ι is the Zariski closure of D ι in X . Thus b H p ( s , λ, α ) is given by b H p ( s , λ, α ) = X ˜ x ∈∪ ι ∈I D ι ( F p ) Z ρ − (˜ x ) H p ( s , g p ) − ψ p ( αx p ) λ S ( αa p ) Z p ( αa p )d g p . Let ˜ x ∈ D ι ( F p ) for some ι ∈ I , but ˜ x / ∈ D ι ′ ( F p ) for any ι ′ ∈ I \ { ι } . Since p is agood reduction place, we can find analytic coordinates y, z such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ρ − (˜ x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z ρ − (˜ x ) H p ( ℜ ( s ) , g p ) − Z p ( αa p )d g p = (cid:18) − p (cid:19) − Z ρ − (˜ x ) H p ( ℜ ( s ) − d , g p ) − Z p ( αa p )d τ X,p = (cid:18) − p (cid:19) − Z m p | y | ℜ ( s ι ) − d ι p Z p ( αy )d y p d z p = 1 p · p − k ( ℜ ( s ι ) − d ι +1) − p − ( ℜ ( s ι ) − d ι +1) , where d τ X,p is the local Tamagawa measure (see [CLT10, Section 2] for the defini-tion). For the construction of such local analytic coordinates, see [Wei82], [Den87],or [Sal98]. If ˜ x ∈ D ι ( F p ) ∩ D ι ′ ( F p ) for ι ∈ I , ι ′ ∈ I , then we can find local analyticcoordinates y, z such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ρ − (˜ x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18) − p (cid:19) Z m p | y | ℜ ( s ι ) − d ι +1 p | z | ℜ ( s ι ′ ) − d ι ′ +1 p Z p ( αy )d y × p d z × p = (cid:18) − p (cid:19) p − k ( ℜ ( s ι ) − d ι +1) − p − ( ℜ ( s ι ) − d ι +1) p − ( ℜ ( s ι ′ ) − d ι ′ +1) − p − ( ℜ ( s ι ′ ) − d ι ′ +1) . If ˜ x ∈ D ι ( F p ) ∩ D ι ′ ( F p ) for ι, ι ′ ∈ I , ι = ι ′ , then we can find analytic coordinates x, y such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ρ − (˜ x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18) − p (cid:19) Z m p | y | ℜ ( s ι ) − d ι +1 p | z | ℜ ( s ι ′ ) − d ι ′ +1 p Z p ( αyz ) d y × p d z × p ≤ (cid:18) − p (cid:19) Z m p | yz | min {ℜ ( s ι ) − d ι +1 , ℜ ( s ι ′ ) − d ι ′ +1 } p Z p ( αyz ) d y × p d z × p = (cid:18) − p (cid:19) (cid:18) ( k − p − kr − p − r + p − ( k +1) r (1 − p − r ) (cid:19) , where r = min {ℜ ( s ι ) − d ι + 1 , ℜ ( s ι ′ ) − d ι ′ + 1 } . It follows from these inequalities and Lemma 9.4 in [CLT02] that there exists aconstant
C >
0, independent of p , satisfying the inequality in the statement.Next assume that p is a bad reduction place. Choose an open covering { U η } of ∪ ι ∈I D ι ( Q p ) such that ( ∪ η U η ) ∩ ( ∪ ι ∈I D ι ( Q p )) = ∅ , and each U η has analytic coordinates y η , z η . Moreover, we can assume that theboundary divisor is defined by y η = 0 or y η · z η = 0 on U η . Let V be the complementof ∪ ι ∈I D ι ( Q p ), and consider the partition of unity for { U η , V } which we denoteby { θ η , θ V } . If k is sufficiently large, then { N Z p ( αa ) = 1 } ∩ Supp( θ V ) = ∅ . Hence if k is sufficiently large, then | b H p ( s , λ, α ) | ≤ X η Z U η H p ( ℜ ( s ) , g p ) − N Z p ( αa p ) · θ η d g p . When U η meets only one component D ι ( Q p ) for ι ∈ I , then Z U η ≤ Z Q p | y η | ℜ ( s ι ) − d ι p c Z p ( αy η )Φ( s , y η , z η ) d y η,p d z η,p ≪ p − k ( ℜ ( s ι ) − d ι +1) , as k → ∞ , where c is some rational number and Φ is a smooth function withcompact support. Other integrals are treated similarly. (cid:3) We record the following useful lemma (see, e.g., [CLT09, Lemma 2.3.1]):
Lemma 5.7.
Let d be a positive integer and a ∈ Q p . If | a | p > p and p ∤ d , then Z Z × p ψ p ( ax d ) d x × p = 0 . Moreover, if | a | p = p and d = 2 , then Z Z × p ψ p ( ax d )d x × p = ( √ p − p − or i √ p − p − if pa is a quadratic residue , −√ p − p − or − i √ p − p − if pa is a quadratic non-residue. Lemma 5.8.
Let | α | p = p − k < . Consider an open convex cone Ω ǫ in Pic( X ) R ,defined by the following relations: s ι − d ι + 1 > ι ∈ I s ι − d ι + 2 + ǫ > ι ∈ I s ι − d ι + 1 > ι ∈ I where < ǫ < / . Then, for any compact set in Ω ǫ , there exists a constant C > such that | b H p ( s , λ, t, α ) | < C | α | − (1+2 ǫ ) p , for ℜ ( s ) in that compact set.Proof. First assume that p is a good reduction place and that p ∤ e ι , for any ι ∈ I .We have b H p ( s , λ, α ) = X ˜ x ∈X ( F p ) Z ρ − (˜ x ) H p ( s , g p ) − ψ p ( αx p ) λ S ( αa p ) Z p ( αa p ) d g p . EIGHT ZETA FUNCTIONS 31
A formula of J. Denef (see [Den87, Theorem 3.1] or [CLT10, Proposition 4.1.7]) andLemma 9.4 in [CLT02] give us an uniform bound: | X ˜ x/ ∈∪ ι ∈I D ι ( F p ) | ≤ X ˜ x/ ∈∪ ι ∈I D ι ( F p ) Z ρ − (˜ x ) H p ( ℜ ( s ) , g p ) − d g p . Hence we need to study X ˜ x ∈∪ ι ∈I D ι ( F p ) Z ρ − (˜ x ) H p ( s , g p ) − ψ p ( αx p ) λ S ( αa p ) Z p ( αa p ) d g p . Let ˜ x ∈ D ι ( F p ) for some ι ∈ I , but ˜ x / ∈ D ι ′ ( F p ) ∪ E ( F p ) for any ι ′ ∈ I \ { ι } , where E is the Zariski closure of E in X . Then we can find local analytic coordinates y, z such that Z ρ − (˜ x ) = (cid:18) − p (cid:19) − Z m p | y | s ι − d ι p ψ p ( αf /y e ι ) λ S ( αy − ) Z p ( αy − ) d y p d z p , where f ∈ Z p [[ y, z ]] such that f (0) ∈ Z × p . Since p does not divide e ι , there exists g ∈ Z p [[ y, z ]] such that f = f (0) g e ι . After a change of variables, we can assumethat f = u ∈ Z × p . Lemma 5.7 implies that Z ρ − (˜ x ) = 1 p Z m p | y | s ι − d ι +1 p λ S ( αy − ) Z Z × p ψ p ( αub e ι /y e ι )d b × p Z p ( αy − )d y × p = 1 p Z p − ( k +1) ≤| y eι | p | y | s ι − d ι +1 p λ S ( αy − ) Z Z × p ψ p ( αub e ι /y e ι ) d b × p d y × p Thus it follows from the second assertion of Lemma 5.7 that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ρ − (˜ x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ p Z p − ( k +1) ≤| y eι | | y | ℜ ( s ι ) − d ι +1 p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Z × p ψ p ( αub e ι /y e ι )d b × p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d y × p ≤ p kp keι (1+ ǫ ) + 1 p p k +1 eι (1+ ǫ ) × ( e ι > √ p − if e ι = 2 ≪ p kp k (1+ ǫ ) . If ˜ x ∈ D ι ( F p ) ∩ E ( F p ), for some ι ∈ I , then we have Z ρ − (˜ x ) = (cid:18) − p (cid:19) − Z m p | y | s ι − d ι p ψ p ( αz/y e ι ) λ S ( αy − ) Z p ( αy − ) d y p d z p = Z m p | y | s ι − d ι +1 p λ S ( αy − ) Z p ( αy − ) Z m p ψ p ( αz/y e ι )d z p d y × p = 1 p Z p − ( k +1) ≤| y | eιp < | y | s ι − d ι +1 p λ S ( αy − ) d y × p . Hence we obtain that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ρ − (˜ x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ p Z p − ( k +1) ≤| y | eιp < | y | ℜ ( s ι ) − d ι +1 p d y × p ≤ kp keι (1+ ǫ ) < kp k (1+ ǫ ) . If ˜ x ∈ D ι ( F p ) ∩ D ι ′ ( F p ) for some ι ∈ I and ι ′ ∈ I , then it follows from Lemma 5.7 Z ρ − (˜ x ) = (cid:18) − p (cid:19) − Z m p | y | s ι − d ι p | z | s ι ′ − d ι ′ p ψ p (cid:18) αuy e ι z e ι ′ (cid:19) λ S ( αy − ) Z p ( αy − ) d y p d z p = (cid:18) − p (cid:19) − Z | y | s ι − d ι p | z | s ι ′ − d ι ′ p λ S ( αy − ) Z Z × p ψ p (cid:18) αub e ι y e ι z e ι ′ (cid:19) d b × p d y p d z p , where the last integral is over the domain { ( y, z ) ∈ m p : p − ( k +1) ≤ | y e ι z e ι ′ | p } . We conclude that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ρ − (˜ x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18) − p (cid:19) − Z p − ( k +1) ≤| y eι z eι ′ | p | y | ℜ ( s ι ) − d ι p | z | ℜ ( s ι ′ ) − d ι ′ p d y p d z p ≤ (cid:18) − p (cid:19) − Z p − k ≤| y eι | p < | y | ℜ ( s ι ) − d ι p d y p Z m p | z | ℜ ( s ι ′ ) − d ι ′ p d z p ≤ kp keι (1+ ǫ ) p − ( ℜ ( s ι ′ ) − d ι ′ +1) − p − ( ℜ ( s ι ′ ) − d ι ′ +1) . If ˜ x ∈ D ι ( F p ) ∩ D ι ′ ( F p ) for some ι, ι ′ ∈ I , then the local integral on ρ − (˜ x ) is: (cid:18) − p (cid:19) − Z m p | y | s ι − d ι p | z | s ι ′ − d ι ′ p ψ p (cid:18) αuy e ι z e ι ′ (cid:19) λ S ( αy − z − ) Z p ( αy − z − ) d y p d z p = (cid:18) − p (cid:19) Z m p | y | s ι − d ι p | z | s ι ′ − d ι ′ p λ S ( αy − z − ) Z Z × p ψ p (cid:18) αub e ι y e ι z e ι ′ (cid:19) d b × p d y × p d z × p . We can assume that e ι ≤ e ι ′ . Then we can conclude that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ρ − (˜ x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z p − k ≤| y eι z eι ′ | p | y e ι z e ι ′ | − eι (1+ ǫ ) p d y × p d z × p + Z p − ( k +1) = | y eι z eι ′ | p | y e ι z e ι ′ | − eι (1+ ǫ ) p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Z × p ψ p (cid:18) αub e ι y e ι z e ι ′ (cid:19) d b × (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d y × p d z × p ≤ k p keι (1+ ǫ ) + kp k +1 eι (1+ ǫ ) × ( e ι > √ p − if e ι = 2 ≪ k p k (1+ ǫ ) . Thus our assertion follows from these estimates and Lemma 9.4 in [CLT02].Next assume that p is a place of bad reduction or that p divides e ι , for some ι ∈ I . Fix a compact subset of Ω ǫ and assume that ℜ ( s ) is in that compact set.Choose a finite open covering { U η } of ∪ ι ∈I D ι ( Q p ) with analytic coordinates y η , z η such that the union of the boundary D ( Q p ) and E ( Q p ) is defined by y η = 0 or y η · z η = 0. Let V be the complement of ∪ ι ∈I D ι ( Q p ), and consider a partition ofunity { θ η , θ V } for { U η , V } . Then it is clear that Z V H p ( s , g p ) − ψ p ( αx p ) λ S,p ( αa p ) N Z p ( αa p ) θ V d g p , EIGHT ZETA FUNCTIONS 33 is bounded, so we need to study Z U η H p ( s , g p ) − ψ p ( αx p ) λ S,p ( αa p ) N Z p ( αa p ) θ U η d g p . Assume that U η meets only one D ι ( Q p ) for some ι ∈ I . Then, the above integrallooks like Z U η = Z Q p | y η | s ι − d ι p ψ p ( αf /y e ι η )) λ S,p ( αg/y η ) N Z p ( αg/y η )Φ( s , y η , z η )d y η,p d z η,p , where f and g are nonvanishing analytic functions, and Φ is a smooth functionwith compact support. By shrinking U η and changing variables, if necessary, wecan assume that f and g are constant. The proof of Proposition 3.8 implies ourassertion for this integral. Other integrals are treated similarly. (cid:3) Lemma 5.9.
For any compact set in an open convex cone Ω ′ , defined by s ι − d ι − > ι ∈ I s ι − d ι + 3 > ι ∈ I s ι − d ι + 1 > ι ∈ I there exists a constant C > such that | b H ∞ ( s , t, α ) | < C | α | (1 + t ) , for ℜ ( s ) in that compact set.Proof. Consider the left invariant differential operators ∂ a = a∂/∂a and ∂ x = a∂/∂x . Assume that ℜ ( s ) ≫
0. Integrating by parts, we have b H ∞ ( s , t, α ) = − t Z G ( R ) ∂ a H ∞ ( s , g ∞ ) − ψ ∞ ( αx ∞ ) | a ∞ | − it ∞ d g ∞ = 1(2 π ) | α | t Z G ( R ) ∂ ∂x ( ∂ a H ∞ ( s , g ∞ ) − ) ψ ∞ ( αx ∞ ) | a ∞ | − it ∞ d g ∞ . According to Proposition 2.2. in [CLT02], ∂ ∂x ( ∂ a H ∞ ( s , g ∞ ) − ) = | a | − ∂ x ∂ a H ∞ ( s , g ∞ ) − = H ∞ ( s − m ( a ) , g ∞ ) − × (a bounded smooth function) . Moreover, Lemma 4.4.1. of [CLT10] tells us that Z G ( R ) H ∞ ( s − m ( a ) , g ∞ ) − d g ∞ , is holomorphic on T Ω ′ . Thus we can conclude our lemma. (cid:3) Lemma 5.10.
The Euler product Y p b H p ( s , λ, t, α ) · b H ∞ ( s , t, α ) is holomorphic on T Ω ′ . Proof.
First we prove that the Euler product is holomorphic on T Ω ′ . To concludethis, we only need to discuss: Y p/ ∈ S ∪ S , | α | p =1 , b H p ( s , λ, t, α ) , where S = { p : p | e ι for some ι ∈ I } . Let p be a prime such that p / ∈ S ∪ S and | α | p = 1. Fix a compact subset of Ω ′ , and assume that ℜ ( s ) is sitting in thatcompact set. From the definition of Ω ′ , there exists ǫ > ( s ι − d ι + 1 > ǫ for any ι ∈ I s ι − d ι + 1 > ǫ for any ι ∈ I . Since we have {| a | p ≤ } = X ( Q p ) \ ρ − ( ∪ ι ∈I D ι ( F p )) , we can conclude that b H p ( s , λ, α ) = X ˜ x/ ∈∪ ι ∈I D ι ( F p ) Z ρ − (˜ x ) H p ( s , g p ) − ψ p ( αx p ) λ S ( a p )d g p . It is easy to see that X ˜ x/ ∈∪ ι ∈I D ι ( F p ) Z ρ − (˜ x ) = Z G ( Z p ) g p = 1 . Also it follows from a formula of J. Denef (see [Den87, Theorem 3.1] or [CLT10,Proposition 4.1.7]) and Lemma 9.4 in [CLT02] that there exists an uniform bound
C > x ∈ ∪ ι ∈I D ι ( F p ) , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ρ − (˜ x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < Z ρ − (˜ x ) H p ( ℜ ( s ) , g p ) − d g p < Cp ǫ . Hence we need to obtain uniform bounds of R ρ − (˜ x ) for˜ x ∈ ∪ ι ∈I D ι ( F p ) \ ∪ ι ∈I ∪I D ι ( F p ) . Let ˜ x ∈ D ι ( F p ) for some ι ∈ I , but ˜ x / ∈ ∪ ι ∈I ∪I D ι ( F p ) ∪ E ( F p ). Then it followsfrom Lemmas 4.2 and 5.7 that Z ρ − (˜ x ) = (cid:18) − p (cid:19) − Z m p | y | s ι − d ι p ψ p ( u/y e ι ) d y p d z p = 1 p − Z m p | y | s ι − d ι p Z Z × p ψ p ( ub e ι /y e ι ) d b × p d y p = ( e ι > − p − ( sι − dι +2) p − if e ι = 1 . EIGHT ZETA FUNCTIONS 35
If ˜ x ∈ D ι ( F p ) ∩ E ( F p ) for some ι ∈ I , then we have Z ρ − (˜ x ) = (cid:18) − p (cid:19) − Z m p | y | s ι − d ι p ψ p ( z/y e ι )d y p d z p = (cid:18) − p (cid:19) − Z m p | y | s ι − d ι p Z m p ψ p ( z/y e ι ) d z p d y p = ( e ι > p − ( s ι − d ι +2) if e ι = 1 . If ˜ x ∈ D ι ( F p ) ∩ D ι ′ ( F p ) for some ι, ι ′ ∈ I , then it follows from Lemma 5.7 that Z ρ − (˜ x ) = (cid:18) − p (cid:19) − Z m p | y | s ι − d ι p | z | s ι ′ − d ι ′ p ψ p (cid:18) uy e ι z e ι ′ (cid:19) d y p d z p = (cid:18) − p (cid:19) − Z m p | y | s ι − d ι p | z | s ι ′ − d ι ′ p Z Z × p ψ p (cid:18) ub e ι y e ι z e ι ′ (cid:19) d b × p d y p d z p = 0 . Thus we can conclude from these estimates and Lemma 9.4 in [CLT02] that thereexists an uniform bound C ′ > (cid:12)(cid:12)(cid:12)b H p ( s , λ, t, α ) − (cid:12)(cid:12)(cid:12) < C ′ p ǫ Our assertion follows from this. (cid:3)
Lemma 5.11.
Let Ω ′ ǫ be an open convex cone, defined by s ι − d ι − − ǫ > ι ∈ I s ι − d ι + 2 + 2 ǫ > ι ∈ I s ι − d ι + 1 > ι ∈ I where ǫ > is sufficiently small. Fix a compact subset of Ω ′ ǫ and ǫ ≫ δ > . Thenthere exists a constant C > such that | Y p b H p ( s , λ, t, α ) · b H ∞ ( s , α, t ) | < C (1 + t ) | β | − ǫ − δ | γ | ǫ − δ , for ℜ ( s ) in that compact set, where α = βγ with gcd( β, γ ) = 1 .Proof. This lemma follows from Lemmas 5.6, 5.8, and 5.9, and from the proof ofLemma 5.10. (cid:3)
Theorem 5.12.
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