Distribution of rational points of bounded height on equivariant compactifications of PGL 2 I

We study the distribution of rational points of bounded height on a one-sided equivariant compactification of PGL 2 using automorphic representation theory of PGL 2 .


Introduction
A driving problem in Diophantine geometry is to find asymptotic formulae for the number of rational points on a projective variety X with respect to a height function.In [BM90], Batyrev and Manin formulated a conjecture relating the generic distribution of rational points of bounded height to certain geometric invariants on the underlying varieties.This conjecture has stimulated several research directions and has lead to the development of tools in analytic number theory, spectral theory, and ergodic theory.Although the strongest form of Manin's conjecture is known to be false (e.g., [BT96b], [BL15], and [LR15]), there are no counterexamples of Manin's conjecture in the class of equivariant compactifications of homogeneous spaces whose stabilizers are connected subgroups.
There are mainly two approaches to the study of the distribution of rational points on equivariant compactifications of homogeneous spaces.One is the method of ergodic theory and mixing, (e.g., [GMO08], [GO11], and [GTBT15]), which posits that the counting function of rational points of bounded height should be approximated by the volume function of height balls.This method has been successfully applied to prove Manin's conjecture for wonderful compactifications of semisimple groups.The other approach is the method of height zeta functions and spectral theory (e.g., [STBT07] and [STBT04] as well as [BT98a], [BT96a], [CLT02], and [ST15]); this method solves cases of toric varieties, equivariant compactifications of vector groups, biequivariant compactifications of unipotent groups, and wonderful compactifications of semisimple groups.
In all of these results, one works with a compactification X of a group G that are bi-equivariant, i.e. the right and left action of G on itself by multiplication extends to X.The study of one-sided equivariant compactifications remains largely open, and the only result in this direction is [TT12] which treats the case of the ax + b-group under some technical conditions.
From a geometric point of view, one-sided equivariant compactifications of reductive groups are different from bi-equivariant compactifications, and their birational geometry is more complicated.For example, in the case of bi-equivariant compactifications of reductive groups, the cone of effective divisors is generated by boundary components.In particular, when reductive groups have no character, the cone of effective divisors is a simplicial cone.However, this feature is absent for one-sided equivariant compactifications of reductive groups, and one can have more complicated cones for these classes of varieties.This has a serious impact on the analysis of rational points.
In all previous cases where spectral theory or ergodic theory are applied, the main term of the asymptotic formula for the counting function associated to the anticanonical class arises from the trivial representation component of the spectral expansion of the height zeta function, assuming that the group considered has no character.The trivial representation component has been studied by Chambert-Loir and Tschinkel in [CLT10].They showed that when the cone of effective divisors is generated by boundary components, the trivial representation component coincides with Manin's prediction.However, if the cone of effective divisors is not generated by boundary components, Date: December 23, 2015.
1 then the trivial representation component does not suffice to account for the main term of the height zeta function.
In this paper, we study a one-sided equivariant compactification of PGL 2 whose cone of effective divisors is not generated by boundary components.We use the height zeta functions method and automorphic representation theory of PGL 2 .A new feature is that for the height function associated with the anti-canonical class, the main pole of the zeta function comes not from the trivial representation, but from constant terms of Eisenstein series; indeed, the contribution of the trivial representation is cancelled by a certain residue of Eisenstein series, c.f. §5 for details.In particular, it would appear as though ergodic theory methods cannot shed light on this situation, as, at least in principle, these methods only study the contribution of one-dimensional representations.
Let us express our main result in qualitative terms: Theorem 1.1.Let X be the blow up of P 3 along a line defined over Q.The variety X satisfies Manin's conjecture, with Peyre's constant, for any big line bundle over Q.
The blow up of P 3 along a line is a toric variety and an equivariant compactification of a vector group and thus our result is covered by previous works on Manin's conjecture.However, our proof is new in the sense that we explicitly used the structure of one-sided equivariant compactifications of PGL 2 , but not biequivariant, and we develop a method using automorphic representation theory of PGL 2 , which can be applied to more general examples of equivariant compactifications of PGL 2 , some of which are not covered by previous works.
Let us describe our result more precisely.Consider the following equivariant compactification of G = PGL 2 : The boundary divisor D is a quadric surface defined by ad − bc = 0. We consider the line l on D defined by We let X be the blowup of P 3 along the line l.Note that the left action of PGL 2 on P 3 acts transitively on lines in the ruling of l, so geometrically their blow ups are isomorphic to each other.The smooth projective threefold X is an equivariant compactification of PGL 2 over Q and the natural right action on PGL 2 extends to X.The variety X extends to a smooth projective scheme over Spec(Z) and the action of PGL 2 also naturally extends to this integral model.We let U be the open set consisting of the image of PGL 2 in X.
We denote the strict transformation of D by D, and the exceptional divisor by E. The boundary divisors D and E generate Pic(X) Q , however, the boundary divisors do not generate the cone of effective divisors Λ eff (X).The cone of effective divisors is generated by E and P = 1 2 D − 1 2 E which corresponds to the projection to P 1 .
Let F be a number field.For an archimedean place v ∈ Val(F), the height functions are defined by For a non-archimedean place v ∈ Val(F), we have Thus the local height pairing is given by For ease of reference, we let The complexified height function: The global height paring is given by The anti-canonical class −K X is equal to 2 D + E, and as such we have A more precise version of the theorem is the following: Theorem 1.2.Let C be a real number defined by Then there is an η > 0 such that as B → ∞, We will show in §7 that this is indeed compatible with the conjecture of Peyre [Pey95].
Our method is based on the spectral analysis of the height zeta function given by Namely, for g ∈ PGL 2 (A), we let For ℜs, ℜw large, Z(.; s, w) is in L 2 (PGL 2 (Q)\PGL 2 (A)) and is continuous on PGL 2 (A).We will then use the spectral theory of automorphic functions to analytically continue Z to a large domain.
The main result will be a corollary of the following general statement: Theorem 1.3.Our height zeta function has the following decomposition: with Φ(s, w) a function holomorphic for ℜs > 2 − ǫ and ℜ(s + w) > 2 for some ǫ > 0.Here Λ is the completed Riemann zeta function defined in §2.1, and E(s, g) is the Eisenstein series defined in §2.4.
As in previous works, the proof of the theorem is based on the the spectral decomposition theorem proved in [STBT04] and some approximation of Ramanujan conjecture [Sar05].But a new idea is needed here.Recall that the method of [STBT04] is based on the analysis of matrix coefficients-what facilitates this is the fact that the height functions considered there are bi-Kinvariant.Namely, we need to find bounds for integrals of the form (1.1) H(g, s) −1 φ(g) dg with φ(g) a cusp form.If H is left-K-invariant, then we may write the integral as The function g → K φ(kg) dk is roughly a linear combination of products of local spherical functions coming from various local components of automorphic representations.Approximations to the Ramanujan conjecture give us bounds for spherical functions, and this in turn gives rise to appropriate bounds for our integrals.
As mentioned above, the height functions we consider here are not bi-K-invariant.We use representation theoretic versions of Whittaker functions, which are the adelic analogues of Fourier expansions of holomorphic modular forms.The idea is to write with ψ : A → C × the standard non-trivial additive character.Using these Whittaker functions we can write the integral (1.1) as the infinite sum Whittaker functions are Euler products, and there are explicit formulae for the local components of these functions expressing their values in terms of the Satake parameters of local representations.Again, approximations to the Ramanujan conjecture are used to estimate these local functions.Even though for the sake of clarity we state our results over Q, everything we do goes through with little or no change for an arbitrary number field F .The local computations of §3 and §4 and the global results of §4 and §5 remain valid.The spectral decomposition of §6 needs to be adjusted in the following way.We have where the (finite) sum is over all unramified Hecke characters χ such that χ 2 = 1.When the class number of the field F is odd, e.g. when F = Q, the sum consists of a single term corresponding to χ = 1.At any rate, Lemma 3.1 shows that the only term that may contribute to the main pole is χ = 1.A computation as in the case of Q shows that the term coming from the trivial representation is cancelled, and we arrive at Theorem 1.3.Theorem 1.1 then immediately follows.
Except for the determination of the value of the constant C, Theorem 1.2 is valid as well.At this point we do not know how to compute the constant C in general as there is no Kronecker Limit Formula available for an arbitrary number field.We expect that our method has more applications to Manin's conjecture, and we plan to pursue these applications in a sequel [TBT15].A limitation of our method is that it can only be applied to the general linear group, as automorphic forms on other reductive groups typically do not possess Whittaker models, e.g.automorphic representations on symplectic groups of rank larger than one corresponding to holomorphic Siegel modular forms [HPS83].This paper is organized as follows.§2 contains some background information.The proof of the main theorem has four basic steps: Step 1, the analysis of the one dimensional representations presented in §3; Step 2, the analysis of cuspidal representations and Step 3, the analysis of Eisenstein series, presented, respectively, in §4 and §5; and finally Step 4, the spectral theory contained in §6 where we put the results of the previous sections together to prove the main theorem of the paper.In §7 we show that our results are compatible with the conjecture of Peyre.

Preliminaries
We assume that the reader is familiar with the basics of the theory of automorphic forms for PGL 2 at the level of [Gel75] or [God70].For ease of reference we recall here some facts and set up some notation that we will be using in the proof of the main theorem.2.2.An integration formula.We will need an integration formula.If H is a unimodular locally compact group, and S and T are two closed subgroups, such that ST covers H except for a set of measure zero, and S ∩ T is compact, then dx = d l s d r t is a Haar measure on H where d l s is a left invariant haar measure on S, and d r t is a right invariant haar measure on T .In particular, if T is unimodular, then dx = d l s dt is a Haar measure.We will apply this to the Iwasawa decomposition.
As we defined in the introduction, let G = PGL 2 .Suppose that F is a number field.For each place v ∈ Val(F ), we denote its completion by F v .Then we have the Iwasawa decomposition: where P is the standard Borel subgroup of G i.e., the closed subgroup of upper triangler matrices, and .) It follows from the integration formula that for any measurable function f on G(F v ), we have If v is a non-archimedean place and f is a function on PGL 2 (F v ) which is invariant on the right under K v then we have where ̟ is an uniformizer of F v .If v is an archimedean place, then instead we have We will use these integration formulae often without comment.
2.3.Whittaker models.Let N be the unipotent radical of the standard Borel subgroup in PGL 2 , i.e., the closed subgroup of upper triangler matrices.For a non-archimedean place v of Q, we let θ v be a non-trivial character of ) to be the space of smooth complex valued functions on PGL 2 (Q v ) satisfying ) is at most one dimensional; if the dimension is one, we say π is generic, and we call the corresponding realization of π as a space of N -quasi-invariant functions the Whittaker model of π.
We recall some facts from [God70], §16.Let π be an unramified principal series representation π = Ind G P (χ ⊗ χ −1 ), with χ unramified, where P is the standard Borel subgroup of G. Then π has a unique K v fixed vector.The image of this K v -fixed vector in the Whittaker model, call it W π , will be K v -invariant on the right, and N -quasi-invariant on the left.By Iwasawa decomposition in order to calculate the values of W π it suffices to know the values of the function along the diagonal subgroup.We have Written compactly we have Also by definition where where Let us also recall the automorphic Fourier expansion [Gel75, P. 85].If φ is a cusp form on PGL 2 we have 2.4.Eisenstein series.By Iwasawa decomposition, any element of PGL 2 (Q v ) can be written as where we have represented an element in A(Q v ) in the form We set We note that for γ ∈ P (Q), we have χ P (γg) = χ P (g) for any g ∈ G(A).Moreover, χ −1 P is the usual height on P 1 (Q) = P (Q)\G(Q) which is used in the study of height zeta functions for generalized flag varieties in [FMT89].Define the Eisenstein series E(s, g) by where χ(s, g) := χ P (g) s+1/2 .For later reference we note the Fourier expansion of the Eisenstein series [Gel75, Equation 3.10] in the following form: e 2πix dx, where w = 1 −1 is a representative for the longest element of the Weyl group.This integral converges when ℜ(s) is sufficiently large, and has an analytic continuation to an entire function of s.We also have the functional equation where g ∈ G(A).We note that The following lemma, generalized by Langlands [Lan66], is well-known: Lemma 2.2.We have , where e ∈ G(A) is the identity.
Proof.For any smooth compactly supported function f on (0, +∞), we define the Mellin transform by Mellin inversion says for σ ≫ 0 We also define for g ∈ G(A) We have As the residue of the Eisenstein series at s = 1/2 does not depend on g, we have We now calculate the integral of θ f a different way.We have where A c is the kernel of the norm N : A × → R × .Since Q has class number one, we conclude that vol(Q × \A c ) = 1.Comparing the two expressions for θ f gives the lemma.
2.5.Spectral expansion.Let f be a smooth bounded right K-and left PGL 2 (Q)-invariant function on PGL 2 (A) all of whose derivatives are also smooth and bounded.Here we recall a theorem from [STBT04] regarding the spectral decomposition of such a function.
We start by fixing a basis of right K-fixed functions for L 2 (G(Q)\G(A)).We write res the trivial representation part, L K cusp the cuspidal part, and L K eis the Eisenstein series part.An orthonormal basis of this space is the constant function The projection of f onto L K res is given by Next, we take an orthonormal basis {φ π } π for L K cusp where π runs over all automorphic cuspidal representations with a K-fixed vector.We have This is possible because dim(π K ) = 1 for any π.Indeed, by Tensor product theorem, we have π ∼ = ′ π v where π v is a local cuspidal representation.Taking the K-invariant part, we conclude Since π has a non-zero K-fixed vector, the local representation π v also has a non-zero K v -fixed vector.This implies that π v is the induced representation Ind G P (χ ⊗ χ −1 ) of some unramified character χ for P (F v ).Now it follows from the Iwasawa decomposition that dim(π Kv v ) = 1.
Finally we consider the projection onto the continuous spectrum.We have We then have as an identity of continuous functions.

Step one: one dimensional automorphic characters
In this step we study the function Z(s, g) res .We have We consider the local integral 3.1.Non-archimedean computation.Here we will use Chambert-Loir and Tschinkel's formula of height integrals.Let F be a number field.We fix the standard integral model G of PGL 2 over O F where O F is the ring of integers for F .Let ω be a top degree invariant form on G defined over O F which is a generator for Ω 3 G/Spec(OF ) .For any non-archimedean place v ∈ Val(F ), we denote its the v-adic completion by F v , the ring of integers by O v , the residue field by F v .We write the cardinality of F v by q v .For any uniformizer ̟,we have We denote this number by a v .Then the normalized Haar measure is .

An integral computation.
For use in a later section we compute a certain type of p-adic integral.Suppose we have a function f given by the following expression: for a fixed complex number τ .Here n ∈ N (F ) and k ∈ K where F is a local field.We would like to compute the integral By the integration formula this is equal to by the integration formula.We state this computation as a lemma: Lemma 3.1.For f as above we have if F is non-archimedean.In the case where F = R, the value of the integral is .

Step two: the cuspidal contribution
In this section π is an automorphic cuspidal representation of PGL 2 with a K-fixed vector.We denote by π K the space of K-fixed vectors in π which is one dimensional, and we let φ π be an orthonormal basis for π K .We let We have By the automorphic Fourier expansion we have The good thing about the use of the Whittaker function is that they have Euler products, so we may write: where π ∼ = ′ π v is the restricted product of local representations.For α ∈ F × v we set In this case, π v is an unramificed principal series representation, so it has the form of Ind G P (χ ⊗ χ −1 ), where χ is an unramified character of P (F v ) which only depends on π v .
We will need the following straightforward lemma: Lemma 4.1.For an unramified quasi-character η and y ∈ F v define where ψ v : F v → S 1 is the standard additive character.
the first integral is trivially zero, and the second integral is zero by the lemma.
We also calculate J πv (α) for α ∈ O × by hand.By what we saw above, Suppose that α ∈ O, i.e., α = ̟ k where k ≥ 0. Using the integration formula, we have Using an explicit computation of Whittaker functions, we have Then we decompose this infinite sum into two parts: where P is a Borel subgroup and dp is a right invariant Haar measure.Similar computation works for the second part, so we have Here each integral is given by where S is the Zariski closure of P in X.This type of integral is studied by the second author and Tschinkel in [TT12].They studied height zeta functions of equivariant compactifications of P under some geometric conditions.
The surface S is isomorphic to P 2 = {c = 0} ⊂ P 3 .The boundary divisors are Let ω be a right invariant top degree form on P .Let We denote the Zariski closures of S, F , D 1 , and E 1 in a smooth integral model X of X over Spec O by S, F , D 1 , and E 1 respectively.They form integral models of S, F , D 1 , and E 1 .Let ρ : S(F v ) → S(F v ) be the reduction map mod ̟.Then we have We analyze J + πv (α, r) following [TT12].When r ∈ G(F v ), we have r∈G(Fv) When r ∈ (D 1 \ E 1 )(F v ), we have where dτ v is the Tamagawa measure.Then there exist analytic local coordinates y, z on ρ −1 (r) ∼ = m 2 v such that , then there exist local analytic coordinates y, z on ρ −1 (r) such that Suppose that r ∈ (D 1 ∩ E 1 )(F v ).In this case there exist local analytic coordinates y, z on ρ −1 (r) such that

Now we need the following lemma:
Lemma 4.2.
Using this lemma, we have .
Then there exist local analytic coordinates such that Putting everything together, we obtain the following ) The same formula holds for J − πv (α) by replacing χ with χ −1 .From these expressions, we conclude the proof of the following lemma: Lemma 4.3.Let 0 < δ < 1 2 be a positive real number such that Then the local integral J πv (α) is holomorphic in the domain ℜ(s) > 1 2 + δ.
We state an approximation of Ramanujan conjecture: Theorem 4.4.[LRS99] There exists a constant 0 < δ < 1 2 such that for any non-archimedean place v and any unramified principal series π v = Ind G P (χ ⊗ χ −1 ) arising as a local representation of an automorphic cuspidal representation π in L 2 cusp (G(Q)\G(A)), we have We proceed to compute J πv (α).
Now using Whittaker functions, we summarize these computations in the following way: To obtain an estimate of this integral, we use the following lemma: Lemma 4.5.Let 0 < δ < 1 2 be a positive real number such that Then we have We come to the conclusion of this section.We let and Lemma 4.6.Let K ⊂ Λ be a compact subset and α ∈ O \ O × .Then there exists a constant C K > 0 which does not depend on v, α and π v such that Proof.Suppose that (s, w) ∈ T K .Let k = v(α) > 0. We apply Lemma 4.5 to bound J πv (α): where C 1 is a constant which does not depend on v or α.Let 0 < ǫ < 1 2 − δ.Then there exists another constant C 2 such that Hence our assertion follows.
4.2.v archimedean.Let π ∞ be an unramified principal series Ind G P (|.| µ ⊗ |.| −µ ) with a K ∞ -fixed vector for some complex number µ.Its Whittaker function satisfies where ψ ∞ : R → S 1 is the standart additive character, and the function W π∞ t 1 is given by where K µ (t) is the modified Bessel function of the second kind.See [GH11, Proposition 7.3.3].However, later we normalize the Whittaker function at the archimedean place by W π∞ (e) = 1, so we may assume that The fact that K µ (t) decays exponentially as t → ∞ gives us the following lemma: Lemma 4.7.The local height integral J π∞ (α) is holomorphic in the domain defined by ℜ(s) > 1 and ℜ(s + w) > 0.
We extend this domain by using the rapidly decaying function K µ .Since our height is invariant under the action of K ∞ = SO 2 (R), using the integration formula we have where S is the Zariski closure of P in X.Let {U η } be a sufficiently fine finite open covering of S(R) and consider a partition of unity θ η subordinate to this covering.Using this partition of unity, we obtain (4.1) Suppose that U η meets with E 1 , but not D 1 .Then the term corresponding to η in (4.1) looks like where Φ(y, z, s, w) is a bounded function with a compact support.Since the Whittaker function decays rapidly, this function is holomorphic everywhere.Note that α is non-zero.
Next assume that U η contains the intersection of E 1 and D 1 .In this case, the term in (4.1) looks like Applying a change of variables by z = z ′ and y = y ′ z ′ , it becomes This integral is absolutely convergent if ℜ(s + w) > 0.
Lemma 4.8.Let ∂ X be a left invariant differential operator on X.Then the function Proof.See the proof of Proposition 2.2 in [CLT02].
Here we use an iterated integration idea.Let and think of them as elements of the universal enveloping algebra of the complexified Lie algebra of PGL 2 (R).The Casimir operator is given by Then we have Ω.W π∞ = λ π W π∞ .As a result for any integrable smooth function f such that f is right K ∞ -invariant and and its iterated derivatives are also integrable, we have Note that Ω N f (g) is right K ∞ -invariant because Ω is an element of the center of the universal enveloping algebra.Hence we have We will apply this simple idea to our height function We define the domain Λ by Lemma 4.9.Fix positive integers N and M .Let K be a compact set in Λ.Then there exists a constant C depending on N, M, K such that we have for all g ∈ G(R) and s, w such that (s, w) ∈ T K .
Proof.First note that ∂/∂x is a RIGHT invariant differential operator on P = a x 1 .Moreover the surface S is a biequivariant compactfication of P so that the differential operator ∂/∂x extends to S. Now our assertion follows from Lemma 4.8.
Combining these statements with the computation of H(g, s, w) −1 1 = Z, 1 ∞ gives the following lemma: Lemma 4.10.Fix positive integers N and M .Let K be a compact set in Λ.Then there exists a constant C only depending on N, M, K, but not π ∞ and α such that Proof.We need to obtain an upper bound for the integral First note that by an approximation of Ramanujan conjecture, there exists 0 < δ < 1/2 such that Suppose that ℜ(w) ≥ ǫ/2.Then the above integral is bounded by It follows from results in Section 8 that we have Finally note that λ π = (1/4 − µ 2 ), so our assertion follows.
Next suppose that there exists a positive integer m such that ǫ/2 ≤ ℜ(w) + m ≤ 1 + ǫ/2.In this situation, we have ℜ(s) − m ≥ 1 + ǫ/2.It follows from Lemma 3.1 that Again it follows from Section 8 that Thus our assertion follows.
4.3.The adelic analysis.As Z is right K-invariant, the only automorphic cuspidal representations that contribute to the automorphic Fourier expansion of Z are those that have a K-fixed vector.Let π = ⊗ ′ v π v be an automorphic cuspidal representation of PGL(2).By a theorem of Jacquet-Langlands every component π v is generic.See [JL70, Chapter 2, Proposition 9.2].Let W πv be the K v -fixed vector in the space of π v normalized so that W πv (e) = 1.Let φ π be the K-fixed vector in the space of π normalized so that φ π , φ π = 1.Then
and K be a compact set in Λ.Then there exists a constant C K,N > 0 independent of π such that Proof.To see this, define a multiplicative function F (α) by where C 1 is a constant in Lemma 4.6.By Lemma 4.6 and Lemma 4.10, we know that Thus we need to discuss the convergence of α F (α) α M .Formally this infinite sum is given by Then the product absolutely converges as soon as M > 1.Thus we need to show the convergence of (4.2) Let ǫ > 0 be a sufficiently small positive real number.Then there exists a constant C 3 such that k p kǫ ≤ C 3 for any k and p.Then the infinite sum (4.2) is bounded by The last infinite sum converges if M > 1 and ǫ is sufficiently small.Now look at the cuspidal contribution Lemma 4.12.The series Z(s) cusp is absolutely uniformaly convergent for ℜ(s) > 3/2 + δ and ℜ(s + w) > 2, and defines a holomorphic function in that region.
Proof.We have by the rapid decay of the K-Bessel function.This last series converges for N large.See Theorem 7.4 of [STBT04].

Step three: Eisenstein contribution
For ease of reference we set where s ∈ C, g ∈ G(A), and χ P , W s are introduced in Section 2.4.Note that we have the Fourier expansion of the Eisenstein series: We will be considering integrals of the form First we examine the inner integral.As usual we have 5.1.The non-constant term.We let We define For any non-archimedean place v, W s,v satisfies the exactly same formula for the Whittaker function W πv of a local unramified principal seires π v = Ind G P (χ ⊗ χ −1 ) by replacing χ by | • | s v , i.e., we have Hence the computation in Section 4.1 can be applied to J t,v (α) without any modification.We summarize this fact as the following lemma: Lemma 5.1.Let v be a non-archimedean place.The function In general, let Λ = {(x, y) ∈ R 2 | x > 3/2, x+ y > 2} and K be a compact subset in Λ.We define ρ = max{−ℜ(w) | (s, w) ∈ T K }.Tnen there exists a constant C K > 0 which does not depend on t, v, α such that For the archimedean place v = ∞, the Whittaker function is given by Moreover for the Casimir operator Ω, we have As the discussion of Section 4.2, we conclude Lemma 5.2.The function J t,∞ (α) is holomorphic in the domain ℜ(s) > 1 and ℜ(s + w) > 0. Fix positive integers N and M .Let K be a compact set in Λ.Then there exists a constant C N,M,K only depending on N, M, K, but not t and α such that Finally the discussion of Section 4.3 leads to the following lemma: Lemma 5.3.The series I nc (s, t) is absolutely convergent for ℜ(s) > 3/2 + ǫ and ℜ(s + w) > 2 + ǫ for any real number ǫ > 0. Furthermore, let K be a compact subset in Λ.Then there is a C K,N > 0 independent of t such that The results of §4 and §5 of [STBT04] show that is holomorphic in the domain ℜ(s) > 3/2 and ℜ(s + w) > 2. Note that it follows from Theorem 8.4 that |ζ(1 + 2it)| −1 ≪ log |t|, so this does not affect convergence.
5.2.The constant term.The issue is now understanding The last equality follows from Lemma 3.1.We then have after using the functional equation for the Eisenstein series.We denote the vertical line defined by ℜ(z) = a by (a) for any real number a ∈ R. Assume ℜs, ℜw ≫ 0. We then have Note that |E(y, g)| ≤ E(ℜy, g).In order to see the convergence of the last integral we rewrite it as This last integral is easily seen to be convergent by Stirling's formula.

Step four: spectral expansion
We start by fixing a basis of right K-fixed functions for L 2 (G(Q)\G(A)).We write res is the trivial representation.An orthonormal basis of this space is the constant function The projection of Z(s, g) onto L K res is given by Z(s, g) res = Z(s, •), φ res φ res (g) = 1 vol (G(Q)\G(A)) Λ(s − 1)Λ(w) Λ(s + w) .
Finally we consider the projection onto the continuous spectrum.We have The discussion in Section 5.2 that with f (s, w, g) a function which is holomorphic in the domain ℜ(s) > 3/2 and ℜ(s + w) > 2.
Consequently, we have proved the following statement: 6.1.The proof of Theorem 1.2.We now proceed to prove Theorem 1.2 without the determination of the constant C. We restrict the function Z(s, w) to the line s = 2w, and determine the order of pole and the leading term at w = 1.We have The function is holomorphic in the domain ℜw > 3/4 + δ/2.The function The asymptotic formula, modulo the determination of the constant C, now follows from Theorem A.1 of [CLT01].A standard computation, as presented in e.g. the proof of Theorem 1 of [DLS05], shows that with g(w) holomorphic for ℜw > 1 − η.As a result, for g(w) holomorphic in an open half plane containing w = 1.So, in order to determine C we need to determine the residue of the function appearing on the left hand side of this equation.
A straightforward computation shows with ϑ 1 (w) holomorphic in a neighborhood of w = 1.Here γ is the Euler constant.
We then recall the Kronecker Limit Formula, Theorem 10.4.6 of [Coh07] and also Chapter 1 of [Sie80], in the following form.We have with ϑ 2 (s) a function holomorphic in a neighborhood of s = 1/2.The function C(g) is described as follows.Since Q has class number 1, we have Given g ∈ PGL 2 (A), we choose a representative g ′ ∈ GL 2 (A), we write g We then set τ (g) = g ∞ • i where the action is given by We let y(g) = ℑτ (g).We then have Multiplying out, we get with ϑ 3 (s) a function holomorphic in a neighborhood of s = 1/2.Hence, with ϑ 4 (w) holomorphic in a neighborhood of w = 1.We also have Also we have the well-known identity Putting everything together we conclude the following lemma: Lemma 6.1.We have where ϑ 5 (w) is a holomorphic function in the domain ℜ(w) > 1−η for some η > 0 and the constant A is given by 7. Manin's conjecture with Peyre's constant 7.1.The anticanonical class.In this section, we verify that the anticanonical class satisfies Manin's conjecture with Peyre's constant, hence finishing the proof of Theorem 1.1 for the anticanonical class.For the definition of Peyre's constant, see [Pey95].The Néron-Severi lattice NS(X) is generated by E and H where H is the pullback of the hyperplane class on P 3 .Let N 1 (X) is the dual lattice of NS(X) which is generated by the dual basis E * and H * .The cone of effective divisors Λ eff (X) ⊂ NS(X) R is generated by E and H − E. We denote the dual cone of the cone of effective divisors by Nef 1 (X).Any element of this dual cone is called a nef class.Let da be the normalized haar measure on N 1 (X) R such that vol(N 1 (X) R /N 1 (X)) = 1.The alpha invariant (see [Tsc09, Definition 4.12.2]) is given by α(X) = Next we compute the Tamagawa number.Let ω ∈ Γ(PGL 2 , Ω 3 PGL2/Spec(Z) ) be a nonzero relative top degree invariant form over Spec(Z).This is unique up to sign.Our height induces a natural metrization on O(K X ), and it follows from the construction that The normalized haar measure dg v is given by |ω| v /a v at non-archimedean places, hence we have τ .
For the infinite place v = ∞, we have For any non-archimedean place v, the local L-function at v is given by Thus the leading constant is 7.2.Other big line bundles: the rigid case.Consider the following Q-divisor: where x, y are rational numbers.The divisor L is big if and only if x > 0 and x + y > 0. We define the following invariant: It follows from Theorem 1.3 that the height zeta function Z(sL) has a pole at s = a(L) of order b(L).Thus, to verify Manin's conjecture, the only issue is the leading constant i.e., the residue of Z(sL).Tamagawa numbers for general big line bundles are introduced by Batyrev and Tschinkel in [BT98b].(See [Tsc09, Section 4.14] as well.)The definition is quite different depending on whether the adjoint divisor a(L)L + K X is rigid or not.Here we assume that a(L)L + K X is a non-zero rigid effective Q-divisor.
In this case, the adjoint divisor a(L)L + K X is proportional to E. This happens if and only if 2y − x > 0. In this situation, we have a(L) = 2/x and Z(sL) has a pole at s = a(L) of order one.By Theorem 1.3, we have where X • = X \E, and τ X • is the Tamagawa measure on X • .Thus the Tamagawa number τ (X, L) in the sense of [BT98b] is given by τ (X, L) = H(x, a(L)L + K X ) −1 dτ X • = 6 π Λ(2y/x) Λ(2 + 2y/x) .
On the other hand, the alpha invariant is given by α(X, L) = 1 2x .
7.3.The non-rigid case.Again, we consider a big Q-divisor L = x D + yE, and assume that a(L)L + K X is not rigid, i.e., some multiple of a(L)L + K X defines the Iitaka fibration.This happens exactly when 2y − x < 0 and a(L) is given by 3/(x + y).The adjoint divisor a(L)L + K X is proportional to D − E which is semiample and defines a morphism f : X → P \G = P 1 .It follows from Theorem 1. H(h ∞ g z , s = 2, w = 1) −1 d|σ| ∞ .
Note that g z ∈ G(Z).Then we have dτ S,∞ .
Let H be the pull back of the ample generator via f : X → P 1 .Then we have a(L)L + K X ∼ 2 x−2y x+y H.We conclude that c(X, L) = z∈P 1 (Q) α(S z , L)H(g z , a(L)L + K X ) −1 τ (S z , L) We have verified Manin's conjecture in this case.

Appendix: Special functions
Here we collect some important facts about special functions which we use in our analysis of the height zeta function.First we state the Stirling formula for the Gamma function: