Manin's conjecture for a quartic del Pezzo surface with A_4 singularity
aa r X i v : . [ m a t h . N T ] J a n MANIN’S CONJECTURE FOR A QUARTIC DEL PEZZOSURFACE WITH A SINGULARITY
T.D. BROWNING AND U. DERENTHAL
Abstract.
The Manin conjecture is established for a split singular delPezzo surface of degree four, with singularity type A . Contents
1. Introduction 12. Calculation of Peyre’s constant 43. Arithmetic functions 64. The universal torsor 85. The main argument 115.1. Real-valued functions 125.2. Estimating N a ( B ) and N b ( B ) — first step 145.3. Estimating N a ( B ) — second step 165.4. Estimating N b ( B ) — second step 215.5. The final step 24References 291. Introduction
The distribution of rational points on del Pezzo surfaces is a challeng-ing topic that has enjoyed a surge of activity in recent years. Guided bythe largely unverified conjectures of Manin [11] and his collaborators, theprimary aim of this paper is to investigate further the situation for splitsingular del Pezzo surfaces of degree 4 in P , that are defined over Q . Ourmain achievement will be a proof of the Manin conjecture for the surface x x − x x = x x + x x + x = 0 , (1.1)which we denote by S ⊂ P . This surface contains a unique singularity oftype A and exactly three lines, all of which are defined over Q .Let U be the Zariski open subset formed by deleting the lines from S , andlet N U,H ( B ) := { x ∈ U ( Q ) | H ( x ) B } , for any B >
1. Here H is the usual height on P , in which the height H ( x )is defined as max {| x | , . . . , | x |} for a point x = ( x : . . . : x ) ∈ U ( Q ),provided that x = ( x , . . . , x ) has integral coordinates that are relativelycoprime. Bearing this in mind, the following is our principal result. Mathematics Subject Classification.
Theorem.
We have N U,H ( B ) = c S,H B (log B ) + O (cid:0) B (log B ) − / (cid:1) , where c S,H = 121600 · ω ∞ · Y p (cid:18) − p (cid:19) (cid:18) p + 1 p (cid:19) and ω ∞ = Z | t | , | t t t | , | t ( t t + t ) | , | t t | , 1. Moreover, theconstants α ( f X ) and ω H ( f X ) are those predicted by Peyre [14]. Note thatthe exponent of log B agrees with the statement of the theorem. We shallverify in § c S,H = α ( e S ) ω H ( e S ) in this result.An overview of progress relating to the Manin conjecture for arbitrarydel Pezzo surfaces can be found in the first author’s survey [3]. The presentpaper should be seen as a modest step on the path to its resolution for thesingular del Pezzo surfaces of degree 4 that are split over Q . Accordingto the classification of such surfaces found in Coray and Tsfasman [6], ittranspires that there are 15 possible singularity types for split singular delPezzo surfaces of degree 4. It follows from the work of Batyrev and Tschinkel[1], la Bret`eche and the first author [2], and the second author’s joint workwith Tschinkel [9], that the Manin conjecture is already known to hold for5 explicit surfaces from this catalogue. In view of our theorem, which dealswith a surface of singularity type A , it remains to deal with the split quarticdel Pezzo surfaces that have singularity types A n for n ∈ { , , } , A , A + A n for n ∈ { , , } . (1.4)Here one should note that there are two types of surfaces that have sin-gularity type A , one containing four lines and one containing five lines.Similarly, there are two types that have 2 A singularities.The surface that we have chosen to focus on in the present investigationsatisfies the property that the cone of effective divisors associated to theminimal desingularisation e S is not merely generated by the divisors thatform a basis for the Picard group Pic( e S ), but requires one further divisor togenerate it. This leads to some additional considerations in the proof, as wewill see shortly.The proof of the theorem uses a universal torsor. For each split del Pezzosurface of degree d , there is one (essentially unique) universal torsor, whichis always an open subset of a (12 − d )-dimensional affine variety. For toric ANIN’S CONJECTURE FOR A QUARTIC DEL PEZZO SURFACE 3 varieties, universal torsors are open subsets of affine space. Salberger [15]has shown how to establish Manin’s conjecture using universal torsors forsplit toric varieties defined over Q . As a step towards handling non-toric delPezzo surfaces that still have a relatively simple universal torsor, the secondauthor [8] has determined which del Pezzo surfaces of degree at least 3 havea universal torsor that can be described as a hypersurface in A − d . Out ofthe singularity types in (1.4), these include those surfaces of type A + A , A + A , 3 A and the A surface with five lines. The surfaces of type D and D considered in [2] and [9] also belong to this class, as does the A surface S considered here. In fact we will see in § η α + η α + η η η η = 0 , (1.5)which is embedded in A ∼ = Spec Q [ η , . . . , η , α , α ]. Note that one of thevariables does not explicitly appear in the equation.Our basic strategy is similar to the one used for the D and D quartic delPezzo surfaces. The first step is to establish an explicit bijection betweenthe rational points outside the lines on S and certain integral points onthe universal torsor. We adopt the approach of Tschinkel and the secondauthor [9] in order to obtain this bijection in an elementary way, motivatedby the structure of the minimal desingularisation e S as a blow-up of P infive points. The integral points on the universal torsor are counted in § η , . . . , η , α , α must satisfy (1.5), together with certaincoprimality and height conditions. The first step is to fix the variables η , . . . , η and to estimate the relevant number of α , α by viewing theequation as a congruence modulo η . The resulting estimate is then summedover the remaining variables.The order in which we handle the remaining variables is crucial and subtle.When it comes to summing over η and η we will run into trouble controllingthe overall contribution from the error term each time, because both η and η can be rather big. Summing the number of α , α over η , for example,leads to an error term that we cannot estimate in a way that is sufficientlysmall when summed over η , . . . , η and large values of η . In line withthis we shall let the order of summation depend on which of η or η haslargest absolute value. When it comes to summing the integral points onthe universal torsor that satisfy | η | > | η | , we sum first over η and thenover η . For the alternative contribution we sum first over η and then over η . This process leads to two main terms that we put back together to getsomething of the general shape M ( η , . . . , η ) := ω H ( e S ) · Bη η η η η , (1.6)where ω H ( e S ) is as in (1.3). The final task is to sum this quantity over theremaining variables η , . . . , η .While essentially routine, it is in this final analysis that a further inter-esting feature of the proof of the theorem is revealed. For k ∈ Z > , define T.D. BROWNING AND U. DERENTHAL the simplex P k := (cid:8) ( x , . . . , x ) ∈ R | x i > , k x + · · · + k x (cid:9) , (1.7)whose volume is easily determined asvol( P k ) = 15! · k · k · k · k · k . In § α ( e S ) = vol( P (2 , , , , ) − vol( P (3 , , , , ), whence α ( e S ) = 15! · · · · · − · · · · · . (1.8)Returning to the summation of (1.6) over η , . . . , η ∈ Z > , which is subjectto η η η η η B , it will transpire that there is a negligible contribu-tion from those η , . . . , η for which η η η η η > B . Summing over the η , . . . , η ∈ Z > that are remaining therefore leads to the final main term (cid:0) vol( P (2 , , , , ) − vol( P (3 , , , , ) (cid:1) · ω H ( e S ) B (log B ) , as expected. Thus the main term in the asymptotic formula is really adifference of two main terms that conspire to give the predicted value for α ( e S ). It would be interesting to see whether the same sort of phenomenonoccurs for other split del Pezzo surfaces of degree 4, with singularity typeamong the list (1.4). Acknowledgements. The authors are extremely grateful to the anony-mous referee for his careful reading of the manuscript and numerous helpfulcomments. While working on this paper the first author was supportedby EPSRC grant number EP/E053262/1 . The second author was partiallysupported by a Feodor Lynen Research Fellowship of the Alexander vonHumboldt Foundation.2. Calculation of Peyre’s constant In this section we wish to show that the value of the constant c S,H ob-tained in our theorem is in agreement with the prediction (1.3) of Peyre[14]. Beginning with the value of ω H ( e S ), whose precise definition we willnot include here but which corresponds to a product of local densities, wehave ω H ( e S ) = ω ∞ Y p (cid:16) − p (cid:17) ω p , (2.1)where ω ∞ and ω p are the real and p -adic densities, respectively. The calcu-lation of ω p is routine and leads to the conclusion that ω p = 1 + 6 p + 1 p . The reader is referred to [2, § 2] for an analogous calculation. We now turnto the calculation of ω ∞ , which needs to agree with (1.2).Recall the equations (1.1) for the surface S , and write f ( x ) = x x − x x and f ( x ) = x x + x x + x . To compute ω ∞ , we parametrise the pointsby writing x , x as functions of x , x , x . Thus we have x = x x x , x = − x x + x x = − x x + x x x , ANIN’S CONJECTURE FOR A QUARTIC DEL PEZZO SURFACE 5 and furthermore, det ∂f ∂x ∂f ∂x ∂f ∂x ∂f ∂x ! = det (cid:18) x x x (cid:19) = x . Since x and − x have the same image in P , we have ω ∞ = 12 Z | x | , ˛˛˛ x x x ˛˛˛ , | x | , | x | , ˛˛˛˛ x x x x x ˛˛˛˛ x − d x d x d x = 12 Z | t | , | t t t | , | t | , | t t | , | t t + t t | d t d t d t , on carrying out the change of variables x = t , x = t and x = t t . Butthe range of integration is symmetric with respect to the transformation( t , t , t ) ( t , − t , − t ), and so we may restrict to the range t > 0. Thistherefore confirms the equality in (1.2).It remains to deal with the constant α ( e S ) that appears in (1.3). As we’vealready commented, the Picard group Pic( e S ) of e S has rank 6. Distinguishedelements of Pic( e S ) are the classes of irreducible curves with negative selfintersection number. As described in § 4, these are the classes of four ex-ceptional divisors E , . . . , E coming from the A -singularity of S and thetransforms E , E , E of the three lines on S . By the work of the second au-thor [8, § E , . . . , E form a basis of Pic( e S ). In terms of this basis we have E = E +2 E + E +2 E − E and − K e S = 2 E +4 E +3 E +2 E +3 E + E .The convex cone in Pic( e S ) R := Pic( e S ) ⊗ Z R generated by classes of ef-fective divisors is generated by E , . . . , E (see [10, Theorem 3.10]). Theintersection of its dual with the hyperplane { x ∈ Pic( e S ) R | ( x, − K e S ) = 1 } is a polytope P whose volume is the constant α ( e S ) defined by Peyre [14].By definition P = (cid:26) ( x , . . . , x ) ∈ Pic( e S ) R | x i > , x + 2 x + x + 2 x − x > , x + 4 x + 3 x + 2 x + 3 x + x = 1 (cid:27) . Eliminating the last coordinate shows that P is isomorphic to P ′ = (cid:26) ( x , . . . , x ) ∈ R | x i > , x + 4 x + 3 x + 2 x + 3 x , x + 6 x + 4 x + 2 x + 5 x > (cid:27) . Analyzing the volume form with respect to which we must compute thevolume of P in order to obtain α ( e S ) (see [10, Section 2], for example), wesee that α ( e S ) = vol( P ′ ) = vol( P (2 , , , , ) − vol( P (3 , , , , ) , in the notation of (1.7). This therefore establishes (1.8).An alternative approach to calculating α ( e S ) is available to us throughrecent work of Joyce, Teitler and the second author [10]. Recall from [7,Table 1] that α ( S ) = 1 / 180 for any non-singular split del Pezzo surface S of degree 4. Since the order of the Weyl group associated to the root T.D. BROWNING AND U. DERENTHAL system A n is ( n + 1)!, as recorded in [10, Table 2], so it follows from [10,Theorem 1.3] that α ( e S ) = 1180 · 15! = 121600 . This completes the verification that our theorem confirms the Manin con-jecture for the split A surface (1.1).3. Arithmetic functions In this section we present some elementary facts about certain arithmeticfunctions and their average order, as required for our argument. Define themultiplicative arithmetic functions φ ∗ ( n ) := Y p | n (cid:16) − p (cid:17) , φ † ( n ) := Y p | n (cid:16) p (cid:17) . Both of these functions have average order O (1), and one has X n x φ † ( n ) j n ≪ j log x, (3.1)for any x > j ∈ Z > . To see this we note that φ † ( n ) P d | n /d ,whence X n x φ † ( n ) j n X n x n X d ,...,d j | n d . . . d j ∞ X d ,...,d j =1 d . . . d j [ d , . . . , d j ] X e x e , where [ d , . . . , d j ] denotes the least common multiple of d , . . . , d j . Therequired bound (3.1) then follows from the estimate ∞ X d ,...,d j =1 d . . . d j [ d , . . . , d j ] ∞ X d ,...,d j =1 d . . . d j ) /j ≪ j . For given positive integers a, b , our work will lead us to work with thefunction f a,b ( n ) := ( φ ∗ ( n ) /φ ∗ (gcd( n, a )) , if gcd( n, b ) = 1 , , if gcd( n, b ) > . (3.2)We begin by establishing the following result. Lemma 1. Let I = [ t , t ] , for t < t . Let α ∈ Z such that gcd( α, q ) = 1 .Then we have X n ∈ I ∩ Z n ≡ α (mod q ) f a,b ( n ) = t − t q c + O (cid:0) ω ( b ) log | I | (cid:1) , where | I | := 2 + max {| t | , | t |} and c = φ ∗ ( b ) φ ∗ (gcd( b, q )) ζ (2) Y p | abq (cid:16) − p (cid:17) − . (3.3) ANIN’S CONJECTURE FOR A QUARTIC DEL PEZZO SURFACE 7 Proof. We will follow the convention that µ ( − n ) = µ ( n ) and µ (0) = 0. Webegin by calculating the Dirichlet convolution( f a,b ∗ µ )( n ) = X d | n f a,b ( d ) µ ( n/d ) = Y p ν k nν > (cid:0) f a,b ( p ν ) − f a,b ( p ν − ) (cid:1) . It is clear that f a,b (1) = 1 and f a,b ( p j ) = f a,b ( p ) = − /p, if p ∤ ab ,1 , if p ∤ b and p | a ,0 , if p | b ,for any j > 1. Hence it follows that( f a,b ∗ µ )( n ) = (cid:26) µ ( n ) gcd( b, n ) / | n | , if gcd( a, n ) | b ,0 , otherwise.In particular X n N | ( f a,b ∗ µ )( n ) | X n N gcd( b, n ) | µ ( n ) || n | ≪ ω ( b ) log N, for any N > 1. Since f a,b = ( f a,b ∗ µ ) ∗ 1, we therefore deduce that X n ∈ I ∩ Z n ≡ α (mod q ) f a,b ( n ) = ∞ X d =1gcd( d,q )=1 ( f a,b ∗ µ )( d ) X m ∈ d − I ∩ Z md ≡ α (mod q ) t − t q ∞ X d =1gcd( d,q )=1 ( f a,b ∗ µ )( d ) d + O (cid:0) ω ( b ) log | I | (cid:1) . Here we have observed that the outer sum in the first line is really a sumover d | I | , making the previous bound applicable for dealing with theerror term. We have then extended the summation over d to infinity, withacceptable error. Finally, it remains to observe that ∞ X d =1gcd( d,q )=1 ( f a,b ∗ µ )( d ) d = Y p (cid:16) − p (cid:17) Y p | abq (cid:16) − p (cid:17) − Y p | bp ∤ q (cid:16) − p (cid:17) = c , as required to complete the proof of the lemma. (cid:3) Rather than Lemma 1, we will actually need a corresponding estimatein which the summand is replaced by f a,b ( n ) g ( n ) , for suitable real-valuedfunctions g . This is supplied for us by the following result. Lemma 2. Let I = [ t , t ] , for t < t , and let g : I → R be any functionsuch that g has a continuous derivative on I which changes its sign only R g ( I ) < ∞ times on I . Let α ∈ Z such that gcd( α, q ) = 1 . Then we have X n ∈ I ∩ Z n ≡ α (mod q ) f a,b ( n ) g ( n ) = c q Z I g ( t ) d t + O (cid:0) ω ( b ) · (log | I | ) · M I ( g ) (cid:1) , with c given by (3.3) and M I ( g ) := (1 + R g ( I )) · sup t ∈ I | g ( t ) | . T.D. BROWNING AND U. DERENTHAL Proof. Let S denote the sum that is to be estimated, and write M ( t ) := X n tn ≡ α (mod q ) f a,b ( n ) , for any t > 0. By partial summation, S = M ( t ) g ( t ) − M ( t ) g ( t ) − Z t t M ( t ) g ′ ( t ) d t. An application of Lemma 1 reveals that M ( t ) = c t/q + O (2 ω ( b ) log(2 + | t | )).Hence partial integration yields S = c q Z I g ( t ) d t + O (cid:0) ω ( b ) · (log | I | ) · ( | g ( t ) | + | g ( t ) | + Z t t | g ′ ( t ) | d t ) (cid:1) . Splitting I into the R g intervals where g ′ has constant sign therefore com-pletes the proof of the lemma. (cid:3) The universal torsor The purpose of this section is to establish a completely explicit bijectionbetween the rational points on the open subset U of our A quartic del Pezzosurface S , and the integral points on the universal torsor above e S which aresubject to a number of coprimality conditions. In doing so we shall followthe strategy of the second author’s joint work with Tschinkel [9].Along the way we will introduce new variables η , . . . , η and α , α . Itwill be convenient to henceforth write η = ( η , . . . , η ) , η ′ = ( η , . . . , η ) , α = ( α , α ) . (4.1)Furthermore, we will make frequent use of the notation η ( k ,k ,k ,k ,k ) := Y i =1 η k i i , (4.2)for any ( k , . . . , k ) ∈ Q .In order to derive the bijection alluded to above, we must begin by col-lecting together some useful information about the geometric structure of S , as defined by equations (1.1). By computing the Segre symbol of S , thedefinition of which can be found in Hodge and Pedoe [13], we see that S contains exactly one singularity. This has type A and is easily determinedas p = (0 : 0 : 0 : 0 : 1). By the classification of singular quartic del Pezzosurfaces found in Coray and Tsfasman [6, Proposition 6.1], S contains ex-actly three lines. Let us call these lines E ′′ , E ′′ and E ′′ , where E ′′ and E ′′ intersect in the singularity p , and E ′′ intersects E ′′ outside p . We easily de-termine these lines as E ′′ = { x = x = x = 0 } , E ′′ = { x = x = x = 0 } and E ′′ = { x = x = x = 0 } .The projection x ( x : x : x ) from E ′′ is a birational map φ : S P , which maps U := S \ ( E ′′ ∪ E ′′ ∪ E ′′ ) = { ( x : . . . : x ) ∈ S | x = 0 } isomorphically to { ( α : η : α ) ∈ P | η = 0 , α η + α = 0 } ⊂ P . ANIN’S CONJECTURE FOR A QUARTIC DEL PEZZO SURFACE 9 The inverse map is ψ : P S given by ψ : ( α : η : α ) ( η : α η : α η : η η : α η ) , (4.3)where η = − ( α η + α ).By [6, Proposition 6.1, Diagram 12], blowing up the singularity p leadsto a minimal desingularisation π : e S → S containing four ( − E , . . . , E (the four exceptional divisors obtained by blowing up p ) andthree ( − E , E , E (the strict transforms of the lines E ′′ , E ′′ , E ′′ on S ). The configuration of these ( − − e S is describedby Figure 1, where the number of edges between two curves is the intersec-tion number, and self intersection numbers are given as upper indices. Thedivisors A , A will be introduced momentarily. A [1]1 CCCCCCCC CCCCCCCC E [ − EEEEEEEE E [ − E [ − E [ − E [ − E [ − yyyyyyyy A [0]2 {{{{{{{{ E [ − Figure 1. Configuration of curves on e S .The surface e S is a blow-up π : e S → P in five points. While thereare several ways to construct e S as such a blow-up of P , we describe amap π that is compatible with the map φ : S P in the sense that φ ◦ π : e S → S P coincides with π where it is defined. Such a map π : e S → P is obtained by contracting E , E , E , E , E on e S in this order.We choose the same coordinates ( α : η : α ) on P as before. Then π maps E , E , E , E , E to (0 : 0 : 1). Furthermore, E is the strict transformof E ′ = { η = − ( α η + α ) = 0 } ⊂ P and E is the strict transform of E ′ = { η = 0 } ⊂ P under π .To describe which points on P we must blow up in order to recover e S ,we introduce A ′ = { α = 0 } ⊂ P and A ′ = { α = 0 } ⊂ P . We note thatits strict transforms A , A under π on e S intersect E , . . . , E as describedby Figure 1, where A , A , E meet in one point which maps under π to(1 : 0 : 0 : 0 : 0) ∈ S . Given E ′ , E ′ , A ′ , A ′ ⊂ P as above, we may nowperform the following sequence of five blow-ups to obtain e S : • blow up the intersection of E , E , A to obtain E ; • blow up the intersection of E , E , E to obtain E ; • blow up the intersection of E , E to obtain E ; • blow up the intersection of E , E to obtain E ; • blow up the intersection of E , E to obtain E .Here we have renamed E ′ i to E i and A ′ j to A j , and we have used the samenames for a divisor and its strict transform in each blow-up in the sequence.We proceed to establish the claim made in § Lemma 3. The surface S is not an equivariant compactification of G .Proof. To establish the lemma we assume for a contradiction that S is of thistype and apply the work of Hassett and Tschinkel [12]. If S is an equivariantcompactification of G then the map φ : S P has to be G -equivariant,resulting in an action of G on P which leaves E ′ = { η = − ( α η + α ) = 0 } invariant. However, we can check that the two distinct G -structures on P (see [12, Proposition 3.2]) do not leave any irreducible quadric curveinvariant. (cid:3) We are now ready to derive the promised bijection between U ( Q ) andintegral points on the universal torsor lying above e S . The map ψ given by(4.3) induces a bijection ψ : ( α , α , η , η ) ( η , α η , α η , η η , α η )between { ( α , η , η ) ∈ Z × Z > × Z =0 | α η + α + η = 0 , gcd( α , α , η ) = 1 } and U ( Q ) = { ( x : . . . : x ) ∈ S ( Q ) | x = 0 } ⊂ S ( Q ) . Note that H ( ψ ( α , α , η , η )) = max i | ψ ( α , α , η , η ) i | gcd( { ψ ( α , α , η , η ) i | i } ) . Motivated by the sequence of blow-ups above, we introduce new variables η := gcd( α , η , η ) , η := gcd( η , η , η ) , η := gcd( η , η ) ,η := gcd( η , η ) , η := gcd( η , η ) , and in each step transform and rename the previous variables accordingly.Observe that this gives a bijection( η ′ , α ) ( η (2 , , , , η , η (1 , , , , η η α , η (2 , , , , α , η (1 , , , , η η , η α ) , which we call Ψ, between T := ( ( η ′ , α ) ∈ Z > × Z =0 × Z (cid:12)(cid:12)(cid:12)(cid:12) η α + η α + η η η η = 0coprimality conditions hold ) and U ( Q ). The coprimality conditions are described by the extended Dynkindiagram of E , . . . , E , A , A in Figure 1, following the rule that any of thevariables η i , α j are coprime if and only if there is no line connecting thedivisors E i , A j in the Dynkin diagram. Once taken in conjunction with theequation T ( η ′ , α ) = η α + η α + η η η η = 0 , that is satisfied by the elements of T , it is easily checked that the coprimalityconditions can be rewritten asgcd( α , η η ) = 1 (4.4)gcd( α , η η η ) = 1 (4.5)gcd( η , η η η η ) = 1 (4.6)gcd( η , η η η η η ) = 1 (4.7)gcd( η , η η η ) = 1 , gcd( η , η ) = 1 , gcd( η , η η ) = 1 . (4.8) ANIN’S CONJECTURE FOR A QUARTIC DEL PEZZO SURFACE 11 In particular it follows that H (Ψ( η ′ , α )) = max i | Ψ( η ′ , α ) i | , since thefive coordinates of Ψ( η ′ , α ) are necessarily coprime for ( η ′ , α ) ∈ T . Theheight conditions may therefore be written asmax n | η (2 , , , , η | , | η (1 , , , , η η α | , | η (2 , , , , α | , | η (1 , , , , η η | , | η α | o B. (4.9)The equation T ( η ′ , α ) = 0 is an embedding of the universal torsor over e S in A . Our argument so far has given us a parametrisation of rational pointsof bounded height in the complement U of the lines in S . This will play apivotal role in our proof of the theorem.5. The main argument In this section we give an overview of the proof of the theorem, and makeour final preparations for its proof. Recall the notation introduced in (4.1)and (4.2) for η , α and η ( k ,k ,k ,k ,k ) . We define the quantities Y := η (2 , , , , B ! / ,Y := (cid:18) B η (2 , − , − , − , − (cid:19) / ,Y := Y − ,Y := (cid:18) B η ( − , − , − , , − (cid:19) / , which clearly depend only on η and B . Using the equation T ( η ′ , α ) = 0, alittle thought reveals that we may write the height condition (4.9) as | Y ( η /Y ) | , (5.1) | Y ( η /Y )( η /Y )( α /Y ) | , (5.2) | Y ( α /Y ) | , (5.3) | Y ( η /Y ) ( η /Y ) | , (5.4) | ( η /Y )(( η /Y ) ( η /Y ) + Y ( α /Y ) ) | , (5.5)with η , . . . , η > 0. For example, eliminating α from | η α | B using T ( η ′ , α ) = 0 gives (5.5). It follows from the contents of § N U,H ( B ) isequal to the number of ( η ′ , α ) ∈ Z > × Z =0 × Z such that (1.5) holds, with(4.4)–(4.8) and (5.1)–(5.5) all holding. As indicated in the introduction itwill be necessary to follow different arguments according to which of η or | η | is biggest in the summation over the variables η ′ . Accordingly, we write N a ( B ) for the overall contribution to N U,H ( B ) from ( η ′ , α ) such that η > | η | , (5.6)and N b ( B ) for the remaining contribution from ( η ′ , α ) such that η < | η | . (5.7)These quantities will be estimated in § § Let us now recall the broad outlines of our approach to estimating N a ( B )and N b ( B ), as discussed in § 1. Thus the idea is to view the torsor equation(1.5) as a congruence modulo η , in order to take care of the summationover the variable α . In § α = ( α , α ). This will lead to a preliminary estimatefor both N a ( B ) and N b ( B ), since it will make no difference whether (5.6)or (5.7) holds. It will then remain to sum this estimate over all of theremaining variables η ′ . We will estimate the overall contribution from theerror term in § N a ( B ), we will sum the main term over η and then over η . This will beundertaken § N b ( B ), we will sum the mainterm over η and then over η . This will be the object of § § η = ( η , . . . , η ).5.1. Real-valued functions. In estimating N a ( B ) and N b ( B ) we will meeta number of real-valued functions, whose basic properties it will be crucialto understand. Let h ( t , t , t , t ) := max {| t t | , | t t t t | , | t t | , | t t t | , | t ( t t + t t ) |} (5.8)Bearing this notation in mind, one notes that the height conditions in (5.1)–(5.5) are equivalent to h ( Y , α /Y , η /Y , η /Y ) 1. Finally, it is easy tosee that ω ∞ = Z h (1 ,t ,t ,t ) ,t > d t d t d t , where ω ∞ is given by (1.2).We define the real-valued functions g ( t , t , t ) := Z h ( t ,t ,t ,t ) t (5.9) g a ( t , t ; η ; B ) := Z Y t > | Y t | ,t > g ( t , t , t ) d t (5.10) g b ( t , t ; η ; B ) := Z | Y t | > max { Y t , } g ( t , t , t ) d t (5.11) g a ( t ; η ; B ) := Z h ( t ,t ,t ,t ) ,Y t > | Y t | > d t d t d t = Z | t | > /Y g a ( t , t ; η ; B ) d t (5.12) g b ( t ; η ; B ) := Z h ( t ,t ,t ,t ) , | Y t | > max { Y t , } ,t > d t d t d t = Z ∞ g b ( t , t ; η ; B ) d t (5.13)We clearly have g ( t ; η ; B ) := g a ( t ; η ; B ) + g b ( t ; η ; B )= Z h ( t ,t ,t ,t ) , | Y t | > ,t > d t d t d t . (5.14) ANIN’S CONJECTURE FOR A QUARTIC DEL PEZZO SURFACE 13 Finally, we define G ( t ) := Z h ( t ,t ,t ,t ) ,t > d t d t d t . The function G : R > → R is intimately related to the real density ω ∞ , asthe following result shows. Lemma 4. We have G ( t ) = ω ∞ t . Proof. This result follows on making the change of variables t = T t − , t = T t − , t = T t . Under this transformation one therefore obtains G ( t ) = 1 t Z h ( t ,T t − ,T t − ,T t ) ,T > d T d T d T , where h ( t , T t − , T t − , T t ) = h (1 , T , T , T ) is independent of t . (cid:3) During the course of our main argument it will be absolutely critical tocontrol the size of the functions (5.9)–(5.11), as t , t , t vary. We may andshall assume that t , t , | t | take only positive values. Lemma 5. Let η ∈ Z > be given. Then the following hold: (1) g ( t , t , t ) ≪ t | t | / . (2) g a ( t , t ; η ; B ) R ∞ g ( t , t , t ) d t ≪ min (cid:8) t | t | / , t (cid:9) . (3) g b ( t , t ; η ; B ) R ∞−∞ g ( t , t , t ) d t ≪ t t / .Proof. Recall the definition (5.8) of h . The upper bound O ( t − ) that appearsin (2) is easy. Indeed, it follows from the inequality h ( t , t , t , t ) | t | /t and | t | /t .For the remaining statements, we distinguish the case | t t | h ( t , t , t , t ) | t t + t t t | . (5.15)Let us begin with the first case, in which case | t t t | 3. We thereforeobtain t ≪ t | t | / , t ≪ | t | / , t ≪ t / . The first of these inequalities implies statement (1), the first and secondimply the first bound in statement (2), and finally, integrating the boundfor g ( t , t , t ) from statement (1) over t ≪ /t / gives statement (3).In the second case | t t | > 2, the inequality (5.15) implies t < t t − t | t | t t t + 1 t | t | . Note that the condition √ x t √ x + y describes an interval for t of length O ( y/x / ). Here, x = ( t t − / ( t | t | ) > t | t | / (2 t ) and y = / ( t | t | ), whence g ( t , t , t ) ≪ t t / | t | / . The inequality t > / / | t | / implies statement (1) and integrating over t > / / | t | / results in the first bound in statement (2). Finally, inte-grating over | t | > / /t / gives statement (3). (cid:3) Estimating N a ( B ) and N b ( B ) — first step. We are now ready tobegin our estimation of N a ( B ) and N b ( B ) in earnest. In what follows, wealways have η , . . . , η ∈ Z > and η ∈ Z =0 .For fixed η ′ = ( η , η , η ) subject to the coprimality conditions (4.6), (4.7)and (4.8), we let N := N ( η ′ ; B ) be the total number of α , α ∈ Z whichsatisfy the equation (1.5), subject to h ( Y , α /Y , η /Y , η /Y ) N = X k | η η µ ( k ) α | α η ≡ − η η η η (mod k η ) ,h ( Y , α /Y , η /Y , η /Y ) , (4.5) holds . It is easy to see that the summand vanishes unless gcd( k , η η η ) = 1. In-deed, if p | k , η then p | η , η η η η , which is forbidden, and furthermore,if p | k , η η then p | η η , α η , which is also forbidden.Let k be a squarefree divisor of η η . Since gcd( η , η ) = 1, we can write k = k k with k | η and k | η . Furthermore such a representation isunique. Writing η = k η ′ we therefore obtain N = X k | η ,k | η gcd( k k ,η η η )=1 µ ( k ) µ ( k ) N ( k , k )where N ( k , k ) := α | α η ≡ − η η k η ′ η (mod k k η ) ,h ( Y , α /Y , η /Y , η /Y ) , (4.5) holds . In view of the congruence we have k | α η , whence k | α sincegcd( k , η ) = 1 and k is squarefree. Writing α = k α ′ , we dividethrough the congruence by k to obtain α ′ k η ≡ − η η k η ′ η (mod k η ) . Using the relation gcd( η , η η ) = 1, we see that gcd( k , k η ) = 1, whencewe can remove a further factor of k in this congruence. It therefore followsthat N ( k , k ) = α ′ | α ′ η ≡ − η η k η ′ η (mod k η ) ,h ( Y , α ′ k /Y , η /Y , η /Y ) , gcd( α ′ , η η η ) = 1 , since gcd( k , η η η ) = 1. ANIN’S CONJECTURE FOR A QUARTIC DEL PEZZO SURFACE 15 Note that gcd( k η , η ) = 1 and gcd( k η , η η k η ′ η ) = 1. It there-fore follows that for each α ′ satisfying the congruence, there is a unique1 ̺ k η , withgcd( ̺, k η ) = 1 , ̺ η ≡ − η η η (mod k η ) , (5.16)such that α ′ ≡ ̺η η ′ (mod k η ) . Thus we obtain N ( k , k ) = X ̺ k η (5.16) holds α ′ | α ′ ≡ ̺η η ′ (mod k η ) ,h ( Y , α ′ k /Y , η /Y , η /Y ) , gcd( α ′ , η η η ) = 1 . We remove gcd( α ′ , η η η ) = 1 by a further application of M¨obius inversion.Writing α ′ = k α ′′ , we see that N ( k , k ) is equal to X ̺ k η (5.16) holds X k | η η η µ ( k ) (cid:26) α ′′ | k α ′′ ≡ ̺η η ′ (mod k η ) ,h ( Y , α ′′ k k /Y , η /Y , η /Y ) (cid:27) . The summand vanished unless gcd( k , k η ) = 1, since p | k , k η implies p | k η , ̺η η ′ , which is forbidden. Thus we may restrict our summationover k to gcd( k , k η ) = 1, and it therefore follows that the number ofavailable α ′′ is Y k k k η g ( Y , η /Y , η /Y ) + O (1) , where g is given by (5.9). Recall the definition of the function φ ∗ from § Lemma 6. We have N = Y η g ( Y , η /Y , η /Y ) ϑ ( η , η , η ) + O ( R ( η , η , η ; B )) with ϑ ( η , η , η ) := φ ∗ ( η ) φ ∗ ( η η η ) φ ∗ (gcd( η , η )) X k | η gcd( k ,η η η )=1 µ ( k ) k φ ∗ (gcd( η , k η )) X ̺ k η (5.16) holds , and X η ,η ,η R ( η , η , η ; B ) ≪ B (log B ) . The final statement in Lemma 6 should be taken to mean that the overallcontribution from the error term in the asymptotic formula for N , oncesummed over all of the available η , η , η , is O ( B (log B ) ). What is cru-cial here is that the exponent of log B is strictly smaller than 5, so thatthis truly is an acceptable error term from the point of view of the maintheorem. In the case of Lemma 6 we need to sum R ( η , η , η ; B ) over all η , η , η which satisfy the height conditions (5.1)–(5.5), and the coprimal-ity conditions (4.6)–(4.8). In the arguments to follow there will be severalpoints at which the overall contribution from various error terms needs to be estimated. In each case we will not stress the precise conditions on thevariables to be summed over, these being invariably self-evident. Proof of Lemma 6. Tracing through our argument above, it follows that N = Y η g ( Y , η /Y , η /Y ) ϑ ( η , η , η ) + O ( R ( η , η , η ; B )) , with ϑ = X k | η ,k | η gcd( k k ,η η η )=1 µ ( k ) µ ( k ) k k X ̺ k η (5.16) holds X k | η η η gcd( k ,k η )=1 µ ( k ) k , and R ( η , η , η ; B ) ≪ ω ( η η η )+ ω ( η ) X k | η | µ ( k ) | X ̺ k η (5.16) holds ≪ ω ( η )+ ω ( η η ) ω ( η η η )+ ω ( η ) ω ( η ) ω ( η )+ ω ( η )+ ω ( η )+ ω ( η ) . We have used here the fact that the congruence in (5.16) has at most2 ω ( k η ) ω ( η η ) solutions ̺ modulo k η .On noting that gcd( η , η η ) = 1 and gcd( η η , k η ) = 1, we deduce that ϑ = X k | η gcd( k ,η η η )=1 µ ( k ) k φ ∗ ( η ) φ ∗ (gcd( η , η )) φ ∗ ( η η η ) φ ∗ (gcd( η , k η )) X ̺ k η (5.16) holds . This completes the proof of the main term in the lemma.To show that R ( η , η , η ; B ) makes a satisfactory contribution once it issummed over all η , η , η satisfying the height conditions in (4.9), we beginby summing over η . Thus it follows that X η ,η ,η R ( η , η , η ; B ) ≪ X η ,η ω ( η ) ω ( η )+ ω ( η )+ ω ( η )+ ω ( η ) B η (1 , , , , η ≪ B (log B ) , as required to complete the proof of the lemma. (cid:3) Estimating N a ( B ) — second step. In this section our task is to sumthe main term in Lemma 6 over all of the relevant η and η , such that (5.6)holds. As we’ve already indicated, we will begin by summing over the η .For fixed η , η satisfying the coprimality conditions (4.7) and (4.8), define N a := N a ( η , η ; B ) to be the sum of the main term in Lemma 6 over all η ∈ Z > such that the coprimality condition (4.6) holds, and furthermore, η > | η | .We begin by noting that it is possible to remove η from (4.6), replac-ing this coprimality condition by gcd( η , η η η ) = 1. Indeed, if p | η , η then (5.16) implies that we must have p | ̺ η , which is forbidden. Sincegcd( η η , k η ) = 1, so there exists a unique integer β ∈ [1 , k η ] such that ̺ η ≡ − η η β (mod k η ) . ANIN’S CONJECTURE FOR A QUARTIC DEL PEZZO SURFACE 17 It therefore follows that N a = Y η φ ∗ ( η η η ) X k | η gcd( k ,η η η )=1 µ ( k ) k φ ∗ (gcd( η , k η )) X ̺ k η gcd( ̺,k η )=1 A, where A = X η ∈ Z > η > | η | η ≡ β (mod k η ) f η ,η η η ( η ) g ( Y , η /Y , η /Y ) . Here f η ,η η η is given by (3.2). Since g ( Y , η /Y , η /Y ) = 0 for η > B ,we may restrict the summation to η in the range | η | η B .We will estimate A using Lemma 2. This produces a main term and anerror term, the latter having size ≪ ω ( η η η ) (log B ) sup t g ( Y , t , η /Y ) , where the supremum is over all t ∈ R such that Y t > | η | . This thereforegives an overall contribution ≪ Y ω ( η )+ ω ( η η η ) (log B ) sup t g ( Y , t , η /Y ) , (5.17)to N a , since 1 k η X ̺ k η gcd( ̺,k η )=1 φ ∗ ( k η ) . The main term in our application of Lemma 2 to A is simplyΘ( η , k ) Y k η Z Y t > | η | ,t > g ( Y , t , η /Y ) d t , with Θ( η , k ) = φ ∗ ( η η η ) ζ (2) φ ∗ (gcd( η η η , k η )) Y p | η η η η η (cid:16) − p (cid:17) − = φ ∗ ( η η η ) ζ (2) φ ∗ (gcd( η , k η )) Y p | η η η η η (cid:16) − p (cid:17) − . Here we have used the fact that gcd( η η , k η ) = 1. Note for future ref-erence that Θ( η , k ) ≪ 1. We are now ready to establish the followingresult. Lemma 7. We have N a = Y Y η g a ( Y , η /Y , η ; B ) ϑ a ( η ) + O ( R a ( η , η ; B )) with ϑ a ( η ) := X k | η gcd( k ,η η η )=1 µ ( k ) k Θ( η , k ) φ ∗ ( η η η η ) , and X η ,η R a ( η , η ; B ) ≪ B (log B ) . Proof. It is clear from our calculations above that the main term in ourestimate for N a is equal to Y Y g a ( Y , η /Y , η ; B ) ϑ a ( η ) /η , with ϑ a ( η ) = X k | η gcd( k ,η η η )=1 µ ( k ) φ ∗ ( η η η ) φ ∗ ( k η ) k φ ∗ (gcd( η η η , k η )) Θ( η , k )= φ ∗ ( η η η η ) X k | η gcd( k ,η η η )=1 µ ( k ) k Θ( η , k ) , since k η is coprime to η η and every divisor of k divides η . Thiscompletes the proof of the main term in the lemma.Turning to the overall contribution from the error term R a ( η , η ; B ),which we have already seen has size (5.17), we conclude from (4.9) and(5.6) that | η (2 , , , , η | B, for the η , η that we need to sum over. We therefore deduce from Lemma 5(1)that X η ,η R a ( η , η ; B ) ≪ log B X η ,η Y ω ( η ) ω ( η )+ ω ( η ) · Y / Y | η | / = log B X η ,η ω ( η ) ω ( η )+ ω ( η ) B / η (1 / , , , , − / | η | / ≪ log B X η ω ( η ) ω ( η )+ ω ( η ) B η (3 / , , / , , ≪ B (log B ) , as required to complete the proof of the lemma. (cid:3) Lemma 7 takes care of the summation of the main term in Lemma 6 overall of the relevant η . We proceed to sum the resulting main term over the η . Thus we let N a := N a ( η ; B ) = X η ∈ Z =0 (4.7) holds Y Y η g a ( Y , η /Y ; η ; B ) ϑ a ( η )We begin with an application of M¨obius inversion to remove the coprimalitycondition (4.7). This gives N a = Y Y η ϑ a ( η ) X k | η η η η η µ ( k ) X | η ′ | > g a ( Y , k η ′ /Y ; η ; B ) , where we have written η = k η ′ . Partial summation now yields N a = Y Y Y η ϑ a ( η ) X k | η η η η η µ ( k ) k Z | t | > k /Y g a ( Y , t ; η ; B ) d t + O (cid:16) Y Y η | ϑ a ( η ) | X k | η η η η η | µ ( k ) | sup | t | > k /Y g a ( Y , t ; η ; B ) (cid:17) . ANIN’S CONJECTURE FOR A QUARTIC DEL PEZZO SURFACE 19 The following result constitutes the final outcome of our summation over η . Lemma 8. We have N a = Y Y Y η g a ( Y , η ; B ) ϑ a ( η ) + O ( R a ( η ; B )) with ϑ a ( η ) := φ ∗ ( η η η η η ) ϑ a ( η ) , and X η R a ( η ; B ) ≪ B (log B ) − / . Proof. The effect of replacing the integral R | t | > k /Y g a ( Y , t ; η ; B ) d t by g a ( Y , η ; B ) in our estimate for N a , is to create an additional term Y Y Y η | ϑ a ( η ) | X k | η η η η η | µ ( k ) | k Z /Y < | t | 3. Let λ > η (3 , , , , < λB in thesummation over the η . Accordingly let E ( λ ) denote the overall contributionfrom the two errors terms once summed over η such that η (3 , , , , < λB, (5.18)and let E ( λ ) denote the remaining contribution from η such that η (3 , , , , > λB. (5.19)Beginning with the estimation of E ( λ ), we employ Lemma 5(2) to con-clude that Z k /Y /Y g a ( Y , t ; η ; B ) d t ≪ Z k /Y /Y Y | t | / d t ≪ Y / Y . Once summed over all η such that (5.18) holds, we use (3.1) to estimate theoverall contribution from the first term in R a ( η ; B ) as ≪ X η X k | η η η η η | µ ( k ) | k φ † ( η ) Y Y Y / η Y ≪ X η φ † ( η η η η η ) φ † ( η ) B / η (1 / , , / , / , / ≪ X η ,η ,η ,η φ † ( η ) φ † ( η ) φ † ( η ) φ † ( η ) λ / B η (1 , , , , ≪ λ / B (log B ) . Turning to the overall contribution from the second term in R a ( η ; B ), weagain deduce from Lemma 5(2) thatsup | t | > k /Y g a ( Y , t ; η ; B ) ≪ sup | t | > k /Y Y | t | / ≪ Y / Y k / . Hence, in this case too, we obtain the overall contribution ≪ X η φ † ( η ) Y Y Y / η Y ≪ X η ,η ,η ,η φ † ( η ) λ / B η (1 , , , , ≪ λ / B (log B ) . Thus far we have shown that E ( λ ) ≪ λ / B (log B ) .It remains to produce a suitable upper bound for E ( λ ). It will be con-venient to record the estimates X n x ω ( n ) φ † ( n ) n ≪ (log x ) , X n>x h ω ( n ) n a ≪ x − a (log x ) h − . The second inequality is valid for any h ∈ Z > and any a > 1, and followson combining partial summation with the bound X n x h ω ( n ) X n x X n = d ...d h ≪ X d ,...,d h − x xd . . . d h − ≪ x (log x ) h − . Beginning with the first term in R a ( η ; B ), we deduce from Lemma 5(2) that Z k /Y /Y g a ( Y , t ; η ; B ) d t ≪ Z k /Y /Y Y d t ≪ k Y Y . Summing over η such that (5.19) holds, we therefore obtain the overallcontribution ≪ X η X k | η η η η η | µ ( k ) | k φ † ( η ) k Y Y η Y ≪ X η ω ( η η η η η ) φ † ( η ) B η (4 , , , , ≪ X η ,...,η ω ( η η η η ) φ † ( η ) B log Bλ η (1 , , , , ≪ λ − B (log B ) , by (3.1). Similarly, for the contribution from the second term in R a ( η ; B ),we may use Lemma 5(2) to deduce the overall contribution ≪ X η ω ( η η η η η ) φ † ( η ) Y Y η Y ≪ λ − B (log B ) . Taken together this shows that E ( λ ) ≪ λ − B (log B ) . We choose λ =(log B ) / , which therefore gives the overall contribution X η R a ( η ; B ) ≪ B (log B ) / , as required. (cid:3) ANIN’S CONJECTURE FOR A QUARTIC DEL PEZZO SURFACE 21 Estimating N b ( B ) — second step. We must now return to the mainterm in Lemma 6, but this time reverse the order of summation for η and η .This will allow us to make use of the inequality (5.7) in our treatment of theerror terms. We begin with the summation over η . For fixed η , η satisfyingthe coprimality conditions (4.6) and (4.8), define N b := N b ( η , η ; B ) to bethe sum of the main term in Lemma 6 over all η ∈ Z =0 such that thecoprimality condition (4.7) holds, and furthermore, | η | > η = max { η , } .Our argument is very similar in spirit to the preceding section. Removing(4.7) with an application of M¨obius inversion, we find that N b = Y η φ ∗ ( η ) φ ∗ ( η η η ) φ ∗ (gcd( η , η )) X k | η gcd( k ,η η η )=1 µ ( k ) k φ ∗ (gcd( η , k η )) × X ̺ k η gcd( ̺,k η )=1 X k | η η η η η gcd( k ,k η )=1 µ ( k ) A, where A = X η ′ ∈ Z =0 ̺ η ≡− η η k η ′ (mod k η ) k | η ′ | >η g ( Y , η /Y , k η ′ /Y ) , and we have written η = k η ′ . Note that we have been able to add theconstraint gcd( k , k η ) = 1 in the sum over k , since A = 0 otherwise.Since gcd( η η k , k η ) = 1, it follows from an easy application of partialsummation that A = Y k k η g b ( Y , η /Y ; η ; B ) + O (cid:16) sup t g ( Y , η /Y , t ) (cid:17) , where the supremum is over t ∈ R such that | t | > η /Y . We may nowestablish the following result. Lemma 9. We have N b = Y Y η g b ( Y , η /Y ; η ; B ) ϑ b ( η ) φ ∗ ( η ) φ ∗ (gcd( η , η )) + O ( R b ( η , η ; B )) with ϑ b ( η ) := φ ∗ ( η η η η ) φ ∗ ( η η η η ) X k | η gcd( k ,η η η )=1 µ ( k ) k φ ∗ (gcd( η , k η )) , and X η ,η R b ( η , η ; B ) ≪ B (log B ) . Proof. It is clear that the main term in the lemma is valid with ϑ b = X k | η gcd( k ,η η η )=1 µ ( k ) φ ∗ ( η η η ) φ ∗ ( k η ) k φ ∗ (gcd( η , k η )) φ ∗ ( η η η η ) φ ∗ (gcd( η η η η , k η ))= X k | η gcd( k ,η η η )=1 µ ( k ) k φ ∗ ( k η η η η ) φ ∗ ( η η η η ) φ ∗ (gcd( η , k η ))= φ ∗ ( η η η η ) φ ∗ ( η η η η ) X k | η gcd( k ,η η η )=1 µ ( k ) k φ ∗ (gcd( η , k η )) , as claimed. We have used here the fact that gcd( k η , η η η ) = 1.For the error term, we deduce from (4.9) that η (2 , , , , η B, for the η , η that we need to sum R b ( η , η ; B ) over. Using Lemma 5(1) to bound g , we easily deduce that X η ,η R b ( η , η ; B ) ≪ X η ,η Y ω ( η )+ ω ( η η η η ) sup | t | >η /Y g ( Y , η /Y , t ) ≪ X η ,η ω ( η )+ ω ( η η η η ) Y Y / Y η / = X η ,η ω ( η )+ ω ( η η η η ) B / η (1 / , , , , − / η / ≪ X η ω ( η )+ ω ( η η η η ) B η (3 / , , / , , ≪ B X η ,η ω ( η ) η η . But this is O ( B (log B ) ), as required. This completes the proof of thelemma. (cid:3) We must now sum the main term in Lemma 9 over all of the relevant η , and then over η , . . . , η . In doing so it will be convenient distinguishbetween values of η , η such that η (2 , , , , B (log B ) A , (5.20)for some A > 0, and those for which this inequality does not hold. We write N b ( B ; A ) and N b ( B ; A ) for the corresponding contributions. The followingresult shows that N b ( B ; A ) makes a negligible contribution to N U,H ( B ). Lemma 10. We have N b ( B ; A ) ≪ A B (log B ) (log log B ) .Proof. Once taken in conjunction with the inequalities for η , η in (4.9), thefailure of (5.20) clearly implies that we must sum over η , η for which η η η η η η B, η < (log B ) A . (5.21) ANIN’S CONJECTURE FOR A QUARTIC DEL PEZZO SURFACE 23 Recalling the definition of the main term from Lemma 9, we see that N b ( B ; A ) ≪ X η ,η (5.21) holds Y Y η g b ( Y , η /Y ; η ; B ) ϑ b ( η ) φ ∗ ( η ) φ ∗ (gcd( η , η )) ≪ X η ,η (5.21) holds Y Y / Y φ † ( η ) Y η η / , using Lemma 5(3). In view of the definitions of the Y i we conclude that N b ( B ; A ) ≪ B / X η ,η (5.21) holds φ † ( η ) η (1 / , , / , / , / η / ≪ B X η ,...,η (5.21) holds φ † ( η ) η (0 , , , , η . This last expression is clearly satisfactory for the lemma by (3.1) with j = 1and the fact that the η summation is over η < (log B ) A . (cid:3) Our focus now shifts to estimating N b ( B ; A ), deemed to be the overallcontribution from the main term in Lemma 9 that arises from η , η for which(5.20) holds. For the moment let N b := N b ( η ; B ) be the quantity obtainedby summing the main term in Lemma 9’s estimate for N b ( η , η ; B ), over all η ∈ Z > such that (4.6) holds. An application of Lemma 2 with α = 0 and q = 1 therefore reveals that N b = Y Y η ϑ b ( η ) X η > f η ,η η η η ( η ) g b ( Y , η /Y ; η ; B )= Y Y Y η g b ( Y ; η ; B ) ϑ b ( η ) φ ∗ ( η η η η ) ζ (2) Y p | η η η η η (cid:16) − p (cid:17) − + O (cid:18) Y Y η | ϑ b ( η ) | (log B )2 ω ( η η η η ) sup t g b ( Y , t ; η ; B ) (cid:19) + O Y Y Y η | ϑ b ( η ) | Z We have N b = Y Y Y η g b ( Y ; η ; B ) ϑ b ( η ) + O ( R b ( η ; B )) with ϑ b ( η ) := ϑ b ( η ) φ ∗ ( η η η η ) ζ (2) Y p | η η η η η (cid:16) − p (cid:17) − , and X η (5.20) holds R b ( η ; B ) ≪ B (log B ) − A/ . Proof. The value of ϑ b ( η ) in the main term for N b is a direct consequence ofour manipulations above. In considering the overall contribution from theerror term it will be convenient to note that ϑ b ( η ) ≪ ϑ b ( η ) ≪ φ † ( η ).Once again the error R b ( η ; B ) is comprised of two basic terms, the firstone involving a supremum of g b over t in an appropriate range, and thesecond involving an integration of g b . We begin with dealing with the firstterm. It is here that we will make critical use of the inequality (5.20), thatunderpins our definition of N b ( B ; A ). The first term in R b ( η ; B ) clearlymakes an overall contribution of ≪ X η ω ( η η η η ) φ † ( η ) Y Y log Bη sup t > /Y g b ( Y , t , η ; B ) , where the summation is restricted to η for which (5.20) holds. UsingLemma 5(3) to estimate g b , we may bound this as ≪ X η ω ( η η η η ) φ † ( η ) Y Y Y / log Bη Y = X η ω ( η η η η ) φ † ( η ) B / log B η (1 / , , / , / , / ≪ (log B ) − A/ X η ,η ,η ,η ω ( η η η η ) φ † ( η ) B η (1 , , , , ≪ B (log B ) − A/ . Turning to the contribution from the second term in R b ( η ; B ), we employLemma 5(3) and (5.20) to derive the overall contribution ≪ X η φ † ( η ) Y Y Y η Z /Y Y t / d t ≪ X η φ † ( η ) Y Y / Y η Y = X η φ † ( η ) B / η (1 / , , / , / , / ≪ (log B ) − A/ X η ,η ,η ,η φ † ( η ) B η (1 , , , , ≪ B (log B ) − A/ . Together these two upper bounds complete the proof of the lemma. (cid:3) The final step. Let us take a moment to compile our work so far. Wesaw at the start of § N U,H ( B ) = N a ( B ) + N b ( B ) . It will be convenient to set B = B/ (log B ) in what follows.The union of Lemmas 6, 7 and 8 shows that N a ( B ) = X η ∈E ( B ) Y Y Y η ϑ a ( η ) g a ( Y , η ; B ) + O (cid:0) B (log B ) − / (cid:1) , ANIN’S CONJECTURE FOR A QUARTIC DEL PEZZO SURFACE 25 where ϑ a ( η ) is as in the statement of Lemma 8, and E ( B ) := (cid:8) η ∈ Z > : (4.8) holds and η (2 , , , , B (cid:9) . Similarly, we can combine Lemmas 6, 9, 10 and 11, taking A = 36 in thelatter two results, to deduce that N b ( B ) = X η ∈E ( B ) η (2 , , , , B Y Y Y η ϑ b ( η ) g b ( Y ; η ; B ) + O (cid:0) B (log B ) (log log B ) (cid:1) , where ϑ b ( η ) is as in the statement of Lemma 11.We would now like to remove the constraint that η (2 , , , , B in ourestimate for N b ( B ). In view of the fact that ϑ b ( η ) ≪ φ † ( η ), it easily followsfrom (5.13) and Lemma 5(3) that X η ∈E ( B ) B < η (2 , , , , B Y Y Y η ϑ b ( η ) g b ( Y ; η ; B ) ≪ X η Y Y Y φ † ( η ) η Z /Y d t Y t / ≪ X η Y Y Y φ † ( η ) η = X η Bφ † ( η ) η (1 , , , , ≪ B (log B ) (log log B ) . In deducing the first bound we have used the fact that g b ( t , t ; η ; B ) = 0unless 0 < t /t , which follows from the definition of (5.8). Thus wemay replace the above formula for N b ( B ; 36) by X η ∈E ( B ) Y Y Y η ϑ b ( η ) g b ( Y ; η ; B ) + O (cid:0) B (log B ) (log log B ) (cid:1) . For given η ∈ E ( B ), define ϑ ( η ) := φ ∗ ( η η η ) φ ∗ ( η η η η η ) φ ∗ ( η η η η ) ζ (2) Q p | η η η η η (1 − /p ) × X k | η gcd( k ,η η η )=1 µ ( k ) k φ ∗ (gcd( η , k η )) . It is easily seen that ϑ ( η ) = ϑ a ( η ), in the notation of Lemmas 7 and 8.Furthermore, on noting that φ ∗ ( η η η ) φ ∗ ( η η η η η ) = φ ∗ ( η η η η ) φ ∗ ( η η η η ) , since gcd( η , η ) = 1, we see that ϑ ( η ) = ϑ b ( η ) also. Thus we may drawtogether our argument so far to conclude that N U,H ( B ) = X η ∈E ( B ) Y Y Y η ϑ ( η ) g ( Y ; η ; B ) + O (cid:0) B (log B ) − / (cid:1) , where g ( t ; η ; B ) is given by (5.14). It turns out that there is a negligi-ble contribution to N U,H ( B ) from summing Y Y Y ϑ ( η ) g ( t ; η ; B ) /η over small values of η ∈ E ( B ). The η that give the dominant contribution belongto the set E ∗ ( B ) := (cid:8) η ∈ Z > : (4.8) holds, η (2 , , , , B and η (3 , , , , > B (cid:9) . We also wish to remove the dependence on η and B from the real-valuedfunction g ( Y ; η ; B ). All of this will be achieved in the following result. Lemma 12. We have N U,H ( B ) = ω ∞ B X η ∈E ∗ ( B ) ϑ ( η ) η (1 , , , , + O (cid:0) B (log B ) − / (cid:1) , where ω ∞ is given by (1.2) .Proof. We begin by showing that M ( B ) := X η ∈E ( B ) η (3 , , , , B Y Y Y η ϑ ( η ) g ( Y ; η ; B ) ≪ B (log B ) . Now it follows from (5.9) and (5.14) that g ( t ; η ; B ) = Z h ( t ,t ,t ,t ) , | Y t | > ,t > d t d t d t Z | t | > /Y Z ∞ g ( t , t , t ) d t d t . Hence Lemma 5(2) yields g ( Y ; η ; B ) ≪ Z | t | > /Y Y | t | / d t ≪ Y / Y . Applying this we deduce that M ( B ) ≪ X η (3 , , , , B φ † ( η ) Y Y Y / η Y ≪ X η (3 , , , , B φ † ( η ) B / η (1 / , , / , / , / ≪ X η ,η ,η ,η φ † ( η ) B η (1 , , , , ≪ B (log B ) , by (3.1), which therefore shows that N U,H ( B ) = X η ∈E ∗ ( B ) Y Y Y η ϑ ( η ) g ( Y ; η ; B ) + O (cid:0) B (log B ) − / (cid:1) . It remains to deal with the real-valued function g ( Y ; η , B ).We will show that M ( B ) := X η ∈E ∗ ( B ) Y Y Y η ϑ ( η ) Z h ( Y ,t ,t ,t ) | Y t | ,t > d t d t d t ≪ B (log B ) . Once achieved, this will suffice to complete the proof of the lemma, since anapplication of Lemma 4 reveals that Z h ( Y ,t ,t ,t ) ,t > d t d t d t = ω ∞ Y , ANIN’S CONJECTURE FOR A QUARTIC DEL PEZZO SURFACE 27 and clearly Y Y Y Y η = B η (1 , , , , . To establish the bound for M ( B ) we appeal to Lemma 5(2), which in asimilar manner to our treatment of M ( B ), implies that M ( B ) ≪ X η ∈E ∗ ( B ) Y Y Y φ † ( η ) η Z | t | /Y d t Y ≪ X η ∈E ∗ ( B ) Y Y φ † ( η ) Y η = X η ∈E ∗ ( B ) φ † ( η ) B η (4 , , , , ≪ X η ,η ,η ,η φ † ( η ) B η (1 , , , , ≪ B (log B ) . This completes the proof of the lemma. (cid:3) Let us redefine the function ϑ ( η ) so that it is equal to zero if η fails tosatisfy the coprimality relations in (4.8). For k = ( k , . . . , k ) ∈ Z > , let∆ k ( n ) := X η ∈ Z > η ( k ,k ,k ,k ,k = n ϑ ( η ) η (1 , , , , . Then Lemma 12 implies that N U,H ( B ) = ω ∞ B X n B (cid:0) ∆ (2 , , , , ( n ) − ∆ (3 , , , , ( n ) (cid:1) + O (cid:0) B (log B ) − / (cid:1) . (5.22)We will want to establish an asymptotic formula for M k ( t ) := X n t ∆ k ( n ) , as t → ∞ . We shall do so by studying the corresponding Dirichlet series F k ( s ) = ∞ X n =1 ∆ k ( n ) n s = X η ∈ Z > ϑ ( η ) η ( k s +1 ,k s +1 ,k s +1 ,k s +1 ,k s +1) , which is absolutely convergent for ℜ e ( s ) > F k ( s ) = Q p F k ,p ( s ) , and a cumbersome computation reveals that the local factors F k ,p ( s ) areequal to(1 − /p ) · (cid:18) (1 + 1 /p ) + 1 − /pp k s +1 − 1+ 1 − /pp k s +1 − (cid:18) (1 − /p ) + 1 − /pp k s +1 − − /pp k s +1 − − /pp k s +1 − (cid:19) + (1 − /p ) p k s +1 − (cid:18) p k s +1 − (cid:19) + 1 − /pp k s +1 − − /pp k s +1 − (cid:19) . Let ε > k ∈ { (2 , , , , , (3 , , , , } . Then it followsthat for all s ∈ C belonging to the half-plane ℜ e ( s ) > − / 12 + ε , we have F k ,p ( s ) Y j =1 (cid:16) − p k j s +1 (cid:17) = 1 + O ε ( p − − ε ) . Thus, on defining E k ( s ) := Y j =1 ζ ( k j s + 1) , G k ( s ) := F k ( s ) E k ( s ) , we may conclude that F k ( s ) has a meromorphic continuation to the half-plane ℜ e ( s ) > − / 12 + ε , with a pole of order 5 at s = 0. It will be usefulto note that G k (0) = Y p (cid:18) − p (cid:19) (cid:18) p + 1 p (cid:19) . (5.23)To estimate M k ( t ) we now have everything in place to apply the followingstandard Tauberian theorem, which is recorded in work of Chambert-Loirand Tschinkel [5, Appendice A]. Lemma 13. Let { c n } n ∈ Z > be a sequence of positive real numbers, and let f ( s ) = P ∞ n =1 c n n − s . Assume that: (1) the series defining f(s) converges for ℜ e ( s ) > ; (2) it admits a meromorphic continuation to ℜ e ( s ) > − δ for some δ > ,with a unique pole at s = 0 of order b ∈ Z > ; (3) there exists κ > such that (cid:12)(cid:12)(cid:12) f ( s ) s b ( s + 2 δ ) b (cid:12)(cid:12)(cid:12) ≪ (1 + ℑ m ( s )) κ , for ℜ e ( s ) > − δ .Then there exists a monic polynomial P of degree b , and a constant δ ′ > such that X n t c n = Θ b ! P (log t ) + O ( t − δ ′ ) , as t → ∞ , where Θ = lim s → s b f ( s ) . In fact [5, Appendice A] deals only with Dirichlet series possessing aunique pole at s = a > 0, but the extension to a pole at s = 0 is straight-forward. We apply Lemma 13 to estimate M k ( t ), for k ∈ { (2 , , , , , (3 , , , , } . We have already seen that the corresponding Dirichlet series F k ( s ) satisfiesparts (1) and (2) of the lemma, with b = 5. The third part follows from theboundedness of G k ( s ) on the half-plane ℜ e ( s ) > − / 12 + ε , and standardupper bounds for the size of the Riemann zeta function in the critical strip.In view of the fact that lim s → s b F ( s ) = G k (0) Q j =1 k j , ANIN’S CONJECTURE FOR A QUARTIC DEL PEZZO SURFACE 29 we therefore conclude that M k ( t ) = G k (0) P (log t )5! · Q j =1 k j + O ( t − δ ) , (5.24)for some δ > P of degree 5.We are now ready to complete the proof of our theorem. Recall thedefinition (2.1) of ω H ( e S ). It therefore follows on combining (1.8), (5.22),(5.23) and (5.24) that N U,H ( B ) = α ( e S ) ω H ( e S ) B (log B ) + O (cid:0) B (log B ) − / (cid:1) , as required. References [1] V.V. Batyrev and Yu. Tschinkel, Manin’s conjecture for toric varieties. J. Alg. Geom. (1998), 15–53.[2] R. de la Bret`eche and T.D. Browning, On Manin’s conjecture for singular del Pezzosurfaces of degree four, I. Michigan Math. J. (2007), 51–80.[3] T.D. Browning, An overview of Manin’s conjecture for del Pezzo surfaces. Analyticnumber theory — A tribute to Gauss and Dirichlet , Clay Math. Proceedings (2007),39–56.[4] A. Chambert-Loir and Yu. Tschinkel, On the distribution of points of bounded heighton equivariant compactifications of vector groups. Invent. Math. (2002), 421–452[5] A. Chambert-Loir and Yu. Tschinkel, Fonctions zˆeta des hauteurs des espaces fibr´es. Rational points on algebraic varieties , Progress in Math. , Birkh¨auser (2001),71–115.[6] D.F. Coray and M.A. Tsfasman, Arithmetic on singular del Pezzo surfaces. Proc.London Math. Soc. (1988), 25–87.[7] U. Derenthal, On a constant arising in Manin’s conjecture for Del Pezzo surfaces. Math. Res. Letters (2007), 481–489.[8] U. Derenthal, Singular Del Pezzo surfaces whose universal torsors are hypersurfaces. arXiv:math.AG/0604194 (2006).[9] U. Derenthal and Yu. Tschinkel, Universal torsors over Del Pezzo surfaces and rationalpoints. Equidistribution in Number Theory, An Introduction , 169–196, NATO Sci. Ser.II Math. Phys. Chem. , Springer, 2006.[10] U. Derenthal, M. Joyce, and Z. Teitler, The nef cone volume of generalized del Pezzosurfaces. Algebra & Number Theory (2008), 157–182.[11] J. Franke, Yu.I. Manin and Yu. Tschinkel, Rational points of bounded height on Fanovarieties. Invent. Math. (1989), 421–435.[12] B. Hassett and Yu. Tschinkel, Geometry of equivariant compactifications of G na . Internat. Math. Res. Notices (1999), 1211–1230.[13] W.V.D. Hodge and D. Pedoe, Methods of algebraic geometry . Vol. 2, CambridgeUniversity Press, 1952.[14] E. Peyre, Hauteurs et nombres de Tamagawa sur les vari´et´es de Fano. Duke Math. J. (1995), 101–218.[15] P. Salberger, Tamagawa measures on universal torsors and points of bounded heighton Fano varieties. Ast´erisque (1998), 91–258. School of Mathematics, University of Bristol, Bristol BS8 1TW E-mail address : [email protected] Institut f¨ur Mathematik, Universit¨at Z¨urich, Winterthurerstrasse 190,8057 Z¨urich, Switzerland E-mail address ::