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0 MANIN’S CONJECTURE FOR QUARTIC DEL PEZZO 1 0 SURFACES WITH A CONIC FIBRATION 2 r by a M R. de la Bretèche & T.D. Browning 1 3 ] T N Abstract. — An asymptotic formula is established for the number of Q-rational . points of bounded height on a non-singular quartic del Pezzo surface with a conic h bundlestructure. t a m [ Contents 2 v 1. Introduction .................................................... 1 6 2. Technical results ................................................ 7 1 3. Preliminary manipulations .................................... 10 6 1 4. Reducing the range of summation .............................. 13 . 5. Parametrisationof the conics .................................. 16 8 0 6. Removing the coprimality conditions .......................... 20 8 7. Lattice point counting .......................................... 25 0 8. The error terms ................................................ 32 : v 9. The main term ................................................ 40 i 10. Peyre’s constant .............................................. 45 X Appendix by U. Derenthal ........................................ 52 r a References ........................................................ 54 1. Introduction Let k be a number field. This investigation centres upon the distribution of k- rationalpointsonconicbundlesurfacesX/P1. Thesearedefinedtobeprojectivenon- k singular surfaces X defined overk, which are equipped with a dominant k-morphism π : X P1, all of whose fibres are conics. A summary of our knowledge concerning → k the arithmetic ofconic bundle surfacescanbe found inthe article by Colliot-Thélène [9]. If K denotes the anticanonical divisor of X, then the degree of X is defined X − 2000 Mathematics Subject Classification. — 11D45(14G25). 2 R. DE LA BRETE`CHE & T.D. BROWNING to be the self-intersection number ( K , K ) = 8 r, where r is the number of X X − − − geometric fibres above π that are degenerate. When 0 6 r 6 3 it is known that the Hasse principle holds for these surfaces, and furthermore, that such X are k-rational as soon as they possess k-rational points. When r = 4, so that X has degree 4, it has been shown by Iskovskikh [17, Proposition 1] that two basic cases arise. Either K is not ample, in which case X is k-birationalto a generalised Chaˆtelet surface, X − or else K is ample, in which case X is a non-singular quartic del Pezzo surface. X − Ourinterestlieswiththequantitativearithmeticofdegree4conicbundlesurfaces. WhenX(k)= andtheanticanonicalheightfunctionH :X(k) R isassociated, >0 6 ∅ → we seek to determine the asymptotic behaviour of the counting function N (B):=# x U(k):H(x)6B , U,H { ∈ } as B , for a suitable Zariski open subset U X. The conjecture that drives → ∞ ⊆ our work is due to Manin [13]. Let PicX be the Picard group of X. Still under the assumption that X(k) is non-empty, this conjecture predicts the existence of a positive constant c such that X,H N (B)=c B(logB)rank(PicX) 1 1+o(1) , (1.1) U,H X,H − as B . Peyre [22] has given a conjectural interpreta(cid:0)tion of c(cid:1)X,H in terms of the →∞ geometryofX. Until veryrecentlywe werenotin possessionofa single conicbundle surface of degree 4 for which this refined conjecture could be established. Henceforthwewillbeinterestedinthecasek=Q. InjointworkwithPeyre[6],the authors have made a study of generalised Chaˆtelet surfaces, ultimately establishing (1.1) for a family of such surfaces that fail to satisfy weak approximation. The aim of the present investigation is to produce a satisfactory treatment of a non-singular del Pezzo surface of degree 4 with a conic bundle structure which admits a section over Q. Such surfaces are defined as the intersection of two quadrics in P4. When Q X(Q)= , we may assume that X P4 is cut out by the system 6 ∅ ⊂ Q Φ (x ,...,x ):=x x x x =0, 1 0 4 0 1 2 3 − (1.2) (Φ2(x0,...,x4)=0, forquadraticformsΦ ,Φ Z[x ,...,x ]suchthattheJacobianmatrix( Φ , Φ ) 1 2 0 4 1 2 ∈ ∇ ∇ has full rank throughout X. Let : R5 R be a norm. Given a point x = [x] P4(Q), with x = k ·k → >0 ∈ Q (x ,...,x ) Z5 such that gcd(x ,...,x ) = 1, we let H(x) := x . Then H is the 0 4 0 4 ∈ k k anticanonical height metrized by the choice of norm. Any line contained in X that is defined over Q will automatically contribute cB2+O(BlogB) to N (B), for an U,H appropriate constant c > 0. Hence it natural to take U X to be the open subset ⊂ formed by deleting the 16 lines from X. The best evidence that we have for (1.1) in the setting of non-singular surfaces of the shape (1.2)is due toSalberger. Inworkcommunicatedatthe conference“Higher dimensional varieties and rational points” at Budapest in 2001, he establishes the upper bound N (B)=O (B1+ε), (1.3) U,H X QUARTIC DEL PEZZO SURFACES WITH A CONIC FIBRATION 3 foranyε>0. Here,asthroughoutourwork,weallowtheimpliedconstanttodepend onthe choiceofε. The Maninconjecturehasreceiveda greatdealofattentioninthe contextofsingulardelPezzosurfacesofdegree3and4. Anaccountofrecentprogress can be found in the second author’s survey [7]. There is general agreement among researchersthatthelevelofdifficultyinestablishingtheexpectedasymptoticformula for del Pezzosurfacesincreasesas the degreedecreasesor as the singularitiesbecome milder. Among the non-singular del Pezzo surfaces, those of degree at least 6 are all toric and so are handled by the work of Batyrev and Tschinkel [1]. In [3] the first author gave the earliest satisfactory treatment of a non-singular del Pezzo surface of degree 5. As highlighted by Swinnerton-Dyer [25, Question 15], it has become somethingofmilestonetoestablishtheManinconjectureforasinglenon-singulardel Pezzo surface of degree 3 or 4. In this paper we will be concernedwith a quartic del Pezzo surface X P4 of the ⊂ Q shape (1.2), with Φ (x):=x2+x2+x2 x2 2x2. (1.4) 2 0 1 2− 3− 4 In particular X contains obvious lines defined over Q, from which it follows that X is Q-rational. This fact is recorded by Colliot-Thélène, Sansuc and Swinnerton-Dyer [10, Proposition2], for example. If L is a line in X defined overQ then it is a simple consequence of the fact that any plane through L must cut out a pair of lines L,L i on each quadric Φ =0 defining X, and the intersection L L meets X in exactly i 1 2 ∩ one further point, which is defined over Q. Let be the norm on R5 given by k·k 2 x :=max x , x , x , x , x . (1.5) 0 1 2 3 4 k k | | | | | | | | 3| | r n o All that is required of this norm is that x = max x , x , x , x for every 0 1 2 3 k k {| | | | | | | |} x = (x ,...,x ) R5 such that [x] X, and furthermore xσ = x , where xσ 0 4 ∈ ∈ k k k k is the vector obtained by permuting the variables x ,...,x and leaving x fixed. It 0 3 4 would be possible to work instead with the norm x := max x ,..., x , but not 0 4 | | {| | | |} withoutintroducingextratechnicaldifficultiesthatwewishtosuppressinthepresent investigation. We are now ready to reveal our main result. Theorem. — We have B(logB)4 N (B)=c B(logB)4+O , U,H X,H loglogB (cid:16) (cid:17) where c >0 is the constant predicted by Peyre. X,H During the final preparationof this paper, the authors have learnt of independent work by Fok-Shuen Leung [18] on the conic bundle surface given by (1.2) and (1.4). This sharpens Salberger’s estimate in (1.3), ultimately providing upper and lower bounds for N (B) that are of the expected order of magnitude. Our result super- U,H sedes this, and confirms the estimate predicted by Manin and Peyre in (1.1) for the non-singular quartic del Pezzo surface under consideration. The fact that the Picard group has rank 5, as needed to verify the power of logB, will be established in due course. 4 R. DE LA BRETE`CHE & T.D. BROWNING The proof of our theorem is long and complicated. We therefore dedicate the remainderofthis introductiontosurveyingsomeofits keyingredientsandindicating someobviouslines for further enquiry. Givenany non-singularsurfacedefined bythe system (1.2), it is possible to define a pair of conic bundle morphisms f : X P1, i → Q for i=1,2. Specifically, for any x X, one takes ∈ [x ,x ], if (x ,x )=(0,0), 0 2 0 2 f (x)= 6 1 ([x3,x1], if (x1,x3)=(0,0), 6 and [x ,x ], if (x ,x )=(0,0), 0 3 0 3 f (x)= 6 2 ([x2,x1], if (x1,x2)=(0,0). 6 Fora givenpointx X(Q)ofheightH(x)6B, it followsfromthe generaltheory of ∈ heightfunctions that there exists anindex i suchthat x∈fi−1(t) for some t∈P1Q(Q) ofheightO(B1/2). The ideais nowtocountrationalpointsofboundedheightonthe fibres f1−1(t) and f2−1(t), uniformly for points t ∈ P1Q(Q) of height O(B1/2). This is the strategy adopted by Salberger in his proof of (1.3). In the present situation, with the quadratic form (1.4), the fibres that we need to examine have the shape C : (a2 b2)x2+(a2+b2)y2 =2z2, (1.6) a,b − for coprime a,b Z. It is clear that C P2 is a non-singular plane conic when ∈ a,b ⊂ Q the discriminant ∆(a,b) = 2(a4 b4) is non-zero. The reduction of the counting − − problemtooneinvolvingthefamilyofconics(1.6)iscarriedoutin 3,wherewehave § avoidedusing the height machinery by doing things in a completely explicit manner. As is well-known there is a group homomorphism PicX Z, which to a divisor → class D PicX associates the intersection number of D with a fibre. The kernel ∈ of this map is generated by the “vertical” divisors, which up to linear equivalence are the irreducible components of the fibres. Since the non-singular fibres are all linearly equivalent, it follows that PicX has rank 2+n, where n is the number of split singular fibres above closed points of P1. In our case there are three closed Q points, corresponding to the irreducible factors a b,a+b and a2 +b2 of ∆(a,b). − Since each singular fibre is split it follows that PicX =Z5, as previously claimed. ∼ Roughly speaking, as one ranges over values of [a,b] P1(Q) of height O(B1/2), ∈ Q oneexpectstheretobeabout(logB)4 rationalpointsonC withheightrestrictedto a,b appropriateintervals. The preliminaryreductionto[a,b] P1(Q) ofheightO(B1/2) ∈ Q is absolutely pivotal here: it is only through this device that we can cover U(Q) with a satisfactorynumber of divisors. Were we chargedinstead with establishing an upper bound like (1.3), our analysis would now be relatively straightforward,thanks tothecontroloverthegrowthrateofrationalpointsonconicsaffordedbythesecond author’s joint work with Heath-Brown [8, Theorem 6]. A key aspect of this estimate is that it is uniform in the height of the conic, becoming sharper as the discriminant grows larger. In 4 we will take advantage of these arguments to eliminate certain § awkwardranges for a,b in (1.6). QUARTIC DEL PEZZO SURFACES WITH A CONIC FIBRATION 5 Obtaininganasymptoticformulaisafarmoreexactingtask. Usingthelargesieve inequality, Serre [24] has shown that most plane conics defined over Q don’t contain rational points. This phenomenon might pose problems for us, given that we want a uniform asymptotic formula for a Zariski dense set of rational points on the fibres. Our choice of surface has been tailored to guarantee that this doesn’t happen, since the corresponding fibres (1.6) always contain the rational point ξ =[1, 1,a]. (1.7) − In the classicalmanner we canuse this pointto parametriseallof the rationalpoints on the conic, which ultimately leads us to evaluate asymptotically the number of points belonging to a 2-dimensionalsublattice Λ Z2 which are constrained to lie in an appropriate region R R2. Both Λ and R de⊂pend on the parameters a and b, so ⊂ this estimate needs to be achieved with a sufficient degree of uniformity. Assuming thatR haspiecewisecontinuousboundary,wewouldideallyliketoapplythefamiliar estimate vol(R) #(Λ R)= +O(∂R+1), (1.8) ∩ detΛ where ∂R denotes the perimeter of R. Unhappily this estimate is too crude for our purposes. We will use Poisson summation to make the error term explicit. The reduction of the problem to a lattice point counting problem is carried out in 5 and § its execution is the subject of 6–8. §§ The one outstanding task, which is the focus of 9, is to evaluate asymptotically § the main term arisingfrom the lattice point counting problem. It turns out that this involves a sum of the shape g(a4 b4 ) | − | , max a,b 2 (a,b)X∈Z2∩R { } gcd(a,b)=1 for a certain multiplicative arithmetic function g that is very similar to the ordinary divisorfunctionτ(n):= 1. Theproblemofdeterminingtheaverageorderofthe dn divisor function as it ranges|over the values of polynomials has enjoyed considerable P attention in the literature. In the setting of binary forms current technology has limited us to handling forms of degree at most 4. When g = τ Daniel [11] has dealt with the case of irreducible binary quartic forms. We need to extend this argument to deal with a more generalclass of arithmetic functions and to binary quartic forms that are no longer irreducible. We have found it convenient to corral the necessary estimates into a separate investigation [5], which is of a more technical nature. The facts that we will need are recalled in 2. § Our main goalin this paper is to outline a generalstrategy for proving the Manin conjecturefornon-singulardelPezzosurfacesofdegree4equippedwithaconicbundle structure admitting a section over Q. In order to minimise the length and technical difficulty of the work, which is already considerable, we have chosen to illustrate the approachbyselectingaconcretesurfacetoworkwith. Itislikelythattheargumentwe present can be adapted to handle other conic bundle surfaces. For example, consider 6 R. DE LA BRETE`CHE & T.D. BROWNING the family of surfaces (1.2), with Φ (x):=c x2+c x2+c x2+c x2+c x2. 2 0 0 1 1 2 2 3 3 4 4 It is easy to check that X will be non-singular if and only if c c =0 and c c = 0 4 0 1 ··· 6 6 c c . As we’ve already mentioned, our success with (1.4) is closely linked to the 2 3 existenceofanobviousrationalpoint(1.7)onallofthefibres(1.6). Thisisequivalent to the map X(Q) P1(Q) being surjective. A necessary and sufficient condition for → Q ensuring this is that the morphismX P1 shouldadmit a sectionoverQ. This fact → Q has a long history and can be traced back to work of Lewis and Schinzel [19]. Even when the conic bundle surface admits a section, however, there remains considerable workto be doneinhandling ageneraldiagonalformΦ asabove. The corresponding 2 fibres will take the shape f (a,b)x2+f (a,b)y2+c z2 =0, 1 2 4 for binary forms f (a,b)= c a2+c b2 and f (a,b)= c a2+c b2. Thus, at the very 1 0 3 2 2 1 least, one needs analogues of our investigation [5] for the case in which f f factors 1 2 over Q as the product of two quadratic forms or splits completely. Notation. — Throughout our work N will denote the set of positive integers. If a,b N then we write gcd(a,b) for the greatest common divisor of a,b and [a,b] = ∈ ab/gcd(a,b) for the least common multiple. We set (a,b)♭ :=2−ν2(gcd(a,b))gcd(a,b) (1.9) for the odd part of the greatest common divisor, and use the symbol ♭ to indicate a summation in which all the variables of summation are restricted to odd integers. P Furthermore, our work will involve the arithmetic functions 1 ϕ(n) 1 ϕ(n):=n 1 , ϕ (n):= , ϕ (n):= 1+ . (1.10) ∗ † − p n p Yp|n(cid:16) (cid:17) Yp|n(cid:16) (cid:17) Finally, in terms of the parameter B, we set Z :=B1/loglogB, Z :=loglogB. (1.11) 1 2 We will reserve c > 0 for a generic absolute positive constant, whose value is always effectively computable. Acknowledgements. — Thisinvestigationwasundertakenwhilethesecondauthor was visiting the Université Paris 6 Pierre et Marie Curie and the Université Paris 7 Denis Diderot. Thehospitalityandfinancialsupportoftheseinstitutionsisgratefully acknowledged. It is a pleasure to thank Olivier Wittenberg for useful conversations relating to the geometry of conic bundle surfaces and the anonymous referee for numerous pertinent remarks. While working on this paper the second author was supported by EPSRC grant number EP/E053262/1. QUARTIC DEL PEZZO SURFACES WITH A CONIC FIBRATION 7 2. Technical results Ourworkrequiresanumberofauxiliaryresults,rangingfrombasicestimatesusing thegeometryofnumberstomoresophisticatedresultsconcerningthedivisorproblem for binary quartic forms. 2.1. Counting rational points on curves. — As made clear in the introduction to this paper, our proof of the theorem uses the conic bundle structure in order to focus the effort on a family of curves of low degree and rather low height. The following result is due to Heath-Brown [16, Lemma 3], and deals with the situation forlines inP2,the rationalpointsonwhichbasicallycorrespondto integerlattices of Q rank 2. Lemma 1. — Let Λ Z2 be a lattice of rank 2 and determinant detΛ, and let R R2 be any region ⊆with piecewise continuous boundary. Then we have ⊂ vol(R) # x Λ R :gcd(x ,x )=1 1+ . { ∈ ∩ 1 2 }≪ detΛ Our next uniform upper bound is extracted from joint work of the second author withHeath-Brown[8,Corollary2],andhandlesthecaseofnon-singularplaneconics. Lemma 2. — Let C P2 be a non-singular conic. Assume that the underlying ⊂ Q quadratic form has matrix of determinant ∆, and that the 2 2 minors have greatest × common divisor ∆ . Then we have 0 # x C Z3 : gcd(x1,x2,x3)=1, τ(∆) 1+ B1B2B3∆30/2 1/3. ∈ ∩ xi 6Bi,(16i63) ≪ | | ∆ (cid:26) | | (cid:27) (cid:16) | | (cid:17) Taking B = B = B = B in Lemma 2, we retrieve the well-known fact that a 1 2 3 non-singular plane conic C P2 contains O (B) rational points of height B. ⊂ Q C 2.2. Generalisation of Nair’s lemma. — In the setting of polynomials in only onevariablethereisawell-knownresultduetoNair[21]whichprovidesupperbounds fortheaverageorderofsuitablenon-negativearithmeticfunctionsastheyrangesover the values of the polynomial. Taking advantage of the authors’ refinement [4] of this work we have the following result. Lemma 3. — Let ε > 0, let a N and let δ 0,1 . Then for x aε there exists ∈ ∈ { } ≫ an absolute constant c>0 such that τ(n)δτ(n2+a) ϕ (a)cx(logx)1+δ. † ≪ n6x X Proof. — Define the multiplicative arithmetic function 2, if ν =1, τ (pν):= ′ ((1+ν)2, if ν >2, 8 R. DE LA BRETE`CHE & T.D. BROWNING foranyprimepowerpν. Wehaveτ(n )τ(n )6τ (n n )foranyn ,n N. LetS (x) 1 2 ′ 1 2 1 2 δ ∈ denote the sum that is to be estimated. On replacing τ by τ we find that ′ Sδ(x)6 τ′(nδ(n2+a)). n6x X Now for any a N it is clear that the polynomial f(t)=tδ(t2+a) has degree 2+δ, ∈ thatit hasdiscriminant∆ = 4a1+2δ andthatithas no fixedprime divisorif δ =0 f − or a is even. If δ = 1 and a is odd then 2 is a fixed prime divisor of f. But then we may break the sum into two sums according to whether n is even or odd and make a corresponding change of variables, absorbing the additional factor τ (2) = 2 into an ′ implied constant. An application of [4, Theorem 2] now reveals that ̺ (p) τ (m)̺ (m) f ′ f S (x) x 1 , δ ≪ − p m pY6x(cid:16) (cid:17)mX6x for x aε, where ̺ (m) denotes the number of roots modulo m of the congruence f ≫ f(n) 0modm. Recallthewell-knowninequalities̺f(pν)63p2νp(∆f) foranyprime ≡ power, and ̺ (pν)63 if p∤∆ . Then it follows that f f τ (m)̺ (m) τ (p)̺ (p) τ (pν)̺ (pν) ′ f 6 1+ ′ f + ′ f m p pν mX6x pY6x(cid:16) νX>2 (cid:17) τ (p)̺ (p) τ (pν)̺ (pν) 6exp ′ f + ′ f p pν (cid:16)pX6x pX6xνX>2 (cid:17) 2̺ (p) 3(1+ν)2p2(1+2δ)σ f exp + ≪ p pν (cid:16)pX6x pσkX4a1+2δνX>2 (cid:17) 2̺ (p) ϕ (a)cexp f , † ≪ p (cid:16)pX6x (cid:17) for a suitable absolute constant c>0. On noting that a ̺ (p)=1+δ+ − , f p (cid:16) (cid:17) for p>2, this therefore concludes the proof of the lemma. WewillalsoneedaversionofNair’slemmaforbinaryformsF Z[x ,x ]. Let F 1 2 ∈ k k denote the maximum modulus of its coefficients and set 1 gcd(n ,n ,m)=1 ̺ (m):= # (n ,n ) (0,m]2 : 1 2 , ∗F ϕ(m) 1 2 ∈ F(n1,n2) 0modm ≡ n o foranym N. Thefollowingestimateisdeducedfrom[4,Corollary1]byspecialising ∈ to the generalised divisor function τ (n):= 1. k n=d1···dk Lemma 4. — Let ε>0 and let F Z[x1,xP2] be a non-zero binary form of degree d, ∈ with Disc(F)=0 and F(1,0)F(0,1)=0. Then we have 6 6 τ (F(n ,n )) F ε B B E+max B ,B 1+ε , k 1 2 d,k 1 2 1 2 | | ≪ k k { } |n1X|6B1|n2X|6B2 (cid:16) (cid:17) QUARTIC DEL PEZZO SURFACES WITH A CONIC FIBRATION 9 where ̺ (p)(k 1) E := 1+ ∗F − . p d<p6mYin{B1,B2}(cid:16) (cid:17) 2.3. The divisor problem for binary forms. — Throughout this section we let i denote a generic element from the set 1,2 . Let L ,Q Z[x ,x ] be binary i 1 2 { } ∈ forms,withdegL =1anddegQ=2,suchthatL ,L arenon-proportionalandQis i 1 2 irreducibleoverQ. LetB [ 1,1]2 beaconvexregionwhoseboundaryisdefinedby ⊆ − apiecewisecontinuouslydifferentiablefunction. AssumethatL (x)>0andQ(x)>0 i foreveryx B. The workhorseinthis paperisanasymptoticformulaforsumsakin ∈ to τ(L (x)L (x)Q(x)) 1 2 , max x , x 2 1 2 gcxd∈(ZxX21∩,xX2)B=1 {| | | |} where τ is the divisor function and XB := Xx:x B . { ∈ } In fact the arguments that appear in our work call for a rather more general type of sum. Suppose that g = h τ is the Dirichlet convolution of the divisor function ∗ with a multiplicative arithmetic function h that satisfies h(d) | | 1. (2.1) d1/4 ≪ d N X∈ Let V [0,1]4 be a regioncut out by a finite number of hyperplanes with absolutely ⊆ bounded coefficients. For any Y >2 we define g(L (x),L (x),Q(x);Y;V):= (1 h)(d). 1 2 ∗ di=gcd(dd|,LL1i((xx))XL),2d(x3=)Qg(cxd)(d,Q(x)) (llooggdY1,llooggdY2,2lologgdY3,logmaxl{og|xY1|,|x2|})∈V Then we will encounter sums of the shape g(L (x),L (x),Q(x);Y;V) 1 2 S (X,Y;V):= . g max x , x 2 1 2 gcxd∈(ZxX21∩,xX2)B=1 {| | | |} If one takes V = [0,1]4 and Y a multiple of X then g(L ,L ,Q;Y;V) = g(L L Q). 1 2 1 2 Moreover,if one takes 1, if n=1, h(n)=U(n)= (0, otherwise, then one arrives at exactly the sum involving τ that was mentioned above. For any prime p and ν ,ν ,ν >0, let 1 2 3 p∤x, ̺†p(ν1,ν2,ν3):=#x∈(Z/pν1+ν2+ν3+1Z)2 : pνikLi(x),  (2.2)  pν3 Q(x)  k and   ̺†p(ν1,ν2,ν3):=p−2(ν1+ν2+ν3+1)̺†p(ν1,ν2,ν3). (2.3) 10 R. DE LA BRETE`CHE & T.D. BROWNING Here, we follow the convention that pν n if and only if ν (n) = ν. The following p k asymptotic formula is established in our companion paper [5, Corollary 3]. Lemma 5. — Let ε>0. Assume that 26X 6Y 6X1/ε. Then we have S (X,Y;V)=4C vol(B)vol(V )(logY)4+O (logX2/Y)(logY)3+(logX)3+ε , g ∗ 0 Li,Q where (cid:0) (cid:1) 1 3 C∗ := 1− p g(pν1+ν2+ν3)̺†p(ν1,ν2,ν3) (2.4) Yp (cid:16) (cid:17) νX∈Z3>0 and 1 V :=V v [0,1]4 :max v ,v ,v 6v 6 . (2.5) 0 1 2 3 4 ∩ ∈ { } 2 n o We will ultimately apply Lemma 5 with L L Q(a,b) equal to the discriminant 1 2 ∆(a,b) of the conic (1.6). The exponent of logY in the lemma reflects the fact that we are dealing with binary forms with three irreducible factors. The attentive reader will observe a correlationwith our description of the rank of PicX in 1. § We will be interested in applications of Lemma 5 when g : N R is the multi- → plicative arithmetic function defined via max 1,ν 1 , if p=2, g(pν):= { − } (2.6) (1+ν(1 1/p)/(1+1/p), if p>2. − It easily follows that g =h τ, with ∗ 0, if p>2 and ν >2, 2/(p+1), if p>2 and ν =1,  − h(pν)=(g∗µ∗µ)(pν)=10,, iiff pp==22 aanndd νν =>34,, (2.7) 0, if p=2 and ν =2, In particular h(n) n−1+ε for any ε>−10,, whence (2i.f1p)=ho2ldas.nd ν =1. | |≪ 3. Preliminary manipulations Recall the definition of the quadratic forms Φ ,Φ from (1.2) and (1.4). We begin 1 2 this section by relating N (B) to the quantity U,H gcd(x ,...,x )=1, max x ,...,x 6B, N1(B):=# x∈N5 : Φ1(x)0=Φ2(x3)=0, x0,{x10 = x23,}x3 , (3.1) (cid:26) { }6 { } (cid:27) inwhichthemaindifferenceisthatthecountisrestrictedtopositiveintegersolutions. This is achieved in the following result. Lemma 6. — We have N (B)=8N (B)+O(B). U,H 1