Counting rational points on quartic del Pezzo surfaces with a rational conic

Upper and lower bounds, of the expected order of magnitude, are obtained for the number of rational points of bounded height on any quartic del Pezzo surface over $\mathbb{Q}$ that contains a conic defined over $\mathbb{Q}$.


Introduction
A quartic del Pezzo surface X over Q is a smooth projective surface in P 4 cut out by a pair of quadrics defined over Q. When X contains a conic defined over Q it may be equipped with a dominant Q-morphism X → P 1 , all of whose fibres are conics, giving X the structure of a conic bundle surface. Let U ⊂ X be the Zariski open set for B 1, where H is the standard height function on P 4 (Q). The Batyrev-Manin conjecture [13] predicts the existence of a constant c 0 such that N (B) ∼ cB(log B) ρ−1 , as B → ∞, where ρ = rank Pic Q (X ) 6. To date, as worked out by de la Bretèche and Browning [2], the only example for which this conjecture has been settled is the surface with Picard rank ρ = 5. For a general quartic del Pezzo surface the best upper bound we have is N (B) = O ε,X (B 3 2 +ε ), for any ε > 0, which appears in forthcoming work of Salberger. In work presented at the conference "Higher dimensional varieties and rational points" at Budapest in 2001, Salberger noticed that one can get much better upper bounds for N (B) when X has a conic bundle structure over Q, ultimately showing that N (B) = O ε,X (B 1+ε ), for all ε > 0. Leung [21] revisited Salberger's argument to promote the B ε to an explicit power of log B. On the other hand, recent work of Frei, Loughran and Sofos [15,Thm. 1.2] provides a lower bound for N (B) of the predicted order of magnitude for any quartic del Pezzo surface over Q with a Q-conic bundle structure and Picard rank ρ 4. (In fact they have results over any number field and for conic bundle surfaces of any degree.) Our main result goes further and shows that the expected upper and lower bounds can be obtained for any conic bundle quartic del Pezzo surface over Q. Theorem 1.1 Let X be a quartic del Pezzo surface defined over Q, such that X (Q) = ∅. If X contains a conic defined over Q then there exist effectively computable constants c 1 , c 2 , B 0 > 0, depending on X , such that for all B B 0 we have It is worth emphasising that this appears to be the first time that sharp bounds are achieved towards the Batyrev-Manin conjecture for del Pezzo surfaces that are not necessarily rational over Q.
Let X be a quartic del Pezzo surface defined over Q, with a conic bundle structure π : X → P 1 . There are 4 degenerate geometric fibres of π and it follows from work of Colliot-Thélène [10] and Salberger [25], using independent approaches, that the Brauer-Manin obstruction is the only obstruction to the Hasse principle and weak approximation. Let δ 0 δ 1 4, where δ 1 is the number of closed points in P 1 above which π is degenerate and δ 0 is the number of these with split fibres. (Recall from [28, Def. 0.1] that a scheme over Q is called split if it contains a non-empty geometrically integral open subscheme.) It follows from [15,Lemma 2.2] that ρ = 2 + δ 0 . (1.1) For comparison, Leung's work [21,Chapter 4] establishes an upper bound for N (B) with the potentially larger exponent 1 + δ 1 . This exponent agrees with the Batyrev-Manin conjecture if and only if X → P 1 is a conic bundle with a section over Q, a hypothesis that our main result avoids. Our proof of the upper bound makes essential use of [29], where detector functions are worked out for the fibres with Q-rational points. Combining this with height machinery and a uniform estimate [7] for the number of rational points of bounded height on a conic, the problem is reduced to finding optimal upper bounds for divisor sums of the shape Here, n = δ 1 and 1 , . . . , n ∈ Z[s, t] are the closed points of P 1 above which π is degenerate, with G 1 , . . . , G n ∈ Z[s, t] being certain associated forms of even degree. Thus far, such sums have only been examined in the special case that G 1 , . . . , G n all have degree zero. In this setting, work of la Bretèche and Browning [1] can be invoked to yield the desired upper bound. Unfortunately, this result is no longer applicable when one of G 1 , . . . , G n has positive degree.
Using [15], we shall see in Sect. 3 that our proof of the lower bound in Theorem 1.1 may proceed for surfaces X → P 1 of Picard rank ρ = 2. In this case the fibre above any degenerate closed point of P 1 must be non-split by (1.1). Ultimately, following the strategy of [15], this leads to the problem of proving tight lower bounds for sums like (1.2) in the special case that none of the characters ( G i (s,t) · ) are trivial. One of the key ingredients in this endeavour is a generalised Hooley -function. Let K /Q be a number field and let ψ K be a quadratic Dirichlet character on K . We define an arithmetic function on integral ideals of K via The main novelty in our work lies in how we overcome the difficulty of divisor sums involving characters without a fixed modulus in (1.2). In Sect. 2.2, drawing inspiration from recent work of Reuss [24], we replace the divisor functions at hand by generalised divisor functions which run over certain integral ideal divisors belonging to the number field obtained by adjoining a root of i , for each 1 i n. Our proof of Theorem 1.1 then relies upon an extension to number fields of work by Nair and Tenenbaum [22] on short sums of non-negative arithmetic functions. This is achieved in an auxiliary investigation [8], the outcome of which is recorded in Sect. 2.1.

Nair-Tenenbaum over number fields
Let K /Q be a number field and let o K be its ring of integers. Denote by I K the set of ideals in o K . We say that a function f : We denote the class of all pseudomultiplicative functions associated to A, B and ε by M Note that any f ∈ M K satisfies the bounds f (a) A K (a) and f (a) (N K a) ε , for any a ∈ I K .
We will need to work with functions supported away from ideals of small norm. To facilitate this, for any ideal a ∈ I K and W ∈ N, we set (2.1) We extend this to rational integers in the obvious way. Similarly, for any f ∈ M K , we define f W (a) = f (a W ).

Remark 2.1
We will always assume that W is of the form for some w > 0 and ν a positive integer. Throughout Sect. 3 we shall take ν to be a large constant depending only on various polynomials that are determined by X , while in Sect. 4 we shall take ν = 1. In either case we have gcd(N K p, W ) = 1 if and only if p > w, if N K p = p f p for some f p ∈ N. Our notation is reminiscent of the "W -trick" that appears in work of Green and Tao [16]. Whereas in their context it is important that the parameter w tends to infinity, in our setting we shall choose w to be a suitably large constant, where the meaning of "suitably large" is allowed to change at various points of the proof.
be the multiplicative span of all prime ideals p ⊂ o K with residue degree f p = 1. For any x > 0 and f ∈ M K we set otherwise.
Suppose now that we are given irreducible binary forms F 1 , . . . , F N ∈ Z[x, y], which we assume to be pairwise coprime. Let i ∈ {1, . . . , N }. Suppose that F i has degree d i and that it is not proportional to y, so that b i = F i (1, 0) is a non-zero integer. It will be convenient to form the homogeneous polynomial This has integer coefficients and satisfiesF i (1, 0) = 1. We let θ i be a root of the monic polynomialF i (x, 1). Then θ i is an algebraic integer and we denote the associated number field of degree d i by K i = Q(θ i ). Moreover, for any (s, t) ∈ Z 2 . If b i = 0, so that F i (x, y) = cy for some non-zero c ∈ Z, we take θ i = −c and K i = Q in this discussion. Our work on Theorem 1.1 requires tight upper bounds for averages of , over primitive vectors (s, t) ∈ Z 2 , for general pseudomultiplicative functions f i ∈ M K i and suitably large w. For any k ∈ N and any polynomial P ∈ Z[x], we set To any non-empty bounded measurable region R ⊂ R 2 , we associate We say that such a region R is regular if its boundary is piecewise differentiable, R contains no zeros of F 1 · · · F N and there exists c 1 > 0 such that vol(R) K c 1 R . Bearing all of this in mind, the following result is [8, Thm. 1.1].

Lemma 2.2
Let R ⊂ R 2 be a regular region, let V = vol(R) and let G ⊂ Z 2 be a lattice of full rank, with determinant q G and first successive minimum Then, for any ε > 0 and w > w 0 ( where the implied constant depends at most on c 1 , Let 1 i n. In the statement of this result we recall the convention that the function f i,W q G is defined in such a way that
Let F, G ∈ Z[x, y] be non-zero binary forms with F irreducible, G of even degree and non-zero resultant Res(F, G). We shall assume that F has degree d and that it is not proportional to y. In particular b = F(1, 0) is a non-zero integer. Let W ∈ N. For any (s, t) ∈ Z 2 prim such that F(s, t) = 0, we define This is a modified version of the functions that appear in (1.2). We recall from (2.4) the associated binary formF(x, y) = b d−1 F(b −1 x, y), with integer coefficients and F(1, 0) = 1. We conclude that for all non-zero integer multiples c of b, we have We henceforth let θ be a root of the polynomial f (x) =F(x, 1). Then θ is an algebraic integer and K = Q(θ ) is a number field of degree d over Q. It follows that . We shall assume that L/K is a quadratic extension and we let D L/K be the ideal norm of the relative discriminant D L/K . Let f = f L/K be the conductor of the extension L/K . Let J f be the group of fractional ideals in K coprime to f and let P f be the group of principal ideals (a) such that a ≡ 1 (mod f) and a totally positive. As explained by Neukirch [23, §VII.10], the Artin symbol ψ(a) = ( L/K a ) gives rise to a character ψ : J f /P f → {±1} of the ray class group J f /P f , with a (mod P f ) → ( L/K a ). This has the property that ψ(p) = 1 if and only if p splits in L, for any unramified prime ideal p ∈ J f .
Note that D is a non-zero integer. Recall the definition (2.3) of P • K of the multiplicative span of degree 1 prime ideals. We shall mainly work with the subset cut out by ideals divisible by at most one prime ideal above each rational prime. It is not hard to see that P K has positive density in I K . The proof of the following result is inspired by an argument found in recent work of Reuss [24,Lemma 4].

Lemma 2.3
Let W ∈ N, let (s, t) ∈ Z 2 prim such that F(s, t) = 0, and let D be given by (2.10). Then the following hold: (i) a ∈ P K for any integral ideal a | (bs − θ t) such that gcd(N a, DW ) = 1; (ii) there exists a bijection between divisors a | (bs − θ t) with N a = k coprime to DW and divisors k |F(bs, t) coprime to DW , in which (k) = K (a) and In particular, when G(s, t) is the constant polynomial 1 in (2.8), then L = K and ψ is just the trivial character in part (iii). We note that K (a) = (N a) and τ K (a) = τ (N a) for any ideal a ∈ P K , where τ K (a) = d|a 1. Similarly, if h : N → R 0 is any arithmetic function, we have for any a ∈ P K . We shall use these facts without further comment in the remainder of the paper.
Part (i) is proved in [8,Lemma 2.3]. Turning to part (ii), it follows from (i) that ( p, n) is a prime ideal for any p | k. Thus there is a bijection between each factorisation |F(bs, t)| = ke, with gcd(k, DW ) = 1, and each ideal factorisation n = ab, with N a = k coprime to DW and N b = e. In order to complete the proof of part (ii) of the lemma, it will suffice to show that where p = ( p, n). Since G has even degree we have Recall the notation g(x) = G(b −1 x, 1). We may suppose that p = ( p, θ −n), for some n ∈ Z/ pZ such that bst − n ≡ 0 (mod p), and we recall from (2.10) that p 2D L/K . We observe that p splits in since n ≡ bst (mod p) and N p = p. Noting that g(bst) = G(st, 1), this completes the proof of part (ii). Finally, part (iii) follows from part (ii).
We close this section with an observation about the condition a | (bs − θ t) that appears in Lemma 2.3, the proof of which is found in [8,Lemma 2.4].

Uniform upper bounds for conics
Let Q ∈ Z[y 1 , y 2 , y 3 ] be a non-singular isotropic quadratic form. Denote its discriminant by Q and the greatest common divisor of the 2 × 2 minors of the associated matrix by D Q . It follows from [26, §IV.2] that there is a quadratic Dirichlet character χ Q such that for any prime p such that p | Q and p 2D Q .
The main aim of this section is to establish the following result.
with an absolute implied constant, where Since C(Q, w) τ ( Q ), this result is a refinement of work due to Browning and Heath-Brown [7,Cor. 2]. In fact, although not needed here, one can show that for any prime p 2D Q , the p-adic factor appearing above is commensurate with the p-adic Hardy-Littlewood density for the conic Q = 0. Furthermore, if this curve has no Q p -points for some prime p 2D Q , then the constant in the upper bound vanishes. Therefore, Lemma 2.5 detects conics with a rational point. This is the point of view adopted in the work of Sofos [29].

Proof of Lemma 2.5
The proof of [7, Cor. 2] relies on earlier work of Heath-Brown [17,Thm. 2]. The latter work produces an upper bound for the number of lattices (with determinant depending on the coefficients of Q) that any non-trivial zero of Q is constrained to lie in. For each prime p such that p ξ Q , it turns out that there are at most L( p ξ ) c p τ ( p ξ ) lattices to consider, where c p = 1 for p > 2.
Suppose that y ∈ Z 3 prim is a non-zero vector for which Q(y) = 0. Let p be a prime such that p ξ Q , with p 2D Q and χ Q ( p) = −1. On diagonalising over Z/ p ξ +1 Z, we may assume that for coefficients a 1 , a 2 ∈ Z such that p a 1 a 2 . In particular, we have Hence L( p ξ ) = 1 when ξ is even, since then y is merely constrained to lie on the lattice {y ∈ Z 3 : y 1 ≡ y 2 ≡ 0 (mod p ξ/2 )}. Likewise, when ξ is odd, there can be no solutions in primitive integers y.
Note that It follows that the total number of lattices emerging is with ξ odd and p 2D Q (resp. otherwise). This completes the proof of the lemma.

Lattice point counting
We will need general results about counting lattice points in an expanding region. Let D ⊂ R 2 \ {0} be a non-empty open disc and put δ(D) = D ∞ , in the notation of Sect. 2.1. Let b, c, q ∈ Z and x 0 ∈ Z 2 such that q 1 and gcd(x 0 , q) = 1. For each e ∈ N such that gcd(e, q) = gcd(b, c, e) = 1, we define the non-empty set We then fix, once and for all, a non-zero vector of minimal Euclidean length within (e) and we call it v(e). We are interested in as x → ∞. We shall prove the following result.
, v(e), N (x) be as above, and assume that |v(e)| δ(D)x. Then The implied constant in this estimate is absolute.
Moreover, using the basic properties of the minimal basis vector, one obtains These inequalities may be used to simplify the error term in Lemma 2.6.

Proof of Lemma 2.6
Our argument is based on a modification of the proof of [29,Lemma 5.3]. We write δ = δ(D) for short and put x 0 = (s 0 , t 0 ). Since gcd(s 0 , t 0 , q) = 1, an application of Möbius inversion gives on making the substitution s = mu and t = mv. The inner sum is empty if m is large enough. Indeed, if it contains any terms then we must have Thus, on using the Möbius function to remove the condition gcd(u, v, e) = 1, we find that Making the substitution u = ds and v = dt, and arguing as before we find that 1.
Now let n ∈ Z be such that n ≡ dm (mod q). Then we can make the change of variables (s, t) = n(s 0 , t 0 ) + q(s , t ) in the inner sum. Noting that (e/d) defines a lattice in Z 2 of determinant e/d, the inner sum is found to be with an absolute implied constant, since the upper bound on d implies that In summary, we have shown that 123 Counting rational points on quartic del Pezzo surfaces The contribution from the error term is The main term equals since (2.12) implies that the extra constraint in m-sum is implied by the constraint in the d-sum. But this is equal to which thereby completes the proof.

Twisted Hooley 1-function over number fields
Adopting the notation of Sect. 1, it is now time to reveal the version of the Hooleyfunction that arises in our work. Let K /Q be a number field and let ψ K be a quadratic Dirichlet character on K . We let : I K → R >0 be the function given by (2.14) for any integral ideal a ∈ I K . We shall put (a) = (a; 1) for the corresponding function in which ψ K is replaced by the constant function 1.
We begin by showing that belongs to the class M K of pseudomultiplicative functions introduced in Sect. 2.1. For coprime ideals a 1 , a 2 ⊂ o K , any ideal divisor d | a 1 a 2 can be written uniquely as Thus the triangle inequality yields (a 1 a 2 ; ψ K ) τ K (a 1 ) (a 2 ; ψ K ), where τ K is the divisor function on ideals of o K . This shows that (·, ψ K ) belongs to M K and an identical argument confirms this for (·).
We shall need the following result proved in [30].

Lemma 2.7 Define the function
for any x 1 and recall the definition (ii) Let ψ K be a quadratic Dirichlet character on K and let W ∈ N. There exists a positive constant c = c(K , ψ K ) such that The implied constant in both estimates is allowed to depend on K and, in the second estimate, also on W and the character ψ K .

The lower bound
In order to prove the lower bound in Theorem 1.1, we first appeal to work of Frei, Loughran and Sofos [15]. It follows from [15, Thm. 1.2] that the desired lower bound holds when ρ 4. Suppose that ρ = 3. Then (1.1) implies that in the fibration π : X → P 1 there is at least one closed point P ∈ P 1 above which the singular fibre X P is split. Since the sum c(π ) defining the complexity of π in [15, Def. 1.5] is at most 4 for conic bundle quartic del Pezzo surfaces, we infer that c(π ) 3 when ρ = 3, so that the lower bound in Theorem 1.1 is a consequence of [15,Thm. 1.7]. Throughout this section, it therefore suffices to assume that ρ = 2 and δ 0 = 0, so that X is a minimal conic bundle surface.
Invoking [15,Thm. 1.6], the lower bound in Theorem 1.1 is a direct consequence of the divisor sum conjecture that is recorded in [14, Con. 1], for the relevant data associated to the fibration π . Note that the principal result in [14] only covers cubic divisor sums, since we still lack the technology to asymptotically evaluate divisor sums of higher degree with a power saving in the error term. The goal of this section is to estimate certain quartic divisor sums, with a logarithmic saving in the error term, which turns out to be sufficient for proving the lower bound in Theorem 1.1. The divisor sums relevant here shall involve complicated quadratic symbols whose modulus tends to infinity, a delicate task that will be the entire focus of this section.
We proceed to explain the particular case of the divisor sum conjecture that is germane here. Assume that we have forms F 1 , . . . , F n , G 1 , . . . , G n ∈ Z[x, y] with For each i such that F i (1, 0) = 0, we define the associated binary formF i (x, y) (1, 0). For such i we let θ i ∈ Q be a fixed root ofF i (x, 1) = 0. If, on the other hand, F i (x, y) is proportional to y, we define θ i = −F i (0, 1). We may assume that and that G i (θ i , 1) / ∈ Q(θ i ) 2 for every i, because in the correspondence outlined in [15], the binary forms F 1 , . . . , F n are equal to the closed points 1 , . . . , n from Sect. 1. Indeed, under this correspondence, the statement G i (θ i , 1) / ∈ Q(θ i ) 2 is equivalent to the singular fibre above i being non-split, which holds for any i since we are working with minimal conic bundle surfaces. Let We need to prove that there exists a finite set of primes S bad = S bad (F i , G i ) such that for all W ∈ N, all (s 0 , t 0 ) ∈ Z 2 prim , and all non-empty compact discs D ⊂ R 2 , which together satisfy the conditions Here, we recall the notation m W = p W p ν p (m) for all m, W ∈ N. We shall prove this conjectured lower bound when S bad is taken to be the set of all primes up to a constant w = w(F i , G i ). In what follows we shall often write that we need to enlarge w. This statement is to be interpreted as having already taken a very large constant w at the outset of the proof of the conjecture, rather than increasing w within the confines of the lower bound arguments. The primary goal of this section is now to establish the following bound, which directly leads to the lower bound in Theorem 1.1.
Here the implied constant depends on F i , G i , s 0 , t 0 , D, w and W , but not on x.
Suppose that ν > ν p (W ) for all p | W and write W 0 = p|W p ν . Then, since every summand in (3.3) is non-negative and In this way we see that it will suffice to prove the lower bound in Proposition 3.1 under the assumption that W = p|W p ν with In this case the identity ) for any (s, t) appearing in the outer summation of (3.3) and any p | W . Hence, for such (s, t), we can always assume that

Dirichlet's hyperbola trick
Let i ∈ {1, . . . , n}. For any (s, t) ∈ Z 2 appearing in (3.3), let Then, possibly on enlarging w, it follows from Lemma 2.3 that where d runs over integral ideals of K i = Q(θ i ), N i denotes the ideal norm N K i /Q and P i = P K i , in the notation of (2.11). Furthermore, for all (s, t) in (3.3), we have for some positive constant c i that depends at most on F i and D. We define X = x max c Dirichlet's hyperbola trick implies that We proceed by introducing the quantity for some α > 0 that will be determined in due course. (When n > 1 we shall take α to be a large constant, but when n = 1 it will be important to restrict to 0 < α < 1.) For (s, t) appearing in (3.3), we proceed by defining and r (∞) n (s, t) = d|(b n s−θ n t), d∈P n gcd(N n d,W )=1 As before, we may now write r n (s, t) = r (∞) n (s, t) + r (0) n (s, t) + r (1) n (s, t). (3.8) For each j = ( j 1 , . . . , j n ) ∈ {0, 1} n , we define in which we recall the definition (3. 2) of f . (Here, we recall our convention that products over empty sets are equal to 1.) Injecting (3.6) and (3.8) into (3.3) yields The validity of Proposition 3.1 is therefore assured, provided we can show that and We shall devote Sects. 3.2-3.4 to the proof of (3.10) and Sect. 3.5 to the proof of (3.9).

The generalised Hooley 1-function
In this section we initiate the proof of (3.10). Define It immediately follows that we use Cauchy's inequality to arrive at Recall the definition (2.14) of the twisted Hooley -function (a; ψ n ) associated to the Dirichlet character ψ n and any integral ideal a. Putting In summary, we have shown that Therefore, in order to prove (3.10), it will be sufficient to prove that there exists a constant δ > 0, that depends only on the data given at the start of Sect. 3, such that and We shall call B ∞ (x) the interval sum and H ∞ (x) the Bretèche-Tenenbaum sum.

The interval sum
By recycling work of la Bretèche and Tenenbaum [4, § 7.4], the case n = 1 is easy to handle. Indeed, in this case F 1 is an irreducible quartic form and (3.12) becomes Note that assumption (C2) ensures that |F 1 (s, t)| 1 whenever (s, t) ∈ D. Increasing w so that every prime factor of b 1 also divides W , shows that Thus it follows from (3.4) thatF 1 (s, t) W |F 1 (s, t)|, for implied constants that depend on F 1 , s 0 , t 0 , w and W . Hence Therefore, on introducing e through the factorisation de = (b 1 s − θ 1 t) W , we can infer that we must have either Without loss of generality we shall assume that we are in the former setting. Therefore there exist constants c 0 , c 1 > 0 such that But now we can employ the bound [4, Eq. (7.41)], with This implies that for any η ∈ (0, 1 2 ), we have where Q(λ) = λ log λ−λ+1. In particular, Q(2η) → 1 as η → 0+ and Q(1+η) > 0 for all η > 0. Recalling the definition (3.7) of L, this means that provided α < 1, we may choose η > 0 small enough (but away from 0), so as to ensure that (3.14) holds when F is irreducible.

First case: (b n s − Â n t) has many prime divisors
We denote by B (1) ∞ (x) the contribution to B ∞ (x) from the set of vectors (s, t) for which n ((b n s − θ n t) W ) > (1 + η) log log x, where n (a) = K n (a) is the total number of prime ideal factors of an ideal a ⊂ o K n . Recall that, as in Sect. 3.1, we denote N K n (a) by N n (a). We have , t), Our plan is now to apply Lemma 2.2 for N = n, with f N (a) = (1 + η) n (a W ) and (2, B, ε). Thus, in the notation of Lemma 2.2, one can take (3.17) When i = N , however, we will show that for every ε > 0 there exists w such that if W is given by (2.2) then η, 1, ε).
Taking w 2 1/ε , so that (1 + η) w ε , yields This means that in the notation of Lemma 2.2 one can take ε N = ε. (3.18) Furthermore, we shall take G = Z 2 and R = xD. Thus q G = 1, R is regular and we have V x 2 and K R x log x, in the notation of the lemma. This means that for large x we can take c 1 = 1, hence by (3.1), (3.17) and (3.18) Therefore, assuming that ε ∈ (0, 1) is fixed, the relevant constant in Lemma 2.2 is ε 0 = max{5, 20 + 12ε}4ε 199ε. This shows that if ε is fixed and 200ε < 1/3 then hence the secondary term of Lemma 2.2 makes a satisfactory contribution. The contribution of the first term of Lemma 2.2 towards the sum in (3.16) is The proof of these estimates is standard and will not be repeated here. (See Heilbronn [18], for example.) Thus B (1) ∞ (x) x 2 (log x) −(1+η) log(1+η)+η . The exponent of the logarithm is strictly negative for all η > 0, which is clearly sufficient for (3.14).

Second case: (b n s − Â n t) has few prime divisors
We denote by B (2) ∞ (x) the contribution to B ∞ (x) from the set of vectors (s, t) for which n ((b n s − θ n t) W ) (1 + η) log log x. Recall from the definition (3.11) of A (∞) n (x) that there exists d ∈ P n such that d | (b n s − θ n t), with gcd(N n d, W ) = 1 and Condition (C3) ensures that N n ((b n s − θ n t) W ) X d n . Defining e via the factorisation de = (b n s − θ n t) W , we can then infer that gcd(N n e, W ) = 1 and e ∈ P n , with L −1 X dn 2 N n e L X dn 2 , where the implied constants depend at most on D and F n . Note that Thus, either n (d) 1 2 (1 + η) log log x, or n (e) 1 2 (1 + η) log log x. We will assume without loss of generality that we are in the latter case.

It follows that
This is a non-archimedean version of Dirichlet's hyperbola trick, where instead of looking at the complimentary divisor to reduce the size, we have tried to reduce the number of prime divisors. Lemma 2.4 implies that the condition e | (b n s −θ n t) defines a lattice in Z 2 of determinant e = N n e, which we shall call G. Hence we may write Let v ∈ Z 2 be such that |v| = max{|v 1 |, |v 2 |} is the first successive minimum of G. Lemma 2.2 can be applied with R = xD, q G = e, N = n − 1, and for 1 i n − 1. For such f i one can take ε i in Lemma 2.2 to be arbitrarily small, whence (Note that h * W (e) = h * (e), since gcd(e, W ) = 1.) We have e = N n e L X dn 2 and so |v| L X dn 2 √ L X, since d n 2. Since F n is irreducible, we note that d n = 1 when F n (v) = 0. Next, we introduce g(e) = {e ∈ P n : N n e = e}. The second term is therefore seen to make the overall contribution which is satisfactory. Next, the overall contribution from the term e .

g(e)h * (e)A − (e) .
Then it follows from Shiu's work [27] that Partial summation now leads to the estimate The exponent of log x is strictly negative for all η ∈ (0, 1), which thereby completely settles the proof of (3.14).

The Bretèche-Tenenbaum sum
We saw in Sect. 2.5 that the Hooley -function defined in (2.14) belongs to M n . The stage is now set for an application of Lemma 2.2 with N = n and G = Z 2 , and with f N (a) = (a; ψ n ) 2 and f i (a) = d|a ψ i (d), for i < N . For such f i one can take ε i in Lemma 2.2 to be arbitrarily small, whence this gives in (3.13). The statement of (3.15) now follows from part (ii) of Lemma 2.7.

Small divisors
In this section we establish (3.9), as required to complete the proof of Proposition 3.1.
When n > 1, the proof follows from the treatment in [15] and will not be repeated here. Thus, provided that one takes α to be sufficiently large in the definition (3.7) of L, one gets an asymptotic formula for D j (x) with a logarithmic saving in the error term. The proof of (3.9) when n = 1 is more complicated. In this case F 1 is an irreducible binary quartic form. In order to simplify the notation, we shall drop the index n = 1 in what follows (and in particular, we shall denote P K 1 = P 1 by P). Our task is to estimate  s, t).
The condition e | (bs − θ t) defines a lattice in Z 2 of determinant N e by Lemma 2.4. Thus we can apply Lemma 2.2, finding that (s,t)∈Z 2 prim ∩xD e|(bs−θt) for any ε > 0, where h * is given by (2.6) with N = 1. Hence we arrive at the overall contribution from N e > y. Taking y = log log x, we therefore conclude that Note that by enlarging w we may assume that any prime factor of b is present in the factorisation of W . We henceforth focus on the case j = 0, the case j = 1 being similar. First, we define for any a ∈ P with gcd(N a, W ) = 1 the set H (a) = (s, t) ∈ Z 2 : a | (bs − θ t) .
By Lemma 2.4 there exists k = k(a) ∈ Z such that a vector (s, t) ∈ Z 2 belongs to H (a) if and only if N a | bs − kt. Therefore, H (a) is a lattice in Z 2 of determinant N a. Recalling the definition of r (0) (s, t) we obtain Arguing as in [15,, once inserted into (3.19), the contribution from the main term (denoted by M ψ in [15]) in Lemma 2.6 is x 2 . This is satisfactory for (3.9). It remains to consider the effect of substituting the error term in Lemma 2.6. Let for any m ∈ N, where we recall that P • is the multiplicative span of prime ideals with residue degree 1. This function is multiplicative and has constant average order. We claim that r * (cd) r * (c)r * (d) for all c, d ∈ N, which we shall keep in use throughout this section. It is enough to consider the case c = p a and d = p b for a rational prime p W with r * ( p) = 0. Letting p 1 , . . . , p m+1 be all the degree 1 prime ideals above p, we easily see that r * ( p k ) = k+m m . We therefore have to verify that for all integers a, b, m 0. This is obvious when m = 0. When m 1 the inequality is equivalent to the validity of which is clear. The error term in Lemma 2.6 is composed of two parts. According to (2.13), the second part contributes we conclude that the second part contributes Writing q = cd and recalling r * (cd) r * (c)r * (d), this is This is satisfactory for any α > 0 in (3.7). Finally, the overall contribution from the first part of the error term of Lemma 2.6 is It is clear that u | d and u | e. Moreover, one easily checks that where with the caveat that v(d ) still depends on d and e. Moreover if there exists d ∈ P with gcd(N d, The contribution from d , d for which |v(d )| x/(log x) ϒ is seen to be by [1]. Here we have used the fact that r * (d ) τ 4 (d ) and where r K are the coefficients in the associated Dedekind zeta function. Once inserted into (3.20) this contributes which is satisfactory, on taking ϒ sufficiently large.
In the opposite case, we plainly have d 2ϒ . Thus it remains to study the contribution where we recall that k depends on d and e. For any d ∈ P with N d = d u and gcd(N d, by Lemma 2.4. On appealing to (3.21) to estimate the u -sum, we are left with the contribution 1.
We will need to restrict the outer sum to a sum over primitive vectors in order to bring Lemma 2.2 into play.
where f is defined to be the greatest common ideal divisor of d 1 and (h). Writing c = f −1 d 1 , we see that Splitting into e-adic intervals the inner sum is easily seen to be where (·) = (·, 1), in the notation of Sect. 2.5. Since there are at most r * (h) ideals f ∈ P such that f | (h) and gcd(N f, W ) = 1, we are left with the final contribution Splitting into dyadic intervals, we now apply Lemma 2.2 with G = Z 2 , combined with part (i) of Lemma 2.7. Noting that one can take ε 1 > 0 in Lemma 2.2 to be arbitrarily small, we deduce that the sum over w can be bounded by for any ε > 0. This leads to the overall bound which thereby completes the proof of (3.9).

The upper bound
This section is concerned with proving the upper bound in Theorem 1.1. Let X be a quartic del Pezzo surface defined over Q, containing a conic defined over Q. We continue to follow the convention that all implied constants are allowed to depend in any way upon the surface X . We appeal to [15,Thm. 5.6 and Rem. 5.9]. This shows that there are binary quadratic forms q where y s,t = max{|s|, |t|} max{|y 1 |, |y 2 |} and s,t is a separable quartic form. The indices i = 1, 2 are related to the existence of the two complimentary conic bundle fibrations. The two cases i = 1, 2 are treated identically and we shall therefore find it convenient to suppress the index i in the notation. It is now clear that we will need a good upper bound for the number of rational points of bounded height on a conic, which is uniform in the coefficients of the defining equation, a topic that was addressed in Sect. 2.2.

Application of the bound for conics
Returning to (4.1), we apply Lemma 2.5 to estimate the inner cardinality. For any (s, t) ∈ Z 2 prim , an argument of Broberg [5,Lemma 7] shows that D Q s,t = O(1). In our work W is given by (2.2), with ν = 1 and w a large parameter depending only on X , which we will need to enlarge at various stages of the argument. In the first instance, we assume that 2D Q s,t < w 1. We deduce that We put for any (s, t) ∈ Z 2 prim . Note that S(s, t) 0. Our work so far shows that Since we are only interested in coprime integers s, t, there is a satisfactory contribution of O(B) to the right hand side from those vectors (s, t) in which one of the components is zero. Hence, by symmetry, Theorem 1.1 will follow from a bound of the shape since (1.1) implies that m + 1 = ρ − 1.

Reduction to divisor sums
For β ∈ C and x, y > 0 we let Consider the divisor function where S(s, t) is given by (4.5). In this section we shall establish (4.6) subject to the following bound for D β (x, y), whose proof will occupy the remainder of the paper.
We proceed to show how (4.6) follows from Proposition 4.1. Since (s, t) is separable, it may contain the polynomial factor t at most once. Therefore there exists c 0 ∈ Q * and pairwise unequal α i , α j ∈ Q such that (s, t) admits the factorisa- If (s, t) ∈ A then (s, t) |t| 4 and it follows that Breaking into dyadic intervals T /2 < |t| T and applying Proposition 4.1 with x = y = T and β = 0, we readily find that the right hand side is O((log B) m+1 ), which is satisfactory for (4.6).
It remains to consider the contribution to (4.6) from (s, t) ∈ Z 2 prim \ A . For each i we define Moreover, the implied constant is effective and only depends on the coefficients of (s, t). The contribution to (4.9) from L is therefore seen to be since for given t there are finitely many integers s in the interval |s − α i t| < 1. This completes the deduction of (4.6) from Proposition 4.1.

Small divisors
The function τ 0 (s, t) in (4.5) is concerned with the contribution to S(s, t) from small primes p w. Our work in Sect. 2.2 only applies to divisor sums supported away from small prime divisors. Hence we shall begin by using the geometry of numbers to deal with the function τ 0 (s, t), before handling the remaining factors in S(s, t).
Following Daniel [11], for any a ∈ N we call two vectors x, y ∈ Z 2 equivalent modulo a if gcd(x, a) = gcd(y, a) = 1 and