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

arXiv:1609.09057v1 [math.NT] 28 Sep 2016

COUNTING RATIONAL POINTS ON QUARTIC DEL PEZZO SURFACES WITH A RATIONAL CONIC T. D. BROWNING AND E. SOFOS Abstract. 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 Q that contains a conic defined over Q.

Contents 1. Introduction 2. Preliminary results 3. Nair–Tenenbaum over number fields 4. The lower bound 5. The upper bound References

1 6 15 29 41 49

1. Introduction 1.1. The Manin conjecture for quartic del Pezzo surfaces. A quartic del Pezzo surface X over Q is a smooth projective surface in P4 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 Ñ P1 , all of whose fibres are conics, giving X the structure of a conic bundle surface. Let U Ă X be the Zariski open set obtained by deleting the 16 lines from X and consider the counting function NpBq “ 7tx P UpQq : Hpxq ď Bu,

for B ě 1, where H is the standard height function on P4 pQq. The Batyrev– Manin conjecture [12] predicts the existence of a constant c ě 0 such that NpBq „ cBplog Bqρ´1 , as B Ñ 8, where ρ “ rank PicQ pXq ď 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 x0 x1 ´ x2 x3 “ x20 ` x21 ` x22 ´ x23 ´ 2x24 “ 0, Date: September 29, 2016. 2010 Mathematics Subject Classification. 11G35 (11G50, 14G05). 1

(1.1)

2

T. D. BROWNING AND E. SOFOS

with Picard rank ρ “ 5. For a general quartic del Pezzo surface the best 3 upper bound we have is NpBq “ Oε,X pB 2 `ε q, for any ε ą 0, which is due to Salberger [24]. 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 NpBq when X has a conic bundle structure over Q, ultimately showing that NpBq “ Oε,X pB 1`ε q, for all ε ą 0. (The surface (1.1) provides an example of this.) Leung [20] 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 [14, Thm. 1.2] provides a lower bound for NpBq of the predicted order of magnitude for any quartic del Pezzo surface over Q with a Q-conic bundle structure and Picard rank ρ ě 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 XpQq ‰ ∅. If X contains a conic defined over Q then there exist constants c1 , c2 ą 0, depending on X, such that c1 Bplog Bqρ´1 ď NpBq ď c2 Bplog Bqρ´1 .

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 Ñ P1 . There are 4 degenerate geometric fibres of π and it follows from work of Colliot-Thélène [10] and Salberger [23], using independent approaches, that the Brauer–Manin obstruction is the only obstruction to the Hasse principle and weak approximation. Let δ0 ď δ1 ď 4, where δ1 is the number of closed points in P1 above which π is degenerate and δ0 is the number of these with split fibres. (Recall from [27, Def. 0.1] that a scheme over Q is called split if it contains a non-empty geometrically integral open subscheme.) A standard calculation (cf. [8, Lemma 1]) shows that ρ “ 2 ` δ0 .

(1.2)

For comparison, Leung’s work [20, Chapter 4] establishes an upper bound for NpBq with the potentially larger exponent 1 ` δ1 . This exponent agrees with the Batyrev–Manin conjecture if and only if X Ñ P1 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 [28], where detector functions are worked out for the fibres with Q-rational points. Combining this with height machinery and a uniform estimate [9] for the number of rational

RATIONAL POINTS ON QUARTIC DEL PEZZO SURFACES

3

points of bounded height on a conic, the problem is reduced to finding optimal upper bounds for divisor sums of the shape n ÿ ź ÿ ˆ Gi ps, tq ˙ . (1.3) di 2 i“1 d |∆ ps,tq ps,tqPZ maxt|s|,|t|uďx

i

i

Here, n “ δ1 and ∆1 , . . . , ∆n P Zrs, ts are the closed points of P1 above which π is degenerate, with G1 , . . . , Gn P Zrs, ts being certain associated forms of even degree. Thus far, such sums have only been examined in the special case that G1 , . . . , Gn all have degree zero. In this setting, work of de la Bretèche and Browning [1], which heralds from pioneering work of Shiu [26], can be invoked to yield the desired upper bound. Unfortunately, this result is no longer applicable when one of G1 , . . . , Gn has positive degree. Using [14], we shall see in §4 that our proof of the lower bound in Theorem 1.1 may proceed for surfaces X Ñ P1 of Picard rank ρ “ 2. Thus the fibre above any degenerate closed point of P1 must be non-split by (1.2). Ultimately, following the strategy of [14], this leads to the problem of proving tight lower bounds for sums like (1.3) in the special case that none of the characters p Gi ps,tq q are trivial. One of the key ingredients in this endeavour ¨ is a generalised Hooley ∆-function. Let K{Q be a number field and let ψK be a quadratic Dirichlet character on K. We define an arithmetic function on integral ideals of K via ˇ ˇ ÿ ˇ ˇ ψK pdqˇ, ∆pa; ψK q “ sup ˇ uPR 0ďvď1

d|a eu ăNK dďeu`v

for any a in the ring of integers oK of K, where NK d “ 7oK {d denotes the ideal norm. When K “ Q this recovers the twisted ∆-function considered by de la Bretèche–Tenenbaum [4] and Br¨ udern [7]. Our treatment of the lower bound requires a second moment estimate for ∆pa; ψK q and this is supplied in a companion paper of Sofos [29]. Remark 1.2. Châtelet surfaces provide the other family of conic bundle surfaces of degree 4. When they are defined over Q, the Batyrev–Manin conjecture also makes a prediction for the distribution of Q-rational points on them. Work of Browning [8] shows that the relevant counting function satisfies an upper bound of the expected size. Although we shall not provide any details here, if we suppose that the Châtelet surface has a Q-rational point, then a lower bound of the proper size follows from the work in this paper, on taking the forms G1 , . . . , Gn to have degree 0 in (1.3).

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1.2. Nair–Tenenbaum over number fields. The main novelty in our work lies in how we overcome the difficulty of divisor sums involving characters without a fixed modulus in (1.3). In §2.1, drawing inspiration from recent work of Reuss [22], we replace the divisor functions at hand by generalised divisor functions which run over integral ideal divisors belonging to the number field obtained by adjoining a root of ∆i , for each 1 ď i ď n. Our proof of Theorem 1.1 then relies upon an extension to number fields of work by Nair and Tenenbaum [21] on short sums of arithmetic functions. Let K{Q be a number field and let oK be its ring of integers. Denote by IK the set of ideals in oK . We say that a function f : IK Ñ Rě0 is pseudomultiplicative if there exists a constant A and a function B “ Bpεq such that ( f pabq ď f paq min AΩK pbq , BpNK bqε , ř for all coprime ideals a, b P IK , where ΩK pbq “ p|b νp pbq. We denote the class of all pseudomultiplicative functions associated to A and B by MK “ MK pA, Bq. When K “ Q, this class contains the class of submultiplicative functions that arose in [26]. Note that any f P MK satisfies f paq ď AΩK paq and f paq !ε pNK aqε , for any a P IK . We will need to work with functions supported away from ideals of small norm. To facilitate this, for any ideal a P IK and W P N, we set ź aW “ pν . (1.4) pν }a gcdpNK p,W q“1

We extend this to rational integers in the obvious way. Next, for any f P MK , we define fW paq “ f paW q. We will always assume that W is of the form ź W “ p, (1.5) pďw

for some w ą 0. Thus gcdpNK p, W q “ 1 if and only if p ą w, if NK p “ pfp for some fp P N. Let PK “ ta Ă oK : p | a ñ fp “ 1u

(1.6)

be the multiplicative span of all prime ideals p Ă oK with residue degree fp “ 1. For any x ą 0 and f P MK we set ¨ ˛ ˚ ‹ ˚ ÿ f ppq ‹ ˚ ‹, Ef px; W q “ exp ˚ ‹ N p ˝pPPK prime K ‚ wăNK pďx fp “1


5

if f is submultiplicative, and Ef px; W q “

ÿ

NK aďx aPPK a square-free gcdpNK a,W q“1

f paq , NK a

otherwise. Suppose now that we are given irreducible binary forms F1 , . . . , FN P Zrx, ys, which we assume to be pairwise coprime. Let i P t1, . . . , Nu. Suppose that Fi has degree di and that it is not proportional to y, so that bi “ Fi p1, 0q is a non-zero integer. It will be convenient to form the homogeneous polynomial F˜i px, yq “ bdi ´1 F pb´1 x, yq. (1.7) i

i

This has integer coefficients and satisfies F˜i p1, 0q “ 1. We let θi be a root of the monic polynomial F˜ px, 1q. Then θi is an algebraic integer and we denote the associated number field of degree di by Ki “ Qpθi q. Moreover, NK {Q pbi s ´ θi tq “ F˜i pbi s, tq “ bdi ´1 Fi ps, tq, (1.8) i

i

2

for any ps, tq P Z . If bi “ 0, so that Fi px, yq “ cy for some non-zero c P Z, we take θi “ ć and Ki “ Q in this discussion. Our work on Theorem 1.1 requires tight upper bounds for averages of f1,W ppb1 s ´ θ1 tqq . . . fN,W ppbN s ´ θN tqq, over primitive vectors ps, tq P Z2 , for general pseudomultiplicative functions fi P MKi and suitably large values of w. More notation is required to state our main result in this direction. For any k P N and any polynomial P P Zrxs, we set We put

ρP pkq “ 7tx pmod kq : P pxq ” 0 pmod kqu. ρi pkq “

and

#

ρFi px,1q pkq 1

if Fi p1, 0q ‰ 0, if Fi p1, 0q “ 0,

˙´1 źˆ ρ1 ppq ` ¨ ¨ ¨ ` ρN ppq 1´ h pkq “ . p ` 1 p|k ˚

To any non-empty bounded measurable region R Ă R2 , we associate volpRq , KR “ 1 ` }R}8 ` BpRq logp1 ` }R}8 q ` 1 ` }R}8

(1.9)

(1.10)

(1.11)

(1.12)

where }R}8 “ suppx,yqPR t|x|, |y|u. We say that such a region R is regular if its boundary is piecewise differentiable, R contains no roots of F1 ¨ ¨ ¨ FN and there exists c0 , c1 ą 0 such that KRc0 ď volpRq ď KRc1 . Bearing all of this in mind, we may now record the following result.

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Theorem 1.3. Let R Ă R2 be a regular region, let V “ volpRq and let G Ă Z2 be a lattice of full rank, with determinant qG and first successive minimum λG . Assume that qG ď V c2 for some c2 ą 0. Let fi P MKi pAi , Bi q, for 1 ď i ď N. Then, for any ε ą 0 and w ą w0 pfi , Fi , Nq, we have ÿ

ps,tqPZ2prim XRXG

KR1`ε h˚W pqG q ź V Ef pV ; W q ` , fi,qG W ppbi s ´ θi tqq! plog V qN qG i“1 i λG i“1

N ź

N

where the implied constant depends at most on c0 , c1 , c2 , Ai , Bi , ε, N, W . It is interesting to compare this result with the principal result in work of de la Bretèche and Tenenbaum [3]. Take G “ Z2 and d1 “ ¨ ¨ ¨ “ dN “ 1. Then bi s ´ θi t “ Fi ps, tq, for 1 ď i ď N. In this setting Theorem 1.3 can be deduced from the special case of [3, Thm. 1.1] in which all of the binary forms are linear. Acknowledgements. We are very grateful to Roger Heath-Brown for useful discussions. While working on this paper the first author was supported by ERC grant 306457. 2. Preliminary results 2.1. Divisor sums over number fields. Let K{Q be a finite extension of degree d. We write o “ oK and N “ NK for the ring of integers and ideal norm, respectively. Let σ1 , . . . , σd : K ãÑ C be the associated embeddings and let tω1 , . . . , ωdu be a Z-basis for o. Let a Ă o be an integral ideal with Z-basis tα1 , . . . , αd u. We henceforth set ∆pα1 , . . . , αd q “ | detpσi pαj qq|2 , and similarly for tω1 , . . . , ωd u. According to [19, Satz 103], we have ∆pα1 , . . . , αd q “ pN aq2 DK ,

(2.1)

where DK “ ∆pω1 , . . . , ωdq is the discriminant of K. Let F, G P Zrx, ys be non-zero binary forms with F irreducible, G of even degree and non-zero resultant RespF, Gq. We shall assume that F has degree d and that it is not proportional to y. In particular b “ F p1, 0q is a non-zero integer. Let W P N. For any ps, tq P Z2prim such that F ps, tq ‰ 0, we define ˆ ˙ ÿ Gps, tq hW ps, tq “ . (2.2) k k|F ps,tq gcdpk,W q“1

This is a modified version of the functions that appear in (1.3). We recall from (1.7) the associated binary form F˜ px, yq “ bd´1 F pb´1 x, yq, with integer coefficients and F˜ p1, 0q “ 1. We conclude that for all non-zero integer multiples


c of b, we have hcW ps, tq “

ÿ

k|F˜ pbs,tq gcdpk,cW q“1

ˆ

Gps, tq k

˙

7

,

since k | F˜ pbs, tq if and only if k | F ps, tq. We henceforth let θ be a root of the polynomial f pxq “ F˜ px, 1q. Then θ is an algebraic integer and K “ Qpθq is a number field of degree d over Q. It follows that Zrθs is an integral ideal of K with discriminant ∆θ “ ∆p1, θ, . . . , θd´1 q. In view of (2.1) we have ∆θ “ ro : Zrθss2DK ,

(2.3)

where o is the ring of integers of K. a We now let L “ Kp gpθqq, where gpxq “ Gpb´1 x, 1q P Qrxs. We shall assume that L{K is a quadratic extension and we let DL{K be the ideal norm of the relative discriminant DL{K . Put m “ 2DL{K and let J m be the group of fractional ideals in K coprime to m. For any prime ideal p P J m we set ψppq “ 1 if p splits in L, and ψppq “ ´1 if p remains inert in L. Then ψ extends to give a character ψ : J m Ñ t˘1u. Let P m “ tpaq principal fractional ideal : a ” 1 pmod mq, a totally positiveu.

Then ψppaqq “ 1 for all paq P P m , and hence we get a Dirichlet character of the ray class group of m. Let

ψ : J m {P m Ñ t˘1u,

D “ 2bDL{K ∆θ . (2.4) Note that D is a non-zero integer. Recall the definition (1.6) of the multiplicative span PK of degree 1 prime ideals. The proof of the following result is based on an argument found in recent work of Reuss [22, Lemma 4]. Lemma 2.1. Let W P N, let ps, tq P Z2prim such that F ps, tq ‰ 0, and let D be given by (2.4). Then the following hold: (i) a P PK for any integral ideal a | pbs ´ θtq such that gcdpN a, DW q “ 1; (ii) there exists a bijection between divisors a | pbs ´ θtq with N a “ k coprime to DW and divisors k | F˜ pbs, tq coprime to DW , in which Ωpkq “ ΩK paq and p Gps,tq q “ ψpaq; k (iii) we have ÿ hDW ps, tq “ ψpaq. a|pbs´θtq gcdpN a,DW q“1

8


In particular, when Gps, tq is the constant polynomial 1 in (2.2), then L “ K and ψ is just the trivial character in part (iii). In fact we have ΩK paq “ ΩpN aq for any ideal a P PK . Moreover, if řa P PK is such that N a is square-free, then τK paq “ τ pN aq, where τK paq “ d|a 1. Note that when a P P fails to have square-free norm, we still have the upper bound τK paq ď τ pN aqd . Similarly, if f : N Ñ Rě0 is any arithmetic function, we have we have ź ź p1 ` hpN pqq ď p1 ` hppqqd , p|a

p|N a

for any a P PK . We shall use these facts without further comment in the remainder of the paper.

Proof of Lemma 2.1. Let ps, tq P Z2prim such that F ps, tq ‰ 0. We form the integral ideal n “ pbs ´ θtq. This has norm N n “ |F˜ pbs, tq|. Let k | F˜ pbs, tq with gcdpk, DW q “ 1. In particular gcdpk, ∆θ q “ 1. Now let p | k. Then p ∤ t since gcdps, tq “ 1 and p ∤ b. We choose t P Z such that tt ” 1 pmod pq. Let p be any prime ideal such that p | ppq and p | n. Consider the group homomorphism π : Z{pZ Ñ pZrθs ` pq{p,

given by m ÞÑ m ` p. Suppose that πpm1 q “ πpm2 q for m1 , m2 P Z{pZ. Then m1 ´ m2 P p, whence N p | pm1 ´ m2 qd . But this implies that p | m1 ´ m2 , since p | ppq, and ř so πi is injective. Next suppose that P pθq ` p P pZrθs ` pq{p, where P pθq “ i ci θ for ci P Z. Since p | n and p ∤ t, we have bst ´ θ P p. Thus P pθq ´ P pbstq P p. Now choose m P Z{pZ such that m ” P pbstq pmod pq. It then follows that πpmq “ P pθq ` p. Thus π is surjective and so it is an isomorphism. Hence rZrθs ` p : ps “ p. In view of (2.3), we also have DK ro : Zrθs ` ps2 rZrθs ` p : Zrθss2 “ ∆θ .

This implies that p ∤ ro : Zrθs ` ps. Since N p is power of p, we readily conclude that N p “ ro : Zrθs ` psrZrθs ` p : ps “ p. This therefore establishes part (i). We now show that pp, nq is a prime ideal for any p | k. Suppose for a contradiction that pp, nq is not prime. Thus there exist distinct prime ideals pi , pj in the ideal factorisation of p such that pi pj | pp, nq. Part (i) implies that N pi “ N pj “ p. Since p ∤ ∆θ , an application of Dedekind’s theorem on factorisation of prime ideals implies that there exist distinct ni , nj P Z{pZ such that f pxq ” px ´ ni qpx ´ nj qgpxq pmod pq, for a polynomial gpxq P pZ{pZqrxs of degree d ´ 2, with pi “ pp, θ ´ ni q and pj “ pp, θ ´ nj q. We conclude from this that bst ´ θ P pi and θ ´ ni P pi ,


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whence bst ´ ni P pi . Similarly, we have bst ´ nj P pj . But then it follows that p “ N pi | bst ´ ni and p “ N pj | bst ´ nj . This implies that ni ” nj pmod pq, which is a contradiction. Our work so far shows that there is a bijection between each factorisation ˜ |F pbs, tq| “ ke, with gcdpk, DW q “ 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 ˆ ˙ Gps, tq “ ψppq, p where p “ pp, nq. Since G has even degree we have ˆ ˙ ˆ ˙ Gps, tq Gpst, 1q “ . p p Recall the notation gpxq “ Gpb´1 x, 1q. We may suppose that p “ pp, θ ´ nq, for some n P Z{pZ such that bst ´ n ” 0 pmod pq, aand we recall from (2.4) that p ∤ 2DL{K . We observe that p splits in L “ Kp gpθqq if and only if gpnq is a square in o{p, since gpθq ” gpnq pmod pq. But this is if and only if ˆ ˙ gpbstq “ 1, p since n ” bst pmod pq and N p “ p. Noting that gpbstq “ Gpst, 1q, 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 | pbs ´ θtq that appears in Lemma 2.1. Lemma 2.2. Let a P PK such that gcdpN a, DK q “ 1 and let ps, tq P Z2prim . Then there exists k P Z such that a | pbs ´ θtq ô bs ” kt pmod N aq. Proof. It suffices to check this when a “ pa , for some p P PK such that p “ N p is unramified, by the Chinese remainder theorem. Since p P PK , there exists k 1 P Z satisfying k 1 ” θ pmod pq, whence there exists k P Z such that k ” θ pmod pa q. Therefore bs ” θt pmod pa q ô bs ´ kt P Z X pa . We claim that the latter condition is equivalent to bs ” kt pmod N pa q. The reverse implication is obvious since N a P a for any integral ideal a. The forward implication follows on noting that νp pbs ´ ktq ě νp ppbs ´ ktqq ě a.

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2.2. Uniform upper bounds for conics. Let Q P Zry1 , y2 , y3s be a nonsingular isotropic quadratic form. Denote its discriminant by ∆Q and the greatest common divisor of the 2ˆ2 minors of the associated matrix by D ´ Qˆ. For ¯ ´∆ Q ´ν pAq any A P Z and any prime p we set Aˆ “ Ap p . We define χQ ppq “ , p

for any odd prime p. Then it follows from [25, §IV.2] that

7ty pmod pq, Qpyq ” 0 pmod pq, p ∤ yu “ ppp ´ 1q p1 ` χQ ppqq ` p ´ 1, for any prime p such that p | ∆Q and p ∤ 2DQ . The main aim of this section is to establish the following result. Proposition 2.3. Let w, B1, B2 , B3 ą 0 be given. Then ¸ ˜ 1 2 ( pB1 B2 B3 q 3 DQ 3 7 y P Zprim : Qpyq “ 0, |yi | ď Bi ! CpQ, wq 1 ` , 1 |∆Q | 3

with an absolute implied constant, where CpQ, wq “

ź

pξ }∆Q p|2DQ

ξ

τ pp q

ź

pξ }∆Q pďw p∤2DQ

ξ

τ pp q

ź

pξ }∆Q pąw p∤2DQ

˜

ξ ÿ

k“0

χQ ppq

k

¸

.

Since CpQ, wq ď τ p∆Q q, this result is a refinement of work due to Browning and Heath-Brown [9, Cor. 2]. In fact, although it is not needed here, one can show that for any prime p ∤ 2DQ , 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 Qp -points for some prime p ∤ 2DQ , then the constant in the upper bound vanishes. Therefore, Proposition 2.3 can be used to detect conics with a rational point. This is the point of view adopted in the work of Sofos [28]. Proof of Proposition 2.3. The proof of [9, Cor. 2] relies on earlier work of Heath-Brown [16, 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 pξ }∆Q , it turns out that there are at most Lppξ q ď cp τ ppξ q lattices to consider, where cp “ 1 for p ą 2. Suppose that y P Z3prim is a non-zero vector for which Qpyq “ 0. Let p be a prime such that pξ }∆Q , with p ∤ 2DQ and χQ ppq “ ´1. On diagonalising over Z{pξ`1 Z, we may assume that a1 y12 ` a2 y22 ` pξ y32 ” 0 pmod pξ`1 q,


11

for coefficients a1 , a2 P Z such that p ∤ a1 a2 . In particular, we have χQ ppq “ p áp1 a2 q “ ´1. Hence Lppξ q “ 1 when ξ is even, since then y is merely constrained to lie on the lattice ty P Z3 : y1 ” y2 ” 0 pmod pξ{2 qu. Likewise, when ξ is odd, there can be no solutions in primitive integers y. Note that # ξ ÿ τ ppξ q if χQ ppq “ 1, χQ ppqk “ 0 if χQ ppq “ ´1 and ξ is odd. k“0 It follows that the total number of lattices emerging is ź ź ź ! 1p∆Q q τ ppξ q τ ppξ q τ ppξ q “ CpQ, wq, pξ }∆Q p|2DQ

pξ }∆Q pďw p∤2DQ

pξ }∆Q χQ ppq“1 pąw p∤2DQ

where 1p∆Q q “ 0 (resp. 1p∆Q q “ 1) if there exists pξ }∆Q such that χQ ppq “ ´1, with ξ odd and p ∤ 2DQ (resp. otherwise). This completes the proof of the lemma. 2.3. Lattice point counting. We will need general results about counting lattice points in an expanding region. Let A P Mat2ˆ2 pZq be a non-singular upper triangular matrix and consider the lattice given by G “ tAy : y P Z2 u. Recall that G is said to be primitive if the only integers m fulfilling G Ă mZ2 are m “ ˘1. We denote its determinant and first successive minimum by detpGq and λG , respectively. Assume that R Ă R2 is a regular region, in the sense of Theorem 1.3. Then, for any x0 P Z2 and q P N such that gcdpdetpGqx0 , qq “ 1, we will require an asymptotic estimate for the counting function NpRq “ 7tx P Z2prim X R X G : x ” x0 pmod qqu. The following basic estimate follows from work of Sofos [28, Lemma 5.3]. Lemma 2.4. Assume that G is primitive. Then ˙´1 ź ˆ ˙´1 ź ˆ volpRq 1 1 NpRq “ 1` 1´ 2 ζp2q detpGqq 2 p|detpGq p p p|q ˙ ˆ τ pdetpGqqKR , Ò λG where KR is given by (1.12) and the implied constant is absolute. This estimate will be used in §3. When it comes to our work in §4, we shall require a special case in which the error term is made more explicit.

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Let D Ă R2 zt0u be a non-empty open disc and put δpDq “ }D}8 , in the notation of §1.2. Let b, c, q P Z and x0 P Z2 such that q ě 1 and gcdpb, cq “ gcdpx0 , qq “ 1.

For each e P N such that gcdpe, bcqq “ 1, we define the non-empty set Λpeq “ tps, tq P Z2 : bs ” ct pmod equ.

We then fix, once and for all, a non-zero vector of minimal Euclidean length within Λpeq and we call it vpeq. We are now interested in ! ) Npxq “ 7 x P Z2prim X xD X Λpeq : x ” x0 pmod qq , as x Ñ 8. We shall prove the following result.

Lemma 2.5. Let D, b, c, x0 , q, Λpeq, vpeq, Npxq be as above, and assume that |vpeq| ď δpDqx. Then ˆ ˙´1 ź ˆ ˙´1 volpDqx2 ź 1 1 Npxq “ 1` 1´ 2 ζp2qeq 2 p|e p p p|q ,˛ $¨ ˛ ¨ . & ÿ ÿ δpDqx ‚ 1 1 |vpdq| ‚, log ` ` O ˝pβ ` γq x ˝ % d|vpe{dq| d|vpe{dq| e d|e d|e

where

volpDq BD , γ“ . q δpDqq 2 The implied constant in this estimate is absolute. β “ δpDq `

In order to compare this result with Lemma 2.4, we note that G “ Λpeq defines a primitive integer lattice of determinant detpGq “ e and first successive minimum |vpeq| “ λG . In particular, the main terms agree since volpxDq “ volpDqx2 . For any d | e, let us denote vpe{dq by px0 , x1 q, temporarily. Then e | pbx0 ´ cx1 q ñ pdx0 , dx1 q P Λpeq, d whence |vpeq| ď d|vpe{dq|. (2.5) Moreover, using the basic properties of the minimal basis vector, one obtains 1ÿ 1 ÿ? τ peq τ peq dď ? ! |vpdq| ! . (2.6) e d|e e d|e e |vpeq| This shows that Lemma 2.5 implies Lemma 2.4 in the special case that R “ xD and G “ Λpeq.


13

Proof of Lemma 2.5. Our argument is based on a modification of the proof of [28, Lemma 5.3]. We write δ “ δpDq for short and put x0 “ ps0 , t0 q. Since gcdps0 , t0 , qq “ 1, an application of Möbius inversion gives ÿ ÿ µpmq Npxq “ 1. x ps,tqP m DXΛpeq gcdpu,v,eq“1 pu,vq”mps0 ,t0 qpmod qq

mPN gcdpm,eqq“1

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 ! x ) δx . 1 ď |vpeq| “ mint|y| : y P Λpeqzt0uu ď max |y| : y P D ď m m Thus, on using the Möbius function to remove the condition gcdpu, v, eq “ 1, we find that ÿ ÿ ÿ Npxq “ µpmq µpdq 1. mPN gcdpm,eqq“1 δx mď |vpeq|

x pu,vqP m DXΛpeq d|u, d|v pu,vq”mps0 ,t0 qpmod qq

d|e

Making the substitution u “ ds and v “ dt, and arguing as before we find that ÿ ÿ ÿ 1. µpmq µpdq Npxq “ mPN gcdpm,eqq“1 δx mď |vpeq|

d|e

δx dď |vpe{dq|m

x DXΛpe{dq ps,tqP dm ps,tq”dmps0 ,t0 qpmod qq

Now let n P Z be such that n ” dm pmod qq. Then we can make the change of variables ps, tq “ nps0 , t0 q ` qps1 , t1 q in the inner sum. Noting that Λpe{dq defines a lattice in Z2 of determinant e{d, the inner sum is found to be ˙ ˙ ˆ ˆ x BD volpDqx2 volpDqx2 x dm “ , Ò 1` Ò β dem2 q 2 q|vpe{dq| dem2 q 2 md|vpe{dq| with an absolute implied constant, since the upper bound on d implies that 1ď

δx . dm|vpe{dq|

In summary, we have shown that Npxq “

ÿ

mPN gcdpm,eqq“1 δx mď |vpeq|

µpmq

ÿ

d|e δx dď |vpe{dq|m

˙˙ ˆ x volpDqx2 . Ò β µpdq dem2 q 2 md|vpe{dq| ˆ

14


The contribution from the error term is ÿ ÿ ÿ 1 1 δx 1 ! βx ! βx log . d|vpe{dq| m d|vpe{dq| d|vpe{dq| δx d|e d|e mď d|vpe{dq|

The main term equals volpDqx2 eq 2

ÿ µpmq m2 mPN

gcdpm,eqq“1

ÿ

d|e δx dď |vpe{dq|m

µpdq , d

since (2.5) implies that the extra constraint in m-sum is implied by the constraint in the d-sum. But this is equal to ˛ ¨ 2 ÿ ÿ ÿ µpmq µpdq volpDqx ˝ volpDqx ¨ 1 |vpe{dq|‚, ` O 2 2 eq 2 d m δq e mPN d|e d|e gcdpm,eqq“1

which thereby completes the proof.

2.4. Twisted Hooley ∆-function over number fields. Adopting the notation of §1.2, it is now time to reveal the version of the Hooley ∆-function that arises in our work. Let K{Q be a number field and let ψK be a quadratic Dirichlet character on K. We let ∆ : IK Ñ Rą0 be the function given by ˇ ˇ ÿ ˇ ˇ (2.7) ψK pdqˇ, ∆pa; ψK q “ sup ˇ uPR 0ďvď1

d|a eu ăNK dďeu`v

for any integral ideal a P IK . We shall put ∆paq “ ∆pa; 1q for the corresponding function in which ψK is replaced by the constant function 1. We begin by showing that ∆ belongs to the class MK of pseudomultiplicative functions introduced in §1.2. For coprime ideals a1 , a2 Ă oK , any ideal divisor d | a1 a2 can be written uniquely as d “ d1 d2 , where di | ai . Therefore ÿ ÿ ÿ ψK pd1 q ψK pd2 q. ψK pdq “ d|a1 a2 eu ăNK dďeu`v

d1 |a1

d2 |a2 eu´log NK d1 ăNn d2 ďeu´log NK d1 ev

Thus the triangle inequality yields ∆pa1 a2 ; ψK q ď τK pa1 q∆pa2 ; ψK q, where τK is the divisor function on ideals of oK . This shows that ∆p¨, ψK q belongs to MK and an identical argument confirms this for ∆p¨q. We shall need the following result, due to Sofos [29, Proposition 2.2]. Lemma 2.6. Define the function d εppxq “

log log logp16 ` xq , log logp3 ` xq


15

for any x ě 1. (i) There exists a positive constant c “ cpKq such that ÿ ∆paq ! plog xq1`cpεpxq . N a K aPP square-free K

NK aďx

(ii) Let ψK be a quadratic Dirichlet character on K and let W P N. There exists a positive constant c “ cpK, ψK q such that ÿ ∆pa; ψK q2 ! plog xq1`cpεpxq . N a K aPP square-free K

gcdpNK a,W q“1 NK aďx

The implied constant in both estimates is allowed to depend on K and, in the second estimate, also on W and the character ψK . 3. Nair–Tenenbaum over number fields This section is devoted to the proof of Theorem 1.3. We assume familiarity with the notation introduced in §1.2. For a given number field K{Q of degree d and given f P MK , it will sometimes be useful to bound sums of the shape ÿ f paq , N a N aďx K K

by a sum restricted to square-free integral ideals supported away from ideals of small norm. This is encapsulated in the following result.

Lemma 3.1. Let f, f : P MK pA, Bq. Assume that f : is multiplicative and that there exists M ą 0 such that M , |f : ppν q ´ 1| ď NK p for all prime ideals p and ν P N. Assume that W is given by (1.5), with w ą 2pA ` Mq. Then ÿ f paW qf : paW q ÿ f pbq !A,B,M,W . N a N b K K N aďx N bďx K

K

b square-free gcdpNK b,W q“1

If f is submultiplicative then the right hand side can be replaced by ¸ ˜ ÿ f ppq . exp N p K wăN pďx K

16


Proof. The final part of the lemma is obvious. To see the first part we note that there is a unique factorisation a “ qaW , where NK q | W 8 . Furthermore we can decompose uniquely aW “ bc where b, c are coprime integral ideals such that b is square-free and c is square-full. The sum in the lemma is at most ÿ ÿ ÿ 1 f pbqf : pbq AΩK pcq f : pcq . NK q NK b NK c N bďx N cďx N qďx K

NK q|W 8

K

K

c square-full gcdpNK c,W q“1


For prime ideals with NK p ą 2pA ` Mq, we have ˙ ˆ ˆ ˙´1 d 8 8 ÿ ÿ AΩK pp q f : ppd q Ad 2A2 M 1` ď 1´ 1` ď1` . d d 2 pN pq pN pq N p pN pq K K K K d“2 d“2

Thus the sum over c converges absolutely. Defining the multiplicative function g : IK Ñ R via gppdq “ pf : ppq ´ 1q NK p, for d P N, we clearly have ÿ gpdq , f : pbq “ N d K b“de with |gpdq| ď M ΩK pdq , for any square-free ideal b. Therefore ÿ ÿ ÿ gpdq f peq f pbqf : pbq “ . NK b NK e N dďx{pN eq pNK dq2 N eďx N bďx K


K

e square-free gcdpNK e,W q“1

K

The sum over d is absolutely convergent, whence ÿ f paW qf : paW q ÿ f peq !A,B,M,W NK a NK e N aďx N eďx K

K

K

d square-free gcdpNK d,W q“1

e square-free gcdpNK e,W q“1

ÿ

NK qďx NK q|W 8

1 NK q

.

ś The inner sum over q is at most NK p|W d p1´ NK1 p q´1 !A,M,W 1, which thereby completes the proof of the lemma. Inspired by the work of Shiu [26], we now initiate the proof of Theorem 1.3. Since F1 , . . . , FN are pairwise coprime it follows that the resultants RespFi , Fj q are all non-zero integers for i ‰ j. Along the way, at certain stages of the argument, we will need to enlarge the size of W in (1.5). For now we assume that w ą maxi‰j t|Di |, | RespFi , Fj q|u, where Di “ 2bi DLi {Ki ∆θi , as in (2.4). ś We let Ni “ NKi and write F “ N i“1 Fi . Let z “ V ω,

(3.1)

where V “ volpRq, for a small constant ω P p0, 1q that will be chosen in due course. (In particular, it will need to be sufficiently small in terms of the choice


17

of ε made in the statement of Theorem 1.3.) For each ps, tq P Z2prim X R X G, it follows from (1.8) that we have a factorisation N ź i“1

| Ni pbi s ´ θi tqqG W | “

N ź i“1

|Fi ps, tq|qG W “ |F ps, tq|qGW “ pα1 1 ¨ ¨ ¨ pαl l ,

with w ă p1 ă ¨ ¨ ¨ ă pl . We define as,t to be the greatest integer of the form α pα1 1 ¨ ¨ ¨ pj j which is bounded by z and we define bs,t “ F ps, tqqGW {as,t . We have gcdpas,t , bs,t q “ 1 and P ´ pbs,t q ą P ` pas,t q. Our lower bound for w ensures that gcdpNi ai , Nj aj q “ 1,

for any ai | pbi s ´ θi tqqG W and aj | pbj s ´ θj tqqG W , with i ‰ j. Part (i) of Lemma 2.1 implies that ai P Pi , for 1 ď i ď N. We sort the sum N ź ÿ fi,qG W ppbi s ´ θi tqq, ps,tqPZ2prim XRXG i“1

in which we are interested, into four distinct contributions E pIq pRq, . . . , E pIV q pRq.

For an appropriate small parameter η ą 0, these sums are determined by which of the following attributes are satisfied by ps, tq: η (I) P ´ pbs,t q ě z 2 ; η (II) P ´ pbs,t q ă z 2 and as,t ď z 1´η ; (III) P ´ pbs,t q ď log z log log z and z 1´η ă as,t ď z; η (IV) log z log log z ă P ´ pbs,t q ă z 2 and z 1´η ă as,t ď z. In what follows, we will allow all of our implied constants to depend on c0 , c1 , c2 , Ai , Bi , ε, N, W , as in the statement of Theorem 1.3, as well as on ω and η, whose values will be indicated during the course of the proof. Any further dependence will be indicated by an explicit subscript. We let Ωi “ ΩKi and recall that Ωi paq “ ΩpNi aq when a P Pi “ PKi . For given ps, tq, the choice of as,t , bs,t that we have made uniquely determines piq piq piq piq coprime ideals as,t , bs,t Ă oi , with pbi s ´ θi tqqG W “ as,t bs,t , such that N ź i“1

piq

Ni as,t “ as,t

and piq

piq

piq

piq

Ωi pas,t q “ ΩpNi as,t q.

In particular we emphasise that as,t , bs,t are supported on prime ideals whose norms are coprime to qG W . We now have everything in place to start estimating the various contributions. Our main tools will be the geometry of numbers and the fundamental lemma of sieve theory.

18

T. D. BROWNING AND E. SOFOS η

3.1. Case I. We begin by considering the case P ´ pbs,t q ě z 2 . Recalling that piq piq fi P MKi , we have fi,qG W ppbi s ´ θi tqq ď fi pas,t qAΩi pbs,t q , by the coprimality of piq piq as,t , bs,t . Hence E

pIq

pRq “

ÿ

N ź

ps,tqPZ2prim XRXG i“1 η

P ´ pbs,t qěz 2

!

where

ÿ

fi,qG W ppbi s ´ θi tqq

U pa1 , . . . , aN q

ai PPi gcdpNi ai ,qG W Nj aj q“1 śN i“1 Ni ai ďz

U pa1 , . . . , aN q “

ÿ

N ź

N ź i“1

(3.2) fi pai q,

AΩi ppbi s´θi tqqG W {ai q .

i“1 ps,tqPZ2prim XRXG pbi s´θi tqqG W PPi ai |pbi s´θi tq pai ,pbi s´θi tq{ai qi “1 η ś 2 p|F ps,tqqG W { N i“1 Ni ai ñpěz

ś Here, the condition N i“1 Ni ai ď z comes from the fact that as,t ď z. Moreover, we write pa, bqi “ 1 if and only if the ideals a, b Ă oi are coprime. ś ś νp pF ps,tqq Defining b through b N , we see that i“1 Ni ai “ p∤qG W p η

degpF q

degpF q ď KR pz 2 qΩpbq ď P ´ pbqΩpbq ď |b| ď |F ps, tq| ! }R}8

.

KRc0

In view of (3.1) and the inequality ď V that is assumed in Theorem 1.3, this shows that Ωpbq ! 1. Noting that ˆ ˙ N ÿ pbi s ´ θi tqqG W Ωi “ Ωpbq, ai i“1 we may therefore conclude that

U pa1 , . . . , aN q ! U η2 pa1 , . . . , aN q,

(3.3)

where for any γ ą 0 we define Uγ pa1 , . . . , aN q to be the cardinality of ps, tq η appearing in the definition of U pa1 , . . . , aN q, with the lower bound z 2 replaced by z γ . Our next concern is with an upper bound for this quantity. Before revealing our estimate for Uγ pa1 , . . . , aN q, recall the definition of h˚ from (1.11) and set ˙´1 źˆ d1 ` ¨ ¨ ¨ ` dN : . (3.4) 1´ h pkq “ p ` 1 p|k


19

Then we have the following result. Lemma 3.2. Let δ, γ ą 0 and let ai P Pi , for 1 ď i ď N, with gcdpNi ai , qG W Nj aj q “ 1

and

N ź i“1

Then

Ni ai ď z.

V h˚W pqG q ź h: pNi ai q KR z 2γ`δ ` , γ N plog zqN qG i“1 Ni ai λG N

Uγ pa1 , . . . , aN q!δ uniformly in γ.

We defer the proof of this result, temporarily, and show how it can be used to complete the treatment of E pIq pRq, via (3.2) and (3.3). We apply Lemma 3.2 with γ “ η2 . The overall contribution from the second term is therefore found to be KR z 1`η`3δ KR z η`2δ 7 ta1 , . . . , aN : N1 a1 ¨ ¨ ¨ NN aN ď zu !δ , !δ λG λG since fi pai q !δ pNi ai qδ{N ď z δ{N , for any fi P MKi . Recalling (3.1) and taking δ and ω small enough in terms of ε shows that this is OpKR1`ε {λG q, which is satisfactory for Theorem 1.3. In view of (3.2) and (3.3), the first term in Lemma 3.2 makes the overall contribution N ÿ V fi pai qh: pNi ai q h˚W pqG q ź !δ . plog zqN qG i“1 Ni ai a PP i

i

gcdpNi ai ,qG W q“1 Ni ai ďz

Since ai P Pi , (3.4) implies that ˙´di źˆ d1 ` ¨ ¨ ¨ ` dN : 1´ h pNi ai q ď “ h;i pai q, N p ` 1 i p|a i

say, where we recall that di “ rKi : Qs. We enlarge w in order to use Lemma 3.1, and thereby obtain the overall contribution h˚W pqG q ź V !δ plog zqN qG i“1 N

ÿ

ai PPi gcdpNi ai ,qG W q“1 ai square-free Ni ai ďz

In view of (3.1), this is clearly sufficient.

fi pai q . Ni ai

20


Proof of Lemma 3.2. Let ci P Pi be given, with gcdpNi ci , qG W Nj cj q “ 1. Define the set ( Λpc1 , . . . , cN q “ ps, tq P Z2 X G : bi s ” θi t pmod ci q, for i “ 1, . . . , N . ś Since gcdpqG , i Ni ci q “ 1, it follows from Lemma 2.2 that this defines a lattice in Z2 of rank 2 and determinant N ź Ni ci . detpΛpc1, . . . , cN qq “ qG i“1

ś

Write P pz0 q “ păz0 p, for any z0 ą 0, with the usual convention that P pz0 q “ 1 if z0 ă 2. This allows us to write ÿ ÿ ÿ Uγ pa1 , . . . , aN q ď 1“ µpdq. ś ps,tqPS ps,tqPS d|F ps,tq{ N i“1 Ni ai pai ,pbi s´θi tq{ai qi “1 pai ,pbi s´θi tq{ai qi “1 gcdpd,qG W q“1 śN p|F ps,tqqG W { i“1 Ni ai ñpěz γ d|P pz γ q

where S “ Z2prim X R X Λpa1 , . . . , aN q. We shall use the fundamental lemma of sieve theory, as presented by Iwaniec and Kowalski [15, § 6.4]. This provides us with a sieve sequence λ` d supported on square-free integers in the interval ` r1, 2z γ s, with λ` “ 1 and |λ 1 d | ď 1, such that ÿ ÿ Uγ pa1 , . . . , aN q ď λ` d. ś ps,tqPS d|F ps,tq{ N i“1 Ni ai pai ,pbi s´θi tq{ai qi “1 gcdpd,qG W q“1 d|P pz γ q

ś Since gcdpas,t , bs,t q “ 1, we note that only d coprime to N i“1 Ni ai appear in the inner sum. Interchanging the order of summation, we find that ÿ ÿ ÿ Uγ pa1 , . . . , aN q ď µ1 pe1 q ¨ ¨ ¨ µN peN q 1 λ` d 1ďdď2z γ ps,tqPZ2prim XRXΛpa1 e1 ,...,aN eN q gcdpd,qG W q“1 d|F ps,tq śN gcdpd, i“1 Ni ai q“1 d|P pz γ q

ei |ai

“

ÿ

ei |ai

µ1 pe1 q ¨ ¨ ¨ µN peN q

λ` d1 ¨¨¨dN Spdq,

d1 ,...,dN PN gcdpdi ,qG W Ni ai q“gcdpdi ,dj Nj aj q“1 d1 ¨¨¨dN ď2z γ d1 ¨¨¨dN |P pz γ q

where if d “ d1 ¨ ¨ ¨ dN , then Spdq “

ÿ

ÿ

pσ,τ q pmod dq gcdpσ,τ,dq“1 Fi pσ,τ q”0 pmod di q

ÿ

1.

ps,tqPZ2prim XRXΛpa1 e1 ,...,aN eN q ps,tq”pσ,τ q pmod dq

If Fi px, yq “ cy for some i, then the condition bi s ” θi t pmod ai ei q should be replaced by t ” 0 pmod ai ei q.


21

Recall the definition (1.10) of ρi and let ˙´1 źˆ 1 hpdq “ 1` . p p|d The number of possible pσ, τ q pmod dq is equal to ϕpdqρ1 pd1 q ¨ ¨ ¨ ρN pdN q. In Spdq the inner sum can be estimated using the geometry of numbers. Calling upon Lemma 2.4, we deduce that ¸ ˜ δ N V hpqG q ź ρi pdi qhpdi qhpNi ai q KR z γ` 2 Spdq “ , ` Oδ ζp2q qG i“1 di Ni ei Ni ai λG

for any δ ą 0. We emphasise that the implied constant in this estimate does δ 2N , not depend on any of R, di , ai or ei . Since |λ` d | ď 1 and τKi pai q !δ pNi ai q we find that the overall contribution to Uγ pa1 , . . . , aN q from the error term is Oδ pKR z 2γ`δ {λG q, on summing trivially over e1 , . . . , eN and d1 , . . . , dN . This is plainly satisfactory for Lemma 3.2. Turning to the contribution from the main term, we set N ´ ¯ hpdq ź gpdq “ 1 d, qG W Ni ai d i“1

ÿ

N ź

d1 ¨¨¨dN “d i“1 gcdpdi ,dj q“1

ρi pdi q,

where 1pd, aq “ 1 if gcdpd, aq “ 1 and 1pd, aq “ 0, otherwise. Since hpdq ď 1 and ϕi pai q ď Ni ai , the main term contributes N ˇ V ź 1 ˇˇ ÿ ˇ ` ! λd gpdqˇ. ˇ qG i“1 Ni ai 1ďdď2z γ d|P pz γ q

η

We may clearly assume without loss of generality that w ă 2z maxtγ, 2 u . For any prime p ∤ W , let ρ ppq ` ¨ ¨ ¨ ` ρN ppq hppq ÿ . ρi ppq “ 1 ´ 1 cp “ 1 ´ p i“1 p`1 N

Recalling that degpFi q “ di for 1 ď i ď N, we see that cp ě 1 ´ for p ∤ W . Next, for y ě 0, define ź Πpyq “ cp , Π1 “ păy p∤W

d1 ` ¨ ¨ ¨ ` dN , p`1 ź

pě2z γ p| N1 a1 ¨¨¨ NN aN

cp ,

Π2 “

ź

pě2z γ p|qG

cp .

22


By the fundamental lemma of sieve theory [15, Lemma 6.3], we find that ÿ

λ` d gpdq

1ďdď2z γ d|P pz γ q

!

ź

pă2z γ

p1 ´ gppqq “ Πp2z

γ

qΠ1 Π2 h˚W

pqG q

N ź i“1

h: pNi ai q,

in the notation of (1.11) and (3.4). It is clear that Πi ď 1 for i “ 1, 2. Noting that Πpyq ! plog yqŃ , this therefore concludes the proof of the lemma. 3.2. Cases II and III. In this section we estimate E pIIq pRq and E pIIIq pRq. For any ps, tq P Z2prim X R and any δ ą 0 we take the trivial bound N ź i“1

fi,qG W ppbi s ´ θi tqq !δ

N ź δ pNi pbi s ´ θi tqq di N !δ }R}δ8 ď KRδ . i“1

Our aim is to show that there exists β ą 0 such that the total number of ps, tq P Z2prim X R that contribute to Case II or Case III is ˙ ˆ 1´β KR1`δ V ` , (3.5) Oβ,δ qG λG

for any δ ą 0. This is clearly sufficient for the purpose of proving Theorem 1.3, on taking δ sufficiently small in these two estimates, since h˚W pqG q ě 1 and Efi pV ; W q ě 1, for each 1 ď i ď N. η In Case II the relevant extra constraints are P ´ pbs,t q ă z 2 and as,t ď z 1´η . η Let p “ P ´ pbs,t q ă z 2 and let ν be such that pν }F ps, tq. We must have pν ě z η , since otherwise z ă pν as,t ă z η z 1´η “ z , which is a contradiction. For each η prime p ∤ qG W with p ă z 2 , we define η

lp “ mintl P Zě0 : pl ě z η u.

Clearly z η ď plp ă z 2 lp , whence lp ě 2 for every prime p. Therefore " * ÿ 1 ÿ ÿ 1 ÿ 1 η 1 1 min ď , ď ` ! z´ 2 . l η 2 η 2 p z p η p η η z η p păz 2 p∤qG W

păz 2 p∤qG W

pďz 2

(3.6)

pąz 2

The number of elements ps, tq satisfying the constraints of Case II is at most ˙ ˆ N N ÿ ÿ ÿ ÿ ÿ KR hpqG q V lp . ` τ pqG q ρi pp q 1! qG plp λG η η 2 i“1 i“1 păz 2 ps,tqPZprim XRXG p∤qG W plp |Fi ps,tq

păz 2 p∤qG W

Here we have split the inner sum into ρi pplp q different lattices of the form tps, tq P G : s ” xt pmod plp qu, where x ranges over solutions of the congruence Fi px, 1q ” 0 pmod plp q, before applying Lemma 2.4 with q “ 1. Hensel’s lemma


23

implies that ρi ppl q “ ρi ppq ď di for each prime p ∤ W and l P N. Taking δ

hpqG q ď 1 and τ pqG q !δ qG2c1 c2 ď V N ÿ ÿ

δ 2c1

δ

ď KR2 , this therefore reveals that δ

δ

ÿ

i“1 păz η2 ps,tqPZ2 XRXG prim p∤qG W plp |Fi ps,tq

η η 1` 1` V ÿ 1 KR 2 z 2 V 1 KR 2 z 2 1 !δ !δ , ` η ` lp qG λG qG z 2 λG η p

păz 2 p∤qG W

η

by (3.6). Recalling (3.1), we see that z ´ 2 ď V ´β , with β “ η 2

ηω 2

ηωc1 2

η 2

ηω . 2

Likewise,

δ 2ωc1

z “V ď KR . Taking ă therefore shows that the second term is 1`δ at most KR {λG , which is satisfactory for (3.5). We now turn to the contribution from Case III, for which the defining constraints are P ´ pbs,t q ď log z log log z and z 1´η ă as,t ď z. We assume that w ą maxi‰j | RespFi , Fj q| in the definition (1.5) of W . For any ps, tq P Z2prim it follows that the integer factors of Fi ps, tqW are necessarily coprime to the factors of Fj ps, tqW for all i ‰ j. Hence the number of elements ps, tq satisfying the constraints of Case III is at most ÿ ÿ ÿ 1 1ď ps,tqPZ2prim XRXG z 1´η ăaďz gcdpa,qG W q“1 a|F ps,tq P ` paqďlog z log log z

ps,tqPZ2prim XRXG P ´ pbs,t qďlog z log log z z 1´η ăas,t ďz

ď

ÿ

ÿ

1.

z 1´η ăa1 ¨¨äN ďz ps,tqPZ2prim XRXG gcdpai ,qG W aj q“1 ai |Fi ps,tq P ` pai qďlog z log log z

ś δ As before, the final sum can be split into at most N i“1 ρi pai q “ Oδ pz q lattices, śN for any δ ą 0, each of determinant qG i“1 ai . Thus the right hand side is ˙ ˆ ÿ V KR δ ` !δ z qG a1 ¨ ¨ ¨ aN λG 1´η z ăa1 ¨¨äN ďz P ` pa1 ¨¨äN qďlog z log log z

!δ z

2δ

ÿ

z 1´η ăaďz P ` paqďlog z log log z

ˆ

V KR ` qG a λG

˙

.

According to Shiu [26, Lemma 1], we have ˆ ˙ ÿ 3 log x 1 ď exp ? !δ xδ , log log x nďx P ` pnqďlog x log log x

24


whence ÿ

1 !δ z

ˆ

3δ

ps,tqPZ2prim XRXG P ´ pbs,t qďlog z log log z z 1´η ăas,t ďz

V KR ` 1´η qG z λG

˙

.

As before, on redefining the choice of δ, one easily checks that the right hand side is bounded by (3.5). This concludes the treatment of Case III. 3.3. Case IV. The final case to consider is characterised by the constraints η log z log log z ă P ´ pbs,t q ă z 2 and z 1´η ă as,t ď z. Arguing as in (3.2) in the treatment of Case I, we find that E

pIV q

pRq !

ÿ

:

ai PPi gcdpNi ai ,qG W Nj aj q“1 ś z 1´η ă N i“1 Ni ai ďz

U pa1 , . . . , aN q

N ź i“1

fi pai q,

where U : pa1 , . . . , aN q is as in the definition of U pa1 , . . . , aN q after (3.2), but ś η 2 with the condition P ´ pF ps, tqqGW { N i“1 Ni ai q ě z replaced by ˜ ¸ η F ps, tq q W log z log log z ă P ´ śN G ă z2. i“1 Ni ai

In particular, in view of the coprimality of as,t and bs,t , we see that ˜ ˜ ¸ ¸ N ź F ps, tq q W P` Ni ai ă P ´ śN G . N a i i i“1 i“1 Now, we may assume without loss of generality that η 2

plog z log log z, z q Ă

kď 0 `1 k“ η2

1

2 η

P Zě2 . Thus

1

pz k`1 , z k s,

where k0 “ rlog z{ logplog z log log zqs satisfies k0 ď log z{ log log z. Notice 1 1 that for any integer b satisfying log b ! log z and z k`1 ă P ´ pbq ď z k we śN must have Ωpbq ! k. Applying this with b “ F ps, tqqG W { i“1 Ni ai , for any ps, tq P Z2prim X R, we deduce that N ź i“1

Ωi p

Ai

pbi s´θi tqq W G ai

q

Ωpbq

ď max Ai 1ďiďN

ď Ak ,


25

for a suitable constant A " max1ďiďN Ai , where Ai is the constant appearing in the definition of MKi “ MKi pAi , Bi q. Hence E pIV q pRq ď

kÿ 0 `1 k“ η2

ÿ

Ak

U

1 k`1

ai PPi gcdpNi ai ,qG W Nj aj q“1 ś z 1´η ă N i“1 Ni ai ďz 1 ś k P `p N i“1 Ni ai qăz

pa1 , . . . , aN q

N ź i“1

fi pai q,

in the notation of Lemma 3.2, which we now use to estimate U 1 pa1 , . . . , aN q. k`1 The overall contribution from the error term is 5 5 k0 `1 k0 `1 2 KR ÿ KR z 3 `3δ KR z 3 `2δ ÿ `2δ k 1` k`1 k !δ A z A !δ ď , λG λG λG 2 2

k“ η

k“ η

since k ě 2 and k0 ď log z{ log log z. Choosing δ “ 19 and ω ă 2cε1 , we see that the right hand side is OpKR1`ε {λG q. This is satisfactory for Theorem 1.3. It remains to consider the contribution from the main term in Lemma 3.2. This is k0 `1 1 V h˚W pqG q ÿ k N 1´η k q, A pk ` 1q Epz , z ! (3.7) plog zqN qG 2 k“ η

where EpS, T q “

ÿ

N ź fi pai qh: pai q i

i“1 ai PPi gcdpNi ai ,W Nj aj q“1 śN i“1 Ni ai ąS ś P `p N i“1 Ni ai qăT

Ni ai

.

ś Note that we have dropped the condition gcdp N i“1 Ni ai , qG q “ 1. Let us define the multiplicative function u : N Ñ Rě0 via upaq “

Note that uppk q “

ÿ

N ź

i“1 ai PPi gcdpNi ai ,Nj aj q“1 śN i“1 Ni ai “a N ÿ

ÿ

i“1 pi Ăoi prime Ni pi “p

fi pai qh:i pai q.

fi ppki qh:i ppki q ď C k ,

(3.8)

(3.9)

26


for an appropriate constant C ą 1 depending on Ai , di and N. We may therefore write ÿ upaq EpS, T q “ . a gcdpa,W q“1 aąS P ` paqăT

Drawing inspiration from the proof of [21, Lemma 2], we shall find an upper bound for EpS, T q in terms of partial sums involving upaq. This is the object of the following result. C

1 , C ´1 log T q. Then Lemma 3.3. Assume that T ą e 10 and let κ P p 10 log S

EpS, T q !κ e´κ log T

ÿ

upbq . b gcdpb,W q“1 bďT

Taking this result on faith for the moment, we return to (3.7) and apply it with κ satisfying eκp1´ηq ą 2A. (Note that the implied constant in Lemma 3.3 depends on κ and so the choice κ “ plog 2Aq{p1 ´ ηq ` 1 is acceptable.) This produces the overall contribution !

k0 `1 h˚W pqG q ÿ V Ak pk ` 1qN plog zqN qG eκkp1´ηq 2 k“ η

k0 `1 V h˚W pqG q ÿ pk ` 1qN ! plog zqN qG 2k 2 k“ η

!

V plog zqN

h˚W pqG q qG

ÿ

upbq b gcdpb,W q“1 1

bďz k

ÿ

upbq b gcdpb,W q“1 bďz

ÿ

upbq . b gcdpb,W q“1 bďz

Recalling (3.8) and enlarging w to enable the use of Lemma 3.1, shows that the last quantity is V h˚W pqG q ź ! plog zqN qG i“1 N

which is sufficient for Theorem 1.3.

ÿ

Ni aďz aPPi a square-free gcdpNi a,W q“1

fi paq , Ni a


Proof of Lemma 3.3. Let β “

κ . log T

27

Then

EpS, T q ď S ´β

ÿ

upaq β a . a gcdpa,W q“1 P ` paqăT

ř Define the multiplicative function ψβ via aβ “ c|a ψβ pcq, for a P N. We observe that ψβ ppk q “ pβk ´ pβpk´1q , for any k P N, whence 0 ă ψβ paq ă aβ for all a P N. We now have ÿ ÿ ψβ pcq upcdq EpS, T q ď S ´β . c gcdpd,W q“1 d gcdpc,W q“1 P ` pcqăT

P ` pdqăT

Writing d “ c0 d1 , with gcdpd1 , cq “ 1 and c0 | c8 , shows that ÿ ÿ ÿ ψβ pcqupcc0 q upd1q ´β EpS, T q ď S . d1 cc0 1 1 c |c8 gcdpd ,W q“1 P ` pd1 qăT

gcdpc,d W q“1 P ` pcqăT

0

After possibly enlarging w, it follows from (3.9) that the sum over c is ˛ ¨ ÿ ψβ ppk quppk`j q ‹ ź ˚ 1 ` ď ‚ ˝ pk`j păT kě1 p∤d1 W

ď

jě0

¨

˛

ÿ ppβk ´ pβpk´1q qC k`j ‹ ź ˚ ˝1 ` ‚ pk`j păT kě1

p∤d1 W

¨ ¨

jě0

˛˛

˚ ˚ ÿ pβ ´ 1 ‹‹ ‹‹ . ˚ ď exp ˚ ‚‚ ˝O ˝ p păT p∤d1 W

log p log p Writing pβ “ expp κlog q “ 1 ` Op κlog q, this is found to be at most T T ˜ ˜ ¸¸ κ ÿ log p exp O !κ 1. log T pďT p

Our argument so far shows that

log S


ÿ

updq . d gcdpd,W q“1 P ` pdqăT

(3.10)

28


Let ε0 P p0, 1q. Observe that each d with P ` pdq ă T can be written uniquely in the form d “ d´ d` for coprime d´ , d` P N such that P ` pd´ q ď T ε0 and P ´ pd` q ą T ε0 . We clearly have P ` pd` q ă T . Thus ÿ

ÿ ÿ updq upd` q upd´ q . ď d d d ´ ` gcdpd ,W q“1 gcdpd,W q“1 gcdpd ,W q“1 `

´

P ` pdqăT

P ´ pd` qąT ε0 P ` pd` qăT

P ` pd´ qďT ε0

ś By (3.9), the inner sum is at most T ε0 ăpăT p1 ` p1 q2C !C,ε0 1. Thus, once combined with (3.10), we deduce that log S


ÿ

upd´ q . d ´ ,W q“1

gcdpd´ P ` pd´ qăT ε0

In order to complete the proof of the lemma, it remains to show that the ÿ ÿ upd´ q updq ! . d´ d gcdpd ,W q“1 gcdpd,W q“1 ´

P ` pd´ qďT ε0

dăT

This is trivial when T ε0 ă 2. Suppose now that T ε0 ą 2. Taking κ “ 2 and pT, T ε0 q in place of pS, T q, it follows from (3.10) that ÿ

upd´ q ´ 2 ! e ε0 d´ ,W q“1

gcdpd´ d´ ąT P ` pd´ qăT ε0

ÿ

upd´ q . d´ ,W q“1


Taking ε0 suitably small, we conclude that ÿ ÿ ÿ upd´ q upd´ q upd´ q “ ` d´ d´ d´ gcdpd ,W q“1 gcdpd ,W q“1 gcdpd ,W q“1 ´

´

P ` pd´ qăT ε0

ď so that

ÿ

P ` pd´ qăT ε0 d´ ąT

ÿ upd´ q upd´ q 1 ` , d 2 d ´ ´ gcdpd ,W q“1 ,W q“1

ÿ

gcdpd´ d´ ďT

´

P ` pd´ qăT ε0

ÿ upd´ q upd´ q ď2 , d´ d´ ,W q“1 gcdpd ,W q“1


as claimed.

´

P ` pd´ qăT ε0 d´ ďT

´

d´ ďT


29

4. The lower bound In order to prove the lower bound in Theorem 1.1, we first appeal to work of Frei, Loughran and Sofos [14]. It follows from [14, Thm. 1.2] that the desired lower bound holds when ρ ě 4. Suppose that ρ “ 3. Then (1.2) implies that in the fibration π : X Ñ P1 there is at least one closed point P P P1 above which the singular fibre XP is split. Since the sum cpπq defining the complexity of π in [14, Def. 1.5] is at most 4 for conic bundle quartic del Pezzo surfaces, we infer that cpπq ď 3 when ρ “ 3, so that the lower bound in Theorem 1.1 is a consequence of [14, Thm. 1.7]. Throughout this section, it therefore suffices to assume that ρ “ 2 and δ0 “ 0, so that X is a minimal conic bundle surface. Invoking [14, Thm. 1.6], the lower bound in Theorem 1.1 is a direct consequence of the divisor sum conjecture that is recorded in [13, Con. 1], for the relevant data associated to the fibration π. We proceed to explain the particular case of the divisor sum conjecture that is germane here. Let f be the arithmetic function ˙ źˆ 2 1´ f pdq “ . (4.1) p p|d Assume that we are given the following data: D Ă R2 ,

w P Rą0 ,

ps0 , t0 q P Z2prim ,

F1 , . . . , Fn , G1 , . . . , Gn P Zrx, ys,

where Fi , Gi are binary forms and D is a disc containing no F1 ¨ ¨ ¨ Fn . śrootsνof p pnq We define W as in (1.5) and we recall the notation nW “ pąw p , for any non-zero integer n. We shall assume that Fi is irreducible over Q and 2 | degpGi q, for 1 ď i ď n. In the correspondence outlined in [14], the binary forms F1 , . . . , Fn are equal to the closed points ř ∆1 , . . . , ∆n introduced in §1.1. In particular, F “ F1 ¨ ¨ ¨ Fn is separable and ni“1 degpFi q “ 4. We shall further assume that F ps0 , t0 q is nonzero. For each i such that Fi p1, 0q ‰ 0, we define the associated binary form F˜i px, yq “ bidi ´1 Fi pb´1 i x, yq, as in (1.7), where di “ deg Fi and bi “ Fi p1, 0q. For such i we let θi P Qi be a fixed root of F˜i px, 1q “ 0. If, on the other hand, Fi px, yq is proportional to cy, we define θi “ ć. For each i P t1, . . . , nu, we make two further assumptions: (i) Gi pθi , 1q R Qpθi q2 ; (ii) for any ps, tq P Z2 X xD such that ps, tq ” ps0 , t0 q pmod W q and x ě 1, we have ˙ ˆ Gi ps, tq “ 1. (4.2) Fi ps, tqW

30


Bearing all of this notation in mind we may now define the generalised divisor sum ¨ ˛ n ź ÿ ˆ Gi ps, tq ˙ ÿ ˝f pFi ps, tqW q ‚. Dpxq “ (4.3) d 2 i“1 d|F ps,tq ps,tqPZprim XxD ps,tq”ps0 ,t0 q pmod W q

i

W

The primary goal of this section is to prove the following bound. Proposition 4.1. We have Dpxq " x2 . This is the divisor sum conjecture formulated in [14], which directly leads to the lower bound in Theorem 1.1. In what follows we are clearly at liberty to take a smaller disc D or a larger value of W , but we cannot change any of the other data. 4.1. Dirichlet’s hyperbola trick. Let i P t1, . . . , nu. For any ps, tq P Z2 appearing in (4.3), let ÿ ˆ Gi ps, tq ˙ . ri ps, tq “ k k|F ps,tq i

W

Then, possibly on enlarging W , it follows from Lemma 2.1 that ÿ ψi pdq, ri ps, tq “ d|pbi s´θi tq gcdpNi d,W q“1 dPPi

where d runs over integral ideals of Ki “ Qpθi q and Pi “ PKi , in the notation of (1.6). Furthermore, for all ps, tq in (4.3), we have Ni d ď Ni pbi s ´ θi tq “ |F˜i pbi s, tq| ď ci xdi , for some positive constant ci that depends at most on Fi and D. We define 1 d

1

X “ x maxtc1 1 , . . . , cndn u, so that the previous inequality becomes Ni d ď X di . On relabelling the indices we may suppose that dn “ min1ďiďn di . In particular, we have dn ď min degp∆i q. 1ďiďn

(4.4)


31

Suppose that n ą 1. Then for each i P t1, . . . , n ´ 1u and ps, tq appearing in (4.3), we set ÿ ÿ p0q p1q ri ps, tq “ ψi peq. ψi pdq, ri ps, tq “ d|pbi s´θi tq, dPPi

e|pbi s´θi tq, ePPi

gcdpNi d,W q“1 Ni dďX

gcdpNi e,W q“1

di 2

Ni eďX ´

di 2

Fi ps,tqW

Dirichlet’s hyperbola trick implies that p0q

p1q

ri ps, tq “ ri ps, tq ` ri ps, tq.

(4.5)

L “ plog xqα ,

(4.6)

Indeed, if pbi s ´ θi tqW denotes the part of the ideal pbi s ´ θi tq that is composed solely of prime ideals whose norms are coprime to W , as in (1.4), then the sum in ri ps, tq is over ideals d, e such that de “ pbi s ´ θi tqW . Recalling (4.2), it follows from part (ii) of Lemma 2.1 that ψi ppbi s ´ θi tqW q “ 1. This concludes the proof of (4.5). We proceed by introducing the quantity for some α ą 0 that will be determined in due course. (When n ą 1 we shall take α to be a large constant, but when n “ 1 it will be important to restrict to 0 ă α ă 1.) For ps, tq appearing in (4.3), we proceed by defining ÿ ÿ ψn peq ψn pdq, rnp1q ps, tq “ rnp0q ps, tq “ e|pbn s´θn tq, ePPn gcdpNn e,W q“1

d|pbn s´θn tq, dPPn gcdpNn d,W q“1 Nn dďL´1 X

dn 2

Nn eďL´1 X ´

and rnp8q ps, tq “

ÿ

d|pbn s´θn tq, dPPn gcdpNn d,W q“1

L´1 X

dn 2

As before, we may now write

ăNn dăLX

dn 2

Fn ps,tqW

ψn pdq.

dn 2

rn ps, tq “ rnp8q ps, tq ` rnp0q ps, tq ` rnp1q ps, tq.

For each j “ pj1 , . . . , jn q P t0, 1un , we define n ÿ ź pj q Dj pxq “ f pFi ps, tqW qri i ps, tq, i“1 ps,tqPZ2prim XxD ps,tq”ps0 ,t0 q pmod W q

and D8 pxq “

ÿ

ps,tqPZ2prim XxD

ps,tq”ps0 ,t0 q pmod W q

rnp8q ps, tq

n´1 ź i“1

ri ps, tq,

(4.7)

32


in which we recall the definition (4.1) of f . (Here, we recall our convention that products over empty sets are equal to 1.) Injecting (4.5) and (4.7) into (4.3) yields ÿ Dpxq ´ Dj pxq ! D8 pxq. jPt0,1un

The validity of Proposition 4.1 is therefore assured, provided we can show that Dj pxq " x2

(4.8)

D8 pxq “ opx2 q.

(4.9)

and We shall devote §§4.2–4.4 to the proof of (4.9) and §4.5 to the proof of (4.8). 4.2. The generalised Hooley ∆-function. In this subsection we initiate the proof of (4.9). Define $ , ps, tq ” ps , t q pmod W q ’ / 0 0 ’ / & . Dd P P such that: n p8q 2 An pxq “ ps, tq P Zprim X xD : . (4.10) ˝ d | pbn s ´ θn tqW ’ / ’ / dn dn % ˝ L´1 X 2 ă Nn d ă LX 2

It immediately follows that

D8 pxq “

ÿ

p8q ps,tqPAn pxq

rnp8q ps, tq

n´1 ź i“1

ri ps, tq.

Defining B8 pxq “

ÿ

p8q ps,tqPAn pxq

n´1 ź i“1

ri ps, tq,

(4.11)

we use Cauchy’s inequality to arrive at ¨ ˛ 21 n´1 ˇ ˇ ź ÿ 2 1 ˇ p8q ˇ D8 pxq ď B8 pxq 2 ˝ ri ps, tq‚ . ˇrn ps, tqˇ p8q

ps,tqPAn

i“1

pxq

Recall the definition (2.7) of the twisted Hooley ∆-function ∆pa; ψn q associated to the Dirichlet character ψn and any integral ideal a. Putting H8 pxq “

ÿ

∆ppbn s ´

ps,tqPZprim XxD ps,tq”ps0 ,t0 q pmod W q

θn tq; ψn q2W

n´1 ź i“1

ri ps, tq,

(4.12)

RATIONAL POINTS ON QUARTIC DEL PEZZO SURFACES dn

33

dn

and partitioning the interval pL´1 X 2 , LX 2 q into at most Oplog log xq e-adic intervals, we deduce that ˇ ˇ2 n´1 ÿ ˇ p8q ˇ ź ri ps, tq ! plog log xq2 H8 pxq. ˇrn ps, tqˇ p8q

ps,tqPAn

i“1

pxq

In summary, we have shown that

D8 pxq ! plog log xq

a

B8 pxqH8 pxq.

Therefore, in order to prove (4.9), it will be sufficient to prove that there exists a constant δ ą 0, that depends only on the data given at the start of §4, such that B8 pxq ! x2 plog xq´δ (4.13)

and

H8 pxq ! x2 plog xqop1q .

(4.14)

We shall call B8 pxq the interval sum and H8 pxq the Hooley–Tenenbaum sum. 4.3. The interval sum. By recycling work of de la Bretèche and Tenenbaum [5, §7.4], the case when F is irreducible is easy to handle. Indeed, in this case, n “ 1 and dn “ 4, so that (4.11) becomes , $ ps, tq ” ps0 , t0 q pmod W q / ’ / ’ . & Dd P P such that: p8q 1 2 . B8 pxq “ 7A1 pxq ď 7 ps, tq P Zprim X xD : ˝ d | pb1 s ´ θ1 tqW / ’ / ’ % ˝ X 2 {L ă N d ă LX 2 1

Replacing D by a smaller disc, we can ensure that |F1 ps, tq| — 1 whenever ps, tq P D. Thus we deduce that if ps, tq P Z2 X xD, then N1 ppb1 s ´ θ1 tqW q “ F˜1 pb1 s, tqW “ |F1 ps, tq| — x4 — X 4 .

Therefore, on introducing e through the factorisation de “ pb1 s ´ θ1 tqW , we can infer that we must have either X 2 {L ! N1 d ! X 2

or

X 2 {L ! N1 e ! X 2 .

Without loss of generality we shall assume that we are in the former setting. Therefore there exist constants c0 , c1 ą 0 such that * " ps, tq ” ps0 , t0 q pmod W q 2 . B8 pxq ! 7 ps, tq P Zprim X xD : Dd | F1 ps, tq s.t. c0 x2 {L ă d ă c1 x2 But now we can now employ the bound [5, Eq. (7.41)], with T “ F1 ,

Ξ “ ξ “ x,

y1 “ c0 x2 {L,

y2 “ c1 x2 ,

and

1 ! σ, ϑ ! 1.

34


This implies that for any η P p0, 12 q, we have ˙ ˆ log log x L 2 ` , B8 pxq ! x plog xqQp2ηq plog xqQp1`ηq

where Qpλq “ λ log λ ´ λ ` 1. In particular, Qp2ηq Ñ 1 as η Ñ 0` and Qp1 ` ηq ą 0 for all η ą 0. Recalling the definition (4.6) of L, this means that provided α ă 1, we may choose η ą 0 small enough (but away from 0), so as to ensure that (4.13) holds when F is irreducible. It remains to establish (4.13) when F is not irreducible. In this case (4.4) implies that dn “ degpFn q ď 2. Fix η P p0, 1q. To estimate B8 pxq, drawing inspiration from [5, §9.3], we shall divide the terms in the sum (4.11) into two categories. p1q

First case: pbn s ´ θn tq has many prime divisors. We denote by B8 pxq the contribution to B8 pxq from ps, tq for which Ωn ppbn s´θn tqW q ą p1`ηq log log x. We have n´1 ÿ ź p1q B8 pxq ď plog xq´p1`ηq logp1`ηq p1 ` ηqΩn ppbn s´θn tqW q ri ps, tq, i“1

ps,tqPZ2prim XxD

since p1 ` ηq´p1`ηq log log x “ plog xq´p1`ηq logp1`ηq . Our plan is now to apply Theorem 1.3 for N “ n, with fN paq “ p1 ` ηqΩn paq and ÿ ψi pdq, fi paq “ d|a dPPi

for i ă N. Furthermore, we shall take G “ Z2 and R “ xD. Thus qG “ 1, R is regular and we have V — x2 and KR — x log x, in the notation of the p1q theorem. Thus the contribution to B8 pxq from the sum over ps, tq is ˛ ¨ ÿ ÿ 1 ` ψi ppq ÿ ˚n´1 x2 ˚ ! exp ` p1 ` ηq ˝ plog xqn Ni p i“1 pPP pPP i

Ni p!x2

n

Nn p!x2

x2 expppn ´ 1q log log x ` p1 ` ηq log log xq plog xqn ! x2 plog xqη .

!

1 ‹ ‹ Nn p ‚

The proof of these estimates is standard and will not be repeated here. (See p1q Heilbronn [17], for example.) Thus B8 pxq ! x2 plog xq´p1`ηq logp1`ηq`η . The exponent of the logarithm is strictly negative for all η ą 0, which is clearly sufficient for (4.13).


35 p2q

Second case: pbn s ´ θn tq has few prime divisors. We denote by B8 pxq the contribution to B8 pxq from ps, tq for which Ωn ppbn s´θn tqW q ď p1`ηq log log x. p8q Recall from the definition (4.10) of An pxq that there exists d P Pn such that d | pbn s ´ θn tq, with gcdpNn d, W q “ 1 and L´1 X

dn 2

ă Nn d ă LX

dn 2

.

As previously, we can ensure that Nn ppbn s´θn tqW q — X dn , by taking a smaller disc inside D. Defining e via the factorisation de “ pbn s ´ θn tqW , we can then dn dn infer that gcdpNn e, W q “ 1 and e P Pn , with L´1 X 2 ! Nn e ! LX 2 , where the implied constants depend at most on D and Fn . Notice that Ωn pdq ` Ωn peq “ Ωn ppbn s ´ θn tqW q ď p1 ` ηq log log x. Thus, either Ωn pdq ď 21 p1 ` ηq log log x, or Ωn peq ď 21 p1 ` ηq log log x. We will assume without loss of generality that we are in the latter case. It follows that ÿ p2q B8 pxq ! Be pxq, ePPn dn dn ´1 2 L X !Nn e!LX 2 1 Ωn peqď 2 p1`ηq log log x gcdpNn e,W q“1

where Be pxq “

ÿ

n´1 ź

i“1 ps,tqPZ2prim XxD ps,tq”ps0 ,t0 q pmod W q e|pbn s´θn tq

ri ps, tq.

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 2.2 implies that the condition e | pbn s ´ θn tq defines a lattice in Z2 of determinant e “ Nn e, which we shall call G. Hence we may write Be pxq “

ÿ

ps,tqPZ2prim XxDXG

ps,tq”ps0 ,t0 q pmod W q

n´1 ź i“1

ri ps, tq.

Let v P Z2 be such that |v| “ maxt|v1 |, |v2 |u is the first successive minimum of G. Theorem 1.3 can be applied with R “ xD, qG “ e, N “ n ´ 1, and ÿ fi paq “ ψi pdq, d|a

36


for 1 ď i ď n ´ 1. This leads to the bound Be pxq ! x2 for any ε ą 0, where

h˚ peq x1`ε ` , e |v|

˙´1 źˆ ρ1 ppq ` ¨ ¨ ¨ ` ρn´1 ppq 1´ h peq “ . p ` 1 p|k ˚

(Note that h˚W peq “ h˚ peq, since gcdpe, W q “ a 1.)

? dn dn We have e “ Nn e ! LX 2 and so |v| ! LX 2 ď LX, since dn ď 2. Since Fn is irreducible, we note that dn “ 1 when Fn pvq “ 0. Define gpeq “ 7te P Pn : Nn e “ eu. The second term is therefore seen to make the overall contribution ÿ ÿ 3 1 ÿ 1 ÿ ! x1`ε gpeq ! x 2 `2ε , gpeq ` x1`ε |v| e|F pvq |v| ? ? ? |v|! LX Fn pvq‰0

|v|! LX Fn pvq“0

n

e!L X

which is satisfactory. Next, the overall contribution from the term x2 h˚ peq{e is Opx2 Σq, where ÿ gpeqh˚ peq . Σ“ e dn dn Letting A “

` 1`η ˘´1 2

L´1 X 2 !e!LX 2 Ωpeqď 12 p1`ηq log log x gcdpe,W q“1

ą 1, we get

Σ ! plog xq

ÿ

log A A

dn

L´1 X 2 !e!LX gcdpe,W q“1

Put Spyq “

ÿ

eďy gcdpe,W q“1

dn 2

gpeqh˚peq ´Ωpeq A . e

gpeqh˚ peqA´Ωpeq .

Then it follows from Shiu’s work [26] that ˛ ˛ ¨ ¨ ÿ gppqh˚ ppq ‹ ÿ ρ ppq ‹ y y ˚ ˚ n Spyq ! exp ˝A´1 exp ˝A´1 ‚! ‚ log y p log y p pďy pďy p∤W

p∤W

! yplog yq

1 ´1 A

.


37

Partial summation now leads to the estimate p2q B8 pxq ! x2 plog log xqplog xq

log A 1 À ´1 A

“ x2 plog log xqplog xq´ logp

1`η 2

` η´1 qp 1`η 2 q 2 .

The exponent of log x is strictly negative for all η P p0, 1q, which thereby completely settles the proof of (4.13). 4.4. The Hooley–Tenenbaum sum. We saw in §2.4 that the Hooley ∆function defined in (2.7) belongs to Mn . The stage is now set for an application 2 2 of Theorem ř 1.3 with N “ n and G “ Z , and with fN paq “ ∆pa; ψn q and fi paq “ d|a ψi pdq, for i ă N. This gives H8 pxq !

x2 E∆p¨;ψn q2 px2 ; W q log x

in (4.12). The statement of (4.14) now follows from part (ii) of Lemma 2.6. 4.5. Small divisors. In this subsection we establish (4.8), as required to complete the proof of Proposition 4.1. When n ą 1, the proof follows from the treatment in [14] and will not be repeated here. To be more precise, the work in [14, §4] provides an asymptotic formula for Dj pxq (denoted by Sψ in [14]), with a power saving in the error term, provided that the total degree is at most 3. The extension to total degree 4 and n ą 1 is standard (cf. [11]). Thus, provided that one takes α to be sufficiently large in the definition (4.6) of L, one gets an asymptotic formula for Dj pxq with a logarithmic saving in the error term. The proof of (4.8) when n “ 1 is more complicated. In this case F “ F1 is an irreducible binary quartic form and, in order to simplify the notation, we shall drop the index n “ 1 in what follows. Our task is to estimate ÿ Dj pxq “ f pF ps, tqW qr pjq ps, tq, ps,tqPZ2prim XxD ps,tq”ps0 ,t0 q pmod W q

for j P t0, 1u. Opening up the definition of f pF ps, tqW q, it follows from parts (i) and (ii) of Lemma 2.1 that f pF ps, tqW q “

ÿ

d|F ps,tq gcdpd,W q“1

τ pdqµpdq “ d

ÿ

e|pbs´θtq gcdpN e,W q“1 ePP

τ peqµpeq , Ne

since τ pN eq “ τK1 peq “ τ peq, say, for any e P P such that N e is square-free.

38


Let y ą 0. The overall contribution to Dj pxq from e such that N e ą y is ÿ ÿ τ peq|µpeq| ! r pjq ps, tq. Ne 4 2 ps,tqPZprim XxD e|pbs´θtq

yăN e!x gcdpN e,W q“1 ePP

The condition e | pbs ´ θtq defines a lattice in Z2 of determinant N e by Lemma 2.2. Thus we can apply Theorem 1.3, finding that ˚ ÿ ε pjq 2 hW pN eq r ps, tq ! x ` x1` 2 , Ne 2 ps,tqPZprim XxD e|pbs´θtq

for any ε ą 0, where h˚ is given by (1.11) with N “ 1. Hence we arrive at the overall contribution ÿ ÿ ε ε x2 ! x2 pN eq´2`ε ` x1` 2 pN eq´1` 8 ! ? ` x1`ε , y 4 N eąy N e!x

from N e ą y. Taking y “ log log x, we therefore conclude that ˆ ˙ ÿ ÿ x2 τ peqµpeq pjq r ps, tq ` O ? Dj pxq “ . Ne log log x 2 N eďlog log x ps,tqPZprim XxD e|pbs´θtq ps,tq”ps0 ,t0 q pmod W q

gcdpN e,W q“1 ePP

We henceforth focus on the case j “ 0, the case j “ 1 being similar. Opening up the definition of r p0q ps, tq, an application of Lemma 2.2 now yields ÿ ÿ ÿ τ peqµpeq 1 ψpdq D0 pxq “ Ne 2 ´1 2 N eďlog log x N dďL X gcdpN d,W q“1 dPP

gcdpN e,W q“1 ePP

Ò

ˆ

?

˙

x2 , log log x

ps,tqPZprim XxD bs”kt pmod M q ps,tq”ps0 ,t0 q pmod W q

(4.15)

for an appropriate k P Z such that gcdpbk, N d N eq “ 1, where M “ rN d, N es is the least common multiple of N d and N e. The inner sum over s, t is now in a form that is suitable for Lemma 2.5, with c “ k, e “ M and 1 ! δpDq ! 1,

β, γ ! 1.

Arguing as in [14, §§4.3–4.5], once inserted into (4.15), the contribution from the main term (denoted by Mψ in [14]) in Lemma 2.5 is " x2 . This is satisfactory for (4.8). It remains to consider the effect of substituting the error term in Lemma 2.5. Let r ˚ pmq “ 7ta P P : N a “ m, gcdpN a, W q “ 1u,


39

for any m P N. This function is multiplicative and has constant average order. We claim that r ˚ pcdq ď r ˚ pcqr ˚ pdq for all c, d P N, which we shall keep in use throughout this subsection. It is enough to consider the case c “ pa and d “ pb for a rational prime p ∤ W . Let p1 , . . . , pm be all the degree 1 prime ideals above p. Then r ˚ ppk q “ 7tk1 , . . . , km P Zě0 : k1 ` ¨ ¨ ¨ ` km “ ku “ τm ppk q. ` ˘ The claim follows on recalling that τm ppk q “ k`m´1 and comparing binomial m´1 coefficients. The error in Lemma 2.5 is composed of two terms. According to (2.6), the second term contributes ÿ τ peq|µpeq| 1 ÿ ? !x d, ¨ Ne M d|M 2 N d!x {L N eďlog log x

with M “ rN d, N es. Taking M ě N d and ¨ ˛ ÿ? ÿ? ? dď˝ d‚τ pN eq N e, d|M

d|N d

we conclude that that the second term contributes ÿ ÿ r ˚ pkq ÿ ? τ peq2 |µpeq| ÿ ? ? !x d ! x log log x d k d|k N e N d d|N d N d!x2 {L k!x2 {L N eďlog log x

ÿ

r ˚ pcqr ˚pdq ? ! x log log x c d 2 cd!x {L c ÿ r ˚ pcq x2 ! x log log x c cL 2 ´ 21

!L

c!x {L

2

x log log x,

in (4.15). This is satisfactory for any α ą 0 in (4.6). Finally, the overall contribution from the first error term Lemma 2.5 is ˙ ˆ ÿ τ peq|µpeq| ÿ 1 x !x log Ne k|vpM{kq| k|vpM{kq| 2 k|M N d!x {L N eďlog log x gcdpN d N e,W q“1

!x

ÿ

d!x2 {L eďlog log x gcdpde,W q“1

˙ ˆ 1 r ˚ pdqr ˚peqτ peq ÿ x , log e k|vprd, es{kq| k|vprd, es{kq| k|rd,es

40


where M “ rN d, N es. Now k | rd, es if and only if k | d and k | e. Writing d “ kd1 and e “ ke1 , we see that rd, es “ krd1 , e1 s and |vprd1 , e1 sq| ě |vpd1 q|. Hence this error is ˙ ˆ ÿ 1 x r ˚ pkq2 r ˚ pd1 qr ˚ pe1 qτ pke1 q log !x ke1 k|vpd1 q| k|vpd1 q| 1 1 k,d ,e PN kd1 !x2 {L ke1 ďlog log x gcdpkd1 e1 ,W q“1

! x log log x

ÿ

d1 !x2 {L

gcdpd1 ,W q“1

r ˚ pd1 q log |vpd1 q|

ˆ

x |vpd1 q|

˙

,

1 on summing over e? ď log log ?x and noting that the sum over k is convergent. 1 We have |vpd q| ! d1 ! x{ L in this sum. On bounding r ˚ by τ4 and taking N sufficiently large, the contribution from d1 for which |vpd1 q| ď x{plog xqN is easily seen to be ÿ 1 ÿ ! xplog xq2 τ4 pd1 q ! x2 plog xqŃ `6 , |v| 1 2 v“pv1 ,v2 qPZ 0ă|v|ďx{plog xqN

d |F pvq

by [1], which is satisfactory. Here we have used Lemma 2.2 to go from the congruence bv1 ” kv2 pmod d1 q to d1 | F pvq. In the opposite case, we have d1 " |vpd1 q|2 ě x2 {plog xq2N , whence logpx{|vpd1 q|q !N log log x. Thus it remains to study the contribution ÿ r ˚ pd1 q !N xplog log xq2 ď xplog log xq2 Epxq, 1 q| |vpd 2 2N 1 2 x {plog xq !d !x {L gcdpd1 ,W q“1

where Epxq “

ÿ

?

x{plog xqN !2i !x{ L

1 2i

ÿ

ÿ

bv1 ”kv2 pmod d1 q v“pv1 ,v2 qPZ2 2i ă|v|ď2i`1 x2 {plog xq2N !d1 !x2 {L gcdpd1 ,W q“1

r ˚ pd1 q.

We will need to restrict the summation over v to a summation over primitive vectors. Write v “ hw for w P Z2prim . Any d1 factorises uniquely as d1 “ dℓ, where ℓ “ gcdpd1 , hq. Since r ˚ pd1 q ď r ˚ pdqr ˚ pℓq, we thus conclude that ÿ ÿ ÿ 1 ÿ ÿ ˚ r ˚ pdq. Epxq ď r pℓq i ? 2 2 i`1 ℓ|h N i bw ”kw pmod dq x{plog xq !2 !x{ L

hď2

wPZprim 1 2 ´1 2 2N ´1 2 |w|ď2i`1 {h ℓ x {plog xq !d!ℓ x {L gcdpd,W q“1


41

The inner sum vanishes unless d “ N c for some ideal c P P. Hence Lemma 2.2 implies that the congruence in the inner sum is equivalent to c | pbw1 ´ θw2 q. Thus ÿ ÿ 1 ÿ ÿ ˚ Epxq ď ∆ppbw1 ´ θw2 qW q, r pℓq ? 2i 2 N i hď2i`1 ℓ|h wPZprim |w|ď2i`1 {h

x{plog xq !2 !x{ L

where ∆p¨q “ ∆p¨, 1q, in the notation of (2.7). In order to complete the proof of (4.8) it is therefore enough to show that ÿ ∆ppbx1 ´ θx2 qW q !ε X 2 plog Xqε , xPZ2prim |x|ďX

for any ε ą 0. For this we apply Theorem 1.3 with G “ Z2 , combined with part (i) of Lemma 2.6. We conclude that ÿ ÿ X2 ∆paq ∆ppbx1 ´ θx2 qW q ! !ε X 2 plog Xqε , log X N a 2 aPP square-free xPZprim |x|ďX

K

N a!X 2 gcdpN a,W q“1

for any ε ą 0. This completes the proof of (4.8). 5. 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 [14, Thm. 5.6 and Rem. 5.9]. This shows that there are binary piq piq piq quadratic forms q1,1 , q1,2 , q2,2 P Zrs, ts, for i “ 1, 2, such that ! ) ÿ ÿ piq 7 y P Z3prim : Qs,t pyq “ 0, }y}s,t ! B , (5.1) NpBq ď i“1,2 ps,tqPZ2prim ? |s|,|t|! B ∆piq ps,tq‰0

where }y}s,t “ maxt|s|, |t|u maxt|y1 |, |y2 |u and piq

piq

piq

piq

Qs,t pyq “ q1,1 ps, tqy12 ` q1,2 ps, tqy1y2 ` q2,2 ps, tqy22 ` y32 . piq

Moreover, the discriminant ∆piq ps, tq of Qs,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

42


bounded height on a conic, which is uniform in the coefficients of the defining equation, a topic that was addressed in §2.1. 5.1. Application of the bound for conics. Returning to (5.1), we apply Proposition 2.3 to estimate the inner cardinality. For any ps, tq P Z2prim , an argument of Broberg [6, Lemma 7] shows that DQs,t “ Op1q. In our work W is given by (1.5), with 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 2DQs,t ă w ! 1. We deduce that ˜ ¸ ÿ B NpBq ! CpQs,t , wq 1 ` , 1 2 3 maxt|s|, |t|u 3 |∆ps, tq| 2 ps,tqPZ prim ? |s|,|t|! B ∆ps,tq‰0

for any w ą 0, where CpQs,t , wq !

ź

pξ }∆ps,tq

τ ppξ q

pξ }∆ps,tq

˜

ξ ÿ

k“0

pąw

pďw

Since s, t !

ź

χQs,t ppqk

¸

.

? B and degp∆q “ 4, we see that 1

2

|∆ps, tq| 3 maxt|s|, |t|u 3 ! maxt|s|, |t|u2 ! B, whence 1` Now let

B 1 3

|∆ps, tq| maxt|s|, |t|u

2 3

∆ps, tq “

!

n ź i“1

B 1 3

2

|∆ps, tq| maxt|s|, |t|u 3 ∆i ps, tq

.

(5.2)

be the factorisation of ∆ps, tq into irreducible factors over Q. Each ∆i is separable and Resp∆i , ∆j q ‰ 0, whenever i ‰ j. We suppose that X has δ0 “ m split degenerate fibres and we re-order the factorisation of ∆ps, tq in such a way that the split degenerate fibres correspond to the closed points ∆1 ps, tq, . . . , ∆m ps, tq, with the non-split fibres corresponding to the closed points ∆m`1 ps, tq, . . . , ∆n ps, tq. We enlarge w so that w ą max | Resp∆i , ∆j q|. i‰j

Loughran, Frei and Sofos [14, Part (5) of Lemma 4.8] have shown that for each i ą m there exists a binary form Gi ps, tq P Zrs, ts of even non-negative degree,


with RespGi , ∆i q non-zero, such that χQs,t ppq “

ˆ

43

˙ Gi ps, tq , p

for all ps, tq P Z2prim with ∆ps, tq ‰ 0, and all primes p ą w with p | ∆i ps, tq. We proceed by introducing the arithmetic functions ÿ ÿ 1, τi ps, tq “ 1, p1 ď i ď mq, (5.3) τ0 ps, tq “ d|∆i ps,tq gcdpd,W q“1

d|∆ps,tq d|W 8

and ri ps, tq “

ÿ

d|∆i ps,tq gcdpd,W q“1

ˆ

Gi ps, tq di

˙

,

pm ă i ď nq.

(5.4)

We put Sps, tq “ τ0 ps, tq for any ps, tq P

Z2prim .

m ź i“1

τi ps, tq

n ź

i“m`1

ri ps, tq,

(5.5)

Note that Sps, tq ě 0. Our work so far shows that NpBq ! B

ÿ

Sps, tq 1

ps,tqPZ2prim ? s,t! B ∆ps,tq‰0

2

|∆ps, tq| 3 |t| 3

.

By symmetry, it will suffice to consider the case |s| ď |t|. Moreover, if s “ 0 then the condition gcdps, tq “ 1 shows that we must have t “ ˘1. The contribution from these ps, tq is plainly negligible. Hence, Theorem 1.1 will follow from a bound of the shape ÿ Sps, tq m`1 , (5.6) 2 ! plog Bq 1 3 |t| 3 |∆ps, tq| 2 ps,tqPZ prim ? 1ď|s|ď|t|ď B ∆ps,tq‰0

since (1.2) implies that m ` 1 “ ρ ´ 1. 5.2. Reduction to divisor sums. For β P C and x, y ą 0 we let

( V “ ps, tq P R2 : 1 ď |s| ď |t| ď x, |s ´ βt| ď y, ∆ps, tq ‰ 0 .

Consider the divisor function

Dβ px, yq “

ÿ

ps,tqPV XZ2prim

Sps, tq,

(5.7)

44


where Sps, tq is given by (5.5). In this subsection we shall establish (5.6) subject to the following bound for Dβ px, yq, whose proof will occupy the remainder of the paper. Proposition 5.1. Let β P C, let η P p0, 1q and assume that xη ď y ď x. Then Dβ px, yq !β,η xy plog xqm . We proceed to show how (5.6) follows from Proposition 5.1. Since ∆ps, tq is separable, it may contain the polynomial factor t at most once. Therefore there exists c0 P Q˚ and pairwise unequal αi , αj P Q such that ∆ps, tq admits ś ś the factorisation c0 t 3i“1 ps ´ αi tq or c0 4i“1 ps ´ αi tq , according to whether t | ∆ps, tq or not, respectively. Putting α“

1 min t|αi ´ αj |, |αk |u , 2 i,j,k

(5.8)

i‰j

the set of integer pairs ps, tq appearing in (5.6) can be partitioned according to whether or not ps, tq belongs to the set ( A “ ps, tq P R2 : |s ´ αi t| ě α|t|, for all i . If ps, tq P A then ∆ps, tq " |t|4 and it follows that ÿ ÿ Sps, tq 2 ! 1 |∆ps, tq| 3 |t| 3 ps,tqPA XZ2 ps,tqPA XZ2 prim ? 1ď|s|ď|t|ď B ∆ps,tq‰0

prim ? 1ď|s|ď|t|ď B ∆ps,tq‰0

Sps, tq . |t|2

Breaking into dyadic intervals T {2 ă |t| ď T and applying Proposition 5.1 with x “ y “ T and β “ 0, we readily find that the right hand side is Opplog Bqm`1 q, which is satisfactory for (5.6). It remains to consider the contribution to (5.6) from ps, tq P Z2prim zA . For each i we define ÿ Sps, tq Si pBq “ 2 . 1 3 |t| 3 |∆ps, tq| 2 ps,tqPZ prim ? 1ď|s|ď|t|ď B ∆ps,tq‰0 |s´αi t|ăα|t|

It now suffices to prove Si pBq “ Opplog Bqm`1 q for each i and each αi . If ps, tq is counted by Si pBq then (5.8) implies that

1 |s ´ αj t| ě |αi ´ αj ||t|, 2 3 for any j ‰ i. Hence ? |∆ps, tq| " |t| |s ´ αi t| in Si pBq. For given S, T satisfying 1 ! S ! T ! B, the overall contribution to Si pBq from s, t such that


45

T {2 ă |t| ď T and S{2 ă |s ´ αi t| ď S is seen to be !

1

5

1 3

S T3

Dαi pT, Sq,

1

in the notation of (5.7). If S " T 10 then Proposition 5.1 shows that this is 2

!

S 3 plog Bqm 2

T3

.

? 1 Summing over dyadic S, T satisfying T 10 ! S ! T ! B shows that this 1 gives an overall contribution Opplog Bqm`1 q. On the other hand, if S ! T 10 , we take Sps, tq ! T ε for any ε ą 0, by the standard estimate for the divisor 1 function, so that Dαi pT, Sq ! ST 1`ε . Taking ε “ 30 , we therefore arrive at the contribution 1 2 2 1 S 3 T 30 ! T ´ 3 ` 10 , ! 2 T3 1 from this?case. Again, summing over dyadic S, T satisfying S ! T 10 and 1 ! T ! B, this shows that we have an overall contribution Op1q, which is plainly satisfactory. This completes the deduction of (5.6) from Proposition 5.1. 5.3. Small divisors. The function τ0 ps, tq in (5.5) is concerned with the contribution to Sps, tq from small primes p ď w. Our work in §2.1 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 τ0 ps, tq, before handling the remaining factors in Sps, tq. Following Daniel [11], for any a P N we call two vectors x, y P Z2 equivalent modulo a if gcdpx, aq “ gcdpy, aq “ 1

and

∆pxq ” ∆pyq ” 0 pmod aq,

and, moreover, there exists λ pmod aq such that x ” λy pmod aq. The set of equivalence classes is denoted by Apaq and the class elements as A . Letting ̺˚ paq “ 7 tpσ, τ q pmod aq : gcdpσ, τ, aq “ 1, ∆pσ, τ q ” 0 pmod aqu ,

we find that ̺˚ paq “ ϕpaq7Apaq. Moreover, we clearly have ̺˚ paq ď ϕpaqpρ∆px,1q paq ` ρ∆p1,xq paqq,

in the notation of (1.9). Since ∆ps, tq is separable, it follows from Huxley [18] 1 that ρ∆px,1q paq ď 4ωpaq | discp∆q| 2 , and similarly for ρ∆p1,xq paq. Hence 7Apaq “

̺˚ paq ! 4ωpaq . ϕpaq

(5.9)

46


For each ps, tq P V X Z2prim , write rps, tq “ Then Dβ px, yq ď ď

ÿ

m ź i“1

q!x4 q|W 8

ÿ

τi ps, tq ÿ

n ź

i“m`1

ri ps, tq.

rps, tq

XZ2prim

ps,tqPV q|∆ps,tq

ÿ

ÿ

rps, tq,

q!x4 A PApqq ps,tqPV XGpA qXZ2prim q|W 8

where GpA q “ tx P Z2 : Dλ P Z Dy P A s.t. x ” λy pmod qqu is the lattice generated by the vectors in A . The determinant of this lattice is q. We shall establish the following result. Proposition 5.2. Let η P p0, 1q and assume that xη ď y ď x. Then ˆ ˙ ÿ plog xqm 1 rps, tq !β,η,N xy ` , q plog xqN 2 ps,tqPV XGpA qXZprim

for any N ą 0, where the implied constant is independent of q. We now show how Proposition 5.1 follows from this result. Employing (5.9), we deduce that ÿ ÿ 4ωpqq xy 4ωpqq . ` Dβ px, yq !β,η,N xyplog xqm N q plog xq 4 4 q!x q|W 8

q!x q|W 8

The first sum is ! plog wq4 ! 1. On the other hand, the second sum is ź ď p16 log x ` Op1qq ! plog xqπpwq . pďw

Choosing N “ πpwq, we therefore conclude the deduction of Proposition 5.1 from Proposition 5.2. 5.4. The final push. The aim of this subsection is to prove Proposition 5.2. Recall from (5.2) that we have a factorisation m n ź ź ∆ps, tq “ ∆i ps, tq ∆i ps, tq, i“1

i“m`1

where each ∆i P Zrs, ts is irreducible and the fibre above the closed point ∆i is split if and only if i ď m. We now want to bring into play the work in §2.1,


47

in order to transform the sum in Proposition 5.2 into one that can be handled by Theorem 1.3. Let i P t1, . . . , nu. Recall from (5.3) and (5.4) that we are interested in the divisor sum ˆ ˙ ÿ Gi ps, tq , d d|∆ ps,tq i

gcdpd,W q“1

where Gi ps, tq P Zrs, ts is a form of even degree (and we allow Gi ps, tq to be identically equal to 1). This is exactly of the form considered in (2.2). Let bi “ ∆i p1, 0q P Z and suppose for the moment that bi ‰ 0. As previously, ˜ i px, 1q, in the notation of (1.7), and write let θi be a root of the polynomial ∆ Ki “ Qpθi q. Let oi denote the ring of integers of Ki . We enlarge a w to ensure that w ą 2bi DLi {Ki ∆θi , where ∆θi is given by (2.3) and Li “ Ki p Gi pb´1 i x, 1qq. Thus # 1 if i ď m, rLi : Ki s “ 2 if i ą m.

Next, let ψi be the quadratic Dirichlet character constructed in §2.1 (taking ψi “ 1 when Gi ps, tq is identically 1). Let Ni denote the ideal norm in Ki . Then it follows from part (iii) of Lemma 2.1 that for any ps, tq P Z2prim such that ∆i ps, tq ‰ 0, we have ˆ ˙ ÿ ÿ Gi ps, tq “ ψi paq. (5.10) d d|∆ ps,tq a|pb s´θ tq i

i

i

gcdpNi a,W q“1

gcdpd,W q“1

Moreover, as in (1.6), if Pi is the multiplicative span of all prime ideals in oi with residue degree 1, then part (i) of Lemma 2.1 implies that a P Pi for any a | pbi s ´ θi tq such that gcdpNi a, W q “ 1. Suppose now that bi “ 0, so that ∆i ps, tq “ ct for some non-zero c P Z. We enlarge w to ensure that w ą c. In this case we have ˆ ˆ ˙ ˙ ˙ ˆ ÿ ÿ ÿ Gi ps, tq Gi p1, 0q Gi ps, tq “ “ , d d d d|t d|t d|∆ ps,tq i

gcdpd,W q“1

gcdpd,W q“1

gcdpd,W q“1

since Gi has even degree and ps, tq P Z2prim . But this is of the shape (5.10), with bi “ 0, θi “ 1, Ki “ Q, and ψi pdq “ p Gi p1,0q q. d Let i P t1, . . . , nu and let c Ă oi be an integral ideal. We define multiplicative functions ti , ri P MKi , in the notation of §1.2, via ÿ ti pcq “ 1, p1 ď i ď mq, aPPi a|c

48


and ri pcq “

ÿ

aPPi a|c

ψi paq,

pm ă i ď nq.

It follows that rps, tq “

m ź i“1

ti,W pbi s ´ θi tq

n ź

i“m`1

ri,W pbi s ´ θi tq

in Proposition 5.2, for any ps, tq P Z2prim . We are now in a position to apply Theorem 1.3 with R “ V , G “ GpA q and qG “ q. In particular it follows that xy ! V “ volpRq ! xy

and

x log x ! KR ! x log x.

According to the statement of Proposition 5.2, we are given η P p0, 1q and x, y such that xη ď y ď x. Thus R is regular. Since q ! x4 , it therefore follows that all the hypotheses of Theorem 1.3 are met. On enlarging W suitably, we deduce that m n ÿ ź xy h˚W pqq ź 2 rps, tq !η,W Eti px ; 1q Eri px2 ; 1q n plog xq q 2 i“1 i“m`1 ps,tqPV XGpA qXZprim

η

` x1` 2 , η

Note that h˚W pqq “ 1, since q | W 8 . Moreover, since x1` 2 !N xyplog xqŃ , for any N ą 0, the second term here is plainly satisfactory for Proposition 5.2. Finally, we have ¨ ¨ ˛ ˛ ˚ ÿ ti ppq ‹ ˚ ÿ 2 ‹ Eti pz; 1q “ exp ˝ ‚ “ exp ˝ ‚ ! plog zq2 , N p N p i i N pďz N pďz i

i

pPPi

for i P t1, . . . , mu, and

¨

pPPi

˛

˛

¨

˚ ÿ ri ppq ‹ ˚ ÿ 1 ` ψi ppq ‹ Eri pz; 1q “ exp ˝ ‚ “ exp ˝ ‚ ! log z, Ni p Ni p N pďz N pďz i

i

pPPi

pPPi

for i P tm ` 1, . . . , nu. Thus the first term makes the overall contribution xyplog xqm ! , q

which thereby completes the proof of Proposition 5.2.


49

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