INTEGRAL POINTS ON ELLIPTIC CURVES AND THE BOMBIERI-PILA BOUNDS DAVEMENDESDACOSTA 3 1 0 2 n Abstract. LetC beanaffine,plane,algebraiccurveofdegreedwithinteger a coefficients. In 1989, Bombieri and Pila showed that if one takes a box with J sidesoflengthN thenCcanobtainnomorethanOd,(cid:15)(cid:0)N1/d+(cid:15)(cid:1)integerpoints within the box. Importantly, the implied constant makes no reference to the 4 coefficients of the curve. Examples of certain rational curves show that this 2 boundistightbutithaslongbeenthoughtthatwhenrestrictedtonon-rational curvesanimprovementshouldbepossiblewhilstmaintainingtheuniformityof ] T thebound. Inthispaperweconsiderthisproblemrestrictedtoellipticcurves andshowthatforalargefamilyofthesecurvestheBombieri-Pilaboundscan N beimproved. Thetechniquesinvolvedincluderepulsionofintegerpoints,the h. theory of heights and the large sieve. As an application we prove a uniform bound for the number of rational points of bounded height on a general del t a Pezzosurfaceofdegree1. m [ 2 1. Background and Motivation v 6 Let C be a plane curve (projective or affine) defined over Q. The most basic 1 numbertheoreticquestiononecanaskaboutC istocharacterisethecardinalityof 1 the sets C(Q) and C(Z) (the latter being of interest when the curve is affine). An 4 aim of Diophantine geometry is to classify the nature of these sets in terms of the . 1 geometry of C. This aim has seen spectacular success in the 20th century with the 0 genus of the curve g(C) emerging as a key player. 3 1 In this paper, we shall be concerned with integer points on affine curves (in partic- : v ular the usual affine patch on an elliptic curve). The main theorem in this area is i X Siegel’s Theorem which teaches us that if g(C) ≥ 1 then #C(Z) is finite. (In fact it says that this is true with Z replaced by the ring of S-integers of any number r a field.) When C is rational (i.e., g(C)=0), examples exist where C(Z) is finite and others where it is infinite. This behaviour is well understood. After one has Siegel’s Theorem, a natural question is to ask for a more effective bound. Such results in this area tend to depend on the curve in question, indeed they must since there is no uniform bound for the number of integral points on a non-rational affine curve. Since one must make reference to the curve in such bounds the next step is to decide which features of the curve we wish to permit ourselves to use. This is a matter of taste and different types of bounds have dif- ferent applications. In this paper we shall be following the tradition of trying to maintain as much 1 2 DAVEMENDESDACOSTA uniformity as possible. As we have noted, if we desire results which are very uni- form but yet still apply to a large number of curves then it is difficult to study the cardinality of C(Z). Thus attention shifts to questions concerning the density of integral points. In particular one may ask how the counting function C(Z,N):=#{(x,y)∈C(Z):|x|,|y|≤N} behaves as N > 0 gets very large. Since we want our results to be uniform, an asymptotic result is not the correct thing to be looking for. Instead we would like a bound on this function which depends on N and on as little to do with the curve as possible. If we were to look to bound C(Z,N) independently of any information aboutC thenthebestwecoulddowouldbethetrivialboundC(Z,N)(cid:28)N2 since there are curves of high degree which subsume all the integer points in the box. Therefore we should ask for a bound in terms of the degree of the curve. The breakthrough result in this area was reported in 1989 when Bombieri and Pila [3] showed that if C is a plane curve of degree d then (1) C(Z,N)(cid:28) N1/d+(cid:15). d,(cid:15) This bound is tight as is seen by considering the curves C :y =xd d The interesting thing about this family is that it consists of rational curves only. Indeed, the only known examples where (1) is tight is when C is rational. From the point of view of Diophantine geometry this is intriguing since it once again points to differing arithmetic behaviours being catagorised by geometry. Indeed, this prompts us to ask the following question: Question 1. If C is a non-rational curve of degree d then is there a δ(d) > 0 depending only upon d such that C(Z,N)(cid:28) N1/d−δ(d)? d Currently, the best evidence that this question has a positive answer is that the corresponding question for rational points does. In 2000, Heath-Brown [8] showed that if one takes a plane projective curve of degree d and counts points of (naive) height less than N (that is, points P = [x : y : z] such that x,y,z are coprime integers of absolute value less than or equal to N) then for the associated counting function one gets C(N,Q)(cid:28) N2/d+(cid:15). d,(cid:15) Once again the only examples where this is tight are the projective versions of the curves we mentioned before and so the same question as above can be asked about non-rational curves. This was answered in the positive in 2005 by Ellenberg and Venkatesh [6]. So there is hope that our question has a positive answer and this answer is in reach even though the methods in [6] do not directly apply. INTEGRAL POINTS ON ELLIPTIC CURVES AND THE BOMBIERI-PILA BOUNDS 3 This paper is concerned with the case where C is a non-singular degree 3 non- rational curve, i.e., an elliptic curve. We shall show that for a large class of such curves we can give a positive answer to the Question. In particular we show the following two results. Firstly we have our Main Theorem. Let E be an elliptic curve given by the equation E : y2 = f(x) where f ∈ Z[x] is a monic cubic with no repeated roots. Then there is a δ > 0, independent of E, such that E(Z,N)(cid:28)N1/3−δ (the implied constant is uniform). Oursecondresultstrengthensandextendsthistoawiderclassofellipticcurves but requires more hypotheses: Theorem 1. Let E be an elliptic curve given by the equation y2 = f(x) where f ∈ Z[x] is a monic cubic with no repeated roots. Suppose further c (E) < 0 and 4 j(E) > (cid:15) for some (cid:15) > 0. Let B be any box in the plane with sides of length N. Then there is a δ >0 (depending only on (cid:15)) such that #(E(Z)∩B)=O (N1/3−δ). (cid:15) 1.1. Acknowledgments. I am glad to acknowledge the useful conversations and insightsthatmanypeoplehavesharedwithme. Inparticular,Iextendmywarmest regards to Tim Browning, Tim Dokchitser, Roger Heath-Brown, Marc Hindry, PierreLeBoudec, DanLoughran, JoeSilvermanandTrevorWooley. Manythanks go to my advisor Harald Helfgott who gave me this problem and has always been generous both as a supervisor and a friend. I would also like to thank the Univer- sity of Bristol and the Ecole Normale Superieure for providing me with wonderful working conditions during my PhD. Finally, dues are paid to the EPSRC Doc- toral Training Award, which has funded my research, with the hope that they may continue to support such pure mathematical endevours. 2. Notations and Conventions WeshallletE denoteanellipticcurvethroughout. Ourproblemisphrasedwith respect to particular embeddings of E and so we shall be assuming that the model of E takes a Weierstrass form. We say that E is in long Weierstrass form when we have E :y2+a xy+a y =x3+a x2+a x+a 1 3 2 4 6 and in short Weierstrass form if we have E :y2 =x3+Ax+B where a ,A,B ∈ Z. We make no assumptions on the minimality of E. We also i associate several standard quantities with E, namely the discriminant ∆ , c (E) E 4 andthej-invariantj(E). InthecaseofacurveinshortWeierstrassformtheseare: A (4A)3 ∆ =−16(4A3+27B2), c (E)=− , j(E)=−1728 . E 4 27 ∆ E In our bounds we shall make use of both Landau’s Big-Oh notation and Vino- gradov’s (cid:28) notation. Given functions f,g :R→R we shall write either 4 DAVEMENDESDACOSTA f(x)=O(g(x)) or f(x)(cid:28)g(x) inthecasethatthereisaconstantC >0suchthat|f(x)|≤C|g(x)|. Inparticular f =O(1) implies f is bounded. The constant C is suppressed by the notation and soisreferredtoastheimplied constant. Ifwewishtodrawnotetothedependency of the implied constant on something (such as the curve E) then we shall do so in subscript writing, for instance, f(x)=O (g(x)). E We shall make reference to several different heights in what follows. Let P = (x,y) ∈ Z2. By the naive height of P we shall mean the maximum of |x| and |y|. When dealing with points on an elliptic curve we shall be studying the canonical height hˆ on E. Despite its canonical nature, there are still several ways of scaling this height which are used in the literature. We shall be using the scaling which corresponds to seeing hˆ as a sum of local heights as defined in [13]. This is half the size of the version defined by Tate. 3. Repulsion techniques Themajorityoftheworkinthispapergoesintoputtingusintoasituationwhere wecanusearesultabouthowintegralpointscanbesaidtorepeloneanother. The idea of rational points repelling each other has a wonderful pedigree. In 1969, Mumford showed that if C is a curve of genus g > 1 then two rational points P,Q ∈ C(Q), which in the Jacobian of C have comparable canonical height, repel each other in the sense that the angle between them in the Mordell-Weil lattice of Jac(C)isatleastarccos(1). Thisobservation,knownasMumford’sGapPrinciple, g allowed Mumford to show that the number of rational points on C of naive height less than N is O (loglogN). C This result was improved by Faltings who showed that #C(Q)=O (1). C Thisboundcanalsobeseenasaconsequenceofrepulsion, thekeydifferencebeing that the condition that the points have to be of comparable canonical height is shown to not be necessary to force repulsion. So repulsion has a lot to say about rational points; how about integral points? In the case of an elliptic curve, Mumford’s Gap Principle says nothing about ratio- nalpointsotherthantheyrepeleachotherbyanangleofatleast0! However,ifwe restrict to integral points then we obtain a non-trivial repulsion of at least 60◦ be- tweenpointswhichhavecomparablecanonicalheight. Thisphenomenonispresent in the work of Silverman who showed how to integrate the notion of (S-)integrality of points into such repulsion techniques in [14] (later refined with Gross in [7]). If INTEGRAL POINTS ON ELLIPTIC CURVES AND THE BOMBIERI-PILA BOUNDS 5 we demand that the points reduce to the same point modulo some prime p, we can increase the minimum angular repulsion between points at the cost of having to partitionthepointsintothefibresofthereductionmap. Toturnthisrepulsioninto aboundonthenumberofintegralpointsofcanonicalheightlessthanh,wesimply slice up the set of integral points into strips with comparable canonical height, i.e., (1−(cid:15))logN ≤hˆ(P)≤(1+(cid:15))logN for some N > 0, and then to each slice apply bounds for the number of vectors which can packed with suitable angular repulsion onto a sphere of dimension r−1 where r is the rank of E(Q). The idea of gaining bounds through slicing and sphere-packing was first seen in the thesis of Helfgott and in the paper [9]. These ideas culminated in the paper [10] of Helfgott and Venkatesh who show how this repulsioncanbeobtainedinauniformmannerandalsohowthechoiceoftheprime p can be optimised to yield the following bound: Theorem 2. [10, Corollary 3.9] Let E be an elliptic curve over Q with integer coefficients and (cid:15)>0. Then there is a δ >0 such that the number of integral points of E of canonical height less than h is O (eh(1−δ)+(cid:15)) (cid:15) where the implied constant does not depend on E. Thus we see that this theorem yields the desired result when h ≤ 1logN and 3 in fact we can let h be a touch larger. In particular, the key to our application of Theorem 2 will be the following result. Proposition 1. Let E be an elliptic curve in short Weierstrass form and N,δ >0 such that |∆ | < N4+6δ and |A| ≤ N4/3+2δ. Then if Q = (x,y) ∈ E(Z) has E |x|≤N2/3+δ then (cid:18) (cid:19) 1 δ hˆ(Q)≤ + logN +O(1) 3 2 where the implied constant is absolute. 4. Proof of Proposition 1 The idea of the proof is to break up the canonical height, hˆ, as a sum of local canonical heights λ , one for each prime p and then bound each part. We shall use p the local heights as they are defined in [13, Chapter VI]. For the finite primes we shall use the following bound, due to Tate. Lemma 1. Let p be a finite prime, λ the local height of E at p and let Q∈E(Z). p Then ord (∆ ) λ (Q)≤ p E logp. p 12 Proof. [11, Chapter II, Theorem 4.5] (cid:3) 6 DAVEMENDESDACOSTA Thus we see that for Q∈E(Z), (cid:88) 1 (2) λ (Q)≤ log|∆ |. p 12 E p(cid:54)=∞ Thisleavesjusttheinfiniteparttoestimate. Letusrefreshourselvesonthetheory of elliptic curves over C. Every elliptic curve E over C is isomorphic to C/Λ for some rank 2 lattice Λ⊂C. The isomorphism is explicit and comes from the Weierstrass ℘ -function which Λ maps C/Λ to E(C) and is defined by 1 (cid:88) ℘ (z)= + (2k+1)G (Λ)z2k Λ z2 2k+2 k>0 where (cid:88) 1 G (Λ)= 2k ω2k 0(cid:54)=ω∈Λ is the weight 2k Eisenstein series associated to Λ. The isomorphism between C/Λ and E(C) is given by z (cid:55)→(℘ (z),℘(cid:48) (z)/2). Λ Λ In this case the equation for E can be recovered from Λ via E :y2 =x3−30G (Λ)x−70G (Λ). 4 6 Two curves C/Λ and C/Λ are isomorphic over C if and only if there is some 1 2 u ∈ C× such that uΛ = Λ . In this way one can normalise the form of Λ to be 1 2 Λ =<1,τ > where τ (cid:26) (cid:27) 1 τ ∈ z ∈C:|Re(z)|≤ ,|z|≥1 =:F. 2 For such a τ let E be the curve associated to Λ . τ τ Another way of characterising two curves being isomorphic is via the j-invariant (4A)3 j =−1728 . E ∆ E Two curves, E and E are isomorphic if and only if j = j . Moreover, if E is 1 2 E E(cid:48) definedoverafieldK thenj ∈K. Thecurvesweareinterestedinaredefinedover E Q and thus over R, so it is sensible to ask which curves in F have real j-invariant. INTEGRAL POINTS ON ELLIPTIC CURVES AND THE BOMBIERI-PILA BOUNDS 7 Lemma 2. Let E/R be an elliptic curve. Then there is a unique τ in the set (cid:40) √ (cid:41) (cid:110) π π(cid:111) 1 3 C ={ib:b≥1}∪ eiθ : ≤θ < ∪ +ib:b> 3 2 2 2 such that j(E )=j(E). τ Proof. [13, Chapter V, Prop 2.1, p.414] (cid:3) If j(E)=j(E ) then we shall say that E is associated to τ and if the isomorphism τ between E and E is defined over a field K we shall say it is associated to τ over τ K. LetuslabelthethreesetswhichmakeupC byC ,C andC respectively. These 1 2 3 regions neatly separate up the elliptic curves defined over R. The curves which are associated to τ ∈C are those with two real components. These curves have A≤0 1 andj ≥1728. Thecurvesassociatedtoτ ∈C haveonerealcomponent,A<0and 3 j <0. Finally,thecurvesassociatedtoτ ∈C haveonerealcomponent,A>0and 2 0≤j <1728. It will be noted that no mention of the B coefficient has been made. This is because we can change the sign of the B coefficient via the isomorphism (x,y)(cid:55)→(−x,iy) It is worth noting that in terms of the underlying lattices, this isomorphism is 1 Λ(cid:55)→ Λ i and so consists of rotating the lattice through 90◦. We shall refer to the image of E under this map as its twist. We note that every elliptic curve defined over R is τ isomorphic over R to either E or its twist for some τ ∈C. τ We shall now look at the set E(R) in a bit more detail. Let E(C) ∼= E (C) and τ consider a fundamental parallelogram for Λ τ P ={x +x τ :x ,x ∈(−1/2,1/2]}. τ 1 2 1 2 ForeveryellipticcurveEwhichhasj(E)∈R,thereisau∈C×suchthatuΛ=uΛ. Let Λ(cid:48) = uΛ. Then ℘ (z) = ℘ (z¯) and so the real points are those which are Λ(cid:48) Λ(cid:48) invariant under the action of complex conjugation on C/Λ(cid:48). For curves E with τ ∈ C or C we already have Λ = Λ . This means that τ 1 3 τ τ in the case of C , the (two) real components are the images of the points 1 (cid:26) (cid:26) (cid:27)(cid:27) 1 x +x τ ∈P :x ∈ 0, 1 2 τ 2 2 and for C the (one) real component is the image of 3 {x +x τ ∈P :x =0}. 1 2 τ 2 8 DAVEMENDESDACOSTA InthecaseofC weneedtomodifyΛ toΛ(cid:48) = 1 Λ. Thishastheeffectofrotating 2 τ τ1/2 the point 1+τ onto the real axis. Thus the real component is the image of (cid:26) (cid:27) 1 1 t(1+τ):− <t≤ . 2 2 The final thing left to note is that if we take the twist of E (which flips the sign τ of the B coefficient), call this twist E(cid:48), then since this rotates the lattice by 90◦ τ it is easy to find the points of P which correspond to points on E(cid:48)(R) (after we τ rotate the lattice). They are the points which, after we rotate Λ(cid:48) by 90◦, map to τ themselves under complex conjugation. These are the points on E(C) which are mapped to E(cid:48)(R) by the twist. Let τ ∈C and let E be associated to τ over R. We want to study λ (u) for those ∞ u∈P whicharemappedtoE(R)andwherethenaiveheightofthex-coordinateis τ bounded. We have now identified the points which map to E(R) and later we shall find conditions on u which correspond to the bound on the x-coordinate. We also wish to study the same points for the twist E(cid:48) of E. Rather than studying λ on ∞ the fundamental domain for E(cid:48), it is sufficient for us to understand the behaviour of λ (u) for those u ∈ P which correspond to points on E(cid:48)(R) after we twist. ∞ τ This is sufficient since the value of λ (u) does not change as we twist E to E(cid:48). ∞ Therefore we shall, in the sequel, study the behaviour of λ (u) for those u ∈ P ∞ τ which map to either E(R) or to the image of E(cid:48)(R) in E(C). We now have enough background to start examining the archimedean local canon- ical height. Take a point u = u +u τ ∈ P , then following [13, p. 468] we can define the 1 2 τ archimedean local canonical height to be (cid:12) (cid:12) 1 (cid:12)(cid:89) (cid:12) λ (u)=− B (u )log|q|−log|1−t|−log(cid:12) (1−qnt)(1−qnt−1)(cid:12) ∞ 2 2 2 (cid:12) (cid:12) (cid:12) (cid:12) n>0 where q = e2πiτ, B (x) = x2 −x+1/6 is the second Bernoulli polynomial and 2 t=e2πiu. In order to analyse the behaviour of λ for points of E((cid:48))(R), we note that there ∞ is a similarity between certain terms in λ and the Jacobi product formula for the ∞ discriminant function when viewed as a function on F, (cid:89) (3) ∆(τ)=∆(E )=(2π)12q (1−qn)24. τ n>0 We can use this similarity to our advantage. Lemma 3. Let τ ∈F. Then 1 λ (u)≤−log|1−t|− log|∆(τ)|+O(1) ∞ 12 INTEGRAL POINTS ON ELLIPTIC CURVES AND THE BOMBIERI-PILA BOUNDS 9 where the implied constant in the O(1) is absolute. Proof. First we note that since λ (u) = λ (−u) for all u ∈ P we can choose u ∞ ∞ τ suchthatu ∈[0,1/2]. Notingthat|q|<1andthatB (u )∈[−1/12,1/6]wehave 2 2 2 1 1 − B (u )log|q|≤− log|q|. 2 2 2 12 This corresponds to the q term in ∆(τ). Next we note that t = e2πiu = e2πiu1qu2, and so (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)(cid:89) (cid:12) (cid:12)(cid:89) (cid:12) (cid:12) (1−qnt)(1−qnt−1)(cid:12) = (cid:12) (1−e2πiu1qn+u2)(1−e−2πiu1qn−u2)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) n>0 n>0 ≥ (cid:89)(1−|q|n+u2)(1−|q|n−u2) n>0 ≥ (cid:89)(1−|q|n)(1−|q|n−1/2) n>0 (cid:89) √ √ ≥ (1−e− 3πn)(1−e− 3π(n−1/2)) n>0 = 0.92984... √ whereinthepenultimatelineweusethefactthatIm(τ)≥ 3. Thereforeλ (u)≤ 2 ∞ − 1 log|q|−log|1−t|+O(1), which only leaves us to show that 12 log|∆(τ)|=log|q|+O(1) and this is the case since the lower bound on Im(τ) allows us to deduce that (cid:12) (cid:12) (cid:12) (cid:89) (cid:12) 21.588≤log(cid:12)(2π)12 (1−qn)2(cid:12)≤22.4554. (cid:12) (cid:12) (cid:12) (cid:12) n>0 Thus we have our result. (cid:3) Here is the strategy for the rest of the proof. Let E be given in short Weier- strass form and suppose that |∆ | ≤ N4+6δ and |A| ≤ N4/3+2δ. There is a τ ∈ C E such that E and E are isomorphic over C. We want to understand the behaviour τ of λ at points of P which correspond to points Q = (x(Q),y(Q)) ∈ E(R) and ∞ τ satisfy |x(Q)| ≤ N2/3+δ. Since E and E are in short Weierstrass form we know τ that there is a w ∈C such that E (C) →∼= E(C) τ (4) (x,y) (cid:55)→ (w2x,w3y) and so we have x(Q)=w2℘ (u) since (℘(u),℘(cid:48)(u)) parameterises E (C). The idea τ τ 10 DAVEMENDESDACOSTA of the rest of the proof is as follows: we can deduce the size of w from the ratio of ∆ and ∆(τ) since the discriminant is a weight 12 modular form and so the E ratio is w12. Next, we use this to get an upper bound on the size of |℘(u)| which is necessary for |x(Q)| to be smaller than N2/3+δ. Finally we can use this upper bound to get a lower bound on |u|. This bound will turn out to be just what we want in order to cancel out the contribution from the local heights at the finite primes. We shall start by understanding ℘ a bit better and since τ 1 (cid:88) ℘ (z)= + (2k+1)G (τ)z2k τ z2 2k+2 k≥1 it makes sense to start with the G . 2k Lemma 4. For τ ∈C and k ≥2 we have |G (τ)|≤80. 2k Proof. We have (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:88) 1 (cid:12) (cid:88)(cid:12) 1 (cid:12) (cid:88)(cid:12) 1 (cid:12) |G2k(τ)|=(cid:12)(cid:12) ω2k(cid:12)(cid:12)≤ (cid:12)(cid:12)ω2k(cid:12)(cid:12)≤ (cid:12)(cid:12)ω4(cid:12)(cid:12) (cid:12)0(cid:54)=ω∈Λτ (cid:12) where the last line is justified by the fact that, for τ in C, the |ω| ≥ 1 for every ω ∈Λ . Thus it only remains to bound this last term for all τ ∈C. τ For τ =bi∈C we have 1 (cid:12) (cid:12) (cid:88) (cid:12) 1 (cid:12) |G2k(τ)| ≤ (cid:12)(cid:12)ω4(cid:12)(cid:12) 0(cid:54)=ω∈Λτ (cid:88) 1 = |m+nbi|4 (m,n)∈Z2−{0} (cid:88) 1 = (m2+(nb)2)2 (cid:88) 1 ≤ (m2+n2)2 ≤ 7. Similarly, if we consider τ = 1 +bi with b≥ 1 then we have 2 2